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Contenido NÚMEROS................................................................................................................................................3 POTENCIAS Y RAÍCES ................................................................................................................................ 11 SUCESIONES Y PROGRESIONES .................................................................................................................... 17 PROPORCIONALIDAD ................................................................................................................................ 25 POLINOMIOS .......................................................................................................................................... 35 ECUACIONES........................................................................................................................................... 41 SISTEMAS DE ECUACIONES......................................................................................................................... 49 RECTA, PARÁBOLA E HIPÉRBOLA ................................................................................................................. 59 FUNCIONES ............................................................................................................................................ 69 ESTADÍSTICA ........................................................................................................................................... 77 PROBABILIDAD ........................................................................................................................................ 91 ÁREAS Y VOLÚMENES.............................................................................................................................. 99 TRANSFORMACIONES EN EL PLANO. MOSAICOS ........................................................................................... 105 APÉNDICE: LUGARES GEOMÉTRICOS........................................................................................................... 111

1

NÚMEROS Naturales : son 0, 1, 2, 3, 4,… Enteros se obtienen añadiendo los opuestos -1,-2,-3,… y los Racionales Reales : son el conjunto de los números con decimales.

dividiendo.

 Orden de las operaciones (jerarquía de las operaciones): 1) Paréntesis.

Regla de los signos

2) Potencias y raíces. 3) Multiplicaciones y divisiones 4) Sumas y restas.

+·+=+ +·-=-·+=-·-=+

Operaciones con fracciones: Suma y resta: reducimos a común denominador, y se suma o restan los numeradores. Multiplicación: se multiplica en línea. División: se multiplica en cruz. Potencia y raíz: se hace la potencia o raíz del numerador y del denominador. (*) Castillo de fracciones: si el numerador o el denominador contienen una fracción Ej:

.

(*) Valor absoluto: es el número sin el signo. |-2| = 2, y |+2| = 2.  Tipos de decimales: Exacto: vienen de una división exacta. Sólo tienen algunos decimales. Periódico: vienen de una división no exacta. Un grupo de decimales (periodo) se repite infinitamente. Son puros si toda la parte decimal es periódica. Si no, son mixtos. Irracionales ( ): no son fracciones, y tienen infinitos decimales no periódicos.  Aproximación: - Truncar: quitamos las cifras decimales que no necesitemos. - Redondear: truncamos, pero hay que aumentar en uno la última cifra si la siguiente cifra era mayor o igual que 5. Error Absoluto: diferencia (sin signo) entre el valor exacto y el aproximado. Error Relativo: error absoluto dividido entre el valor exacto (se expresa en porcentaje).  Notación científica: escribir el número como un decimal entre 1 y 9, por una potencia de 10.

3

NUMBERS Natural : they are 0, 1, 2, 3, 4,… Integer they are formed by adding their opposites -1,-2,-3,… and Rationals Real : they are the set of all decimal numbers.

dividing.

 Order of operations (precedence rules): 1) Parentheses (brackets). 2) Exponents and roots. 3) Multiplication and divisions 4) Addition and subtraction.

Rules of signs

+·+=+ +·-=-·+=-·-=+

Operations with fractions: Add and subtract: reduce to common denominator, and add or subtract numerators. Multiplication: multiply across the top and bottom. . Division: cross multiply. Power and root: just calculate the power or root of numerator and denominator. (*) Complex fraction: when the numerator or denominator contains a fraction. E.g.:

.

(*) Absolute value: it is the number without its sign. |-2| = 2, and |+2| = 2.  Types of decimal numbers: Terminating: they come from a terminating decimal fraction. Decimals stop after a few digits. Periodic (recurring decimal): they come from a recurring decimal fraction. A group of decimals (period) repeat forever. We call those which start their recurring cycle immediately after the decimal point purely recurring. Those that have some extra digits before their cycles are also called mixed recurring (or eventually recurring). Irrational ( ): any number which does not stop and does not end with a recurring pattern (thus they are non-fractional numbers).  Approximation: - Truncate: remove the decimal digits you do not need. - Round: truncate, but increase in one the last digit if the next one was (to some decimal places) greater or equal to 5. Absolute Error: difference (without sign) between the exact value and the approximation. Relative Error: absolute error divided by the exact value (expressed in percentage).  Scientific notation: write the number as a decimal between 1 and 9 times a power of 10.

4

NUMBERS – REVISION EXERCISES 1. Find the LCM and GCD of the following numbers: a) 120 and 150

b) 378 and 528

c) 140 and 350

d) 720 and 1470

e) 79 and 84

f) 240 and 300

g) 168 and 252

h) 80 and 120.

Solutions: a) LCM=600, GCD=30

b) LCM=24·33·7·11, GCD=6

c) LCM=22·52·7, GCD=70

d) LCM=24·32·5·72, GCD=30

e) LCM=22·3·7·79, GCD=1

f) LCM=24·3·52, GCD=60

g) LCM=23·32·72, GCD=22·3·7

h) LCM=24·3·5, GCD=23·5=40.

2. Find the LCM and GCD of the following numbers: a) 40, 105 and 160

b) 72, 120 and 210

c) 54, 126 and 180.

b) LCM=23·32·5·7, GCD=6

c) LCM=22·33·5·7, GCD=18.

Solutions: a) LCM=25·3·5·7, GCD=5

3. Write as a fraction or as a decimal number: a)

b) 2.8

c)

d)

e)

f)

g)

h)

a)

b)

c)

d)

e) 3.1875

f)

g)

h)

Solutions:

.

5

OPERATIONS INVOLVING FRACTIONS – PRACTICE 1. Simplify before multiply: a)

=

d)

b) =

=

c)

e)

g)

=

h)

=

f)

=

i)

2. Calculate. Remember to simplify, whenever it is possible.

=

a)

=

c)

=

e)

b)

1+2=

d)

= =

f)

3. Work out a)

=

c)

b) =

=

d)

e)

=

=

f)

=

4. Work out: a)

=

b)

d)

c)

e)

=

f)

Solutions: 1. [a] 1

6

[b]

[c] 2.

[d] -4

[e] -

[f]

[g] -25

[h]

[i]

.

2. [a]

[b]

[c]

[d]

[e]

[f]

3. [a]

[b]

[c]

[d]

[e]

[f] 18.

4. [a] 2

[b]

[c]

[d]

[e]

[f]

.

.

Numbers - Word Problems 1. A water pitcher (jarra) weighs 0.64 kg when empty and 1.728 kg when

filled with water. How much does the water weigh?

[Sol: 1.088kg]

2. Eva is on a diet which states that she cannot consume more than 600

calories in one meal. Yesterday she had lunch: 125 g of bread, 140 g of asparagus, 45 g of cheese and an apple of 130 g. If 1 g of bread has 3.3 calories, 1 g of asparagus, 0.32, 1 g of cheese, 1.2, and 1 g of an apple 0.52. Did Eva follow her diet? 3. Juan has got €200 in the bank. He pays a bill for 5 books, which cost €30 each. Then, he

earns his pay (paga) of the last seven days (€40 per day). Last, he withdraws (sacar) €200 and buys a €320 game console. Represent the situation of his account using an integer. Does he have money or does he owe money? How much? 4. A gardener (jardinero) fills 1/5 of his vegetable garden with potatoes, 2/3 with

cabbage (col) and the rest, which amounts to 120 square metres, with onions. What portion of the garden occupy the onions?

120m2

What is the area of the garden?

5. A car dealer is selling a new model for €12,000, with one sixth of the price to be paid

upfront (por adelantado) and the rest in forty equal monthly instalments (plazos). How much is paid upfront? How much is each monthly instalment?

[Sol: 2000 and 250]

6. A thrifty (ahorrador) individual has €245,000 in his current account. He invests two-fifths

of the money in shares (acciones) in an insurance company. How much is that? How much is left in the account? [Sol: Invests €98,000. 147,000 left] 7. A shelf (estantería) in a supermarket holds 80 one-quarter litre bottles and 44 one and a

half litre bottles. How many litres are there on the shelf? 8. Juan moves forward

[Sol: 86 litres]

of a metre with each step. How many steps must he take to

complete a 9 kilometre walk?

[Sol: 10,800 steps]

9. Victoria is planning for her holiday. She calculates that if she spends a third of her savings

(ahorros) on a plane ticket and a quarter on a hotel, she will still have €450 left. How much money does she have? What fraction of her money will she spend? What fraction will she keep? How much money does she have? 10. Find the absolute error and relative error when:

a) We say 30 minutes instead of 27 minutes. b) We say €15 instead of €16.

c) We round 3.66 to tenths. d) We truncate 6.7 to units.

7

Use scientific notation to solve the following problems 11. The distance between the Sun and Earth is approximately 150,000,000 km. Write it using

[Sol: 1.5×108km]

scientific notation.

12. The Sun has a Mass of 1,988,000,000,000,000,000,000,000,000,000 kg. Write it using

[Sol: 1.988×1030 kg.]

scientific notation.

13. The radius of the sun is 695 500 km. What is its approximate volume written in Scientific

notation? [Hint: V=

[Sol: 1.409×1018km3.]

]

14. The mass of the Moon is 73,000,000,000,000,000,000,000 kg. What is this written in

[Sol: 7.3 × 1022 kg]

Scientific notation?

15. The speed of light in a vacuum is 299 792 458 m/s. What is this written in Scientific

notation?

[Sol: 2.997 924 58×108 m/s]

16. Photocopy paper is packaged in reams (“muchas páginas”; 500 sheets). The thickness of

the pack is 41 mm. What is the thickness of one sheet of paper written in Scientific Notation using meters? [Sol: 8.2×10-2mm=8.2×10-5 m] 17. The rest mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 910 938 kg.

What is this written in Scientific notation?

[Sol: 9.10938 × 10-31kg]

18. The human eye blinks (parpadear) about 6.25·106 times each year. About how many

times has the eye of a 14 year old blinked? (Use scientific notation).

[Sol: 8.75 × 107]

19. If the average person eats 2.3 slices of pizza per week, how many slices of pizza are

consumed in Alabama (population: 5.8·105) in one week?

[Sol: 1.334 × 106 slices]

20. A tiny space inside a computer chip has been measured to be 0.00000256m wide,

0.00000014m long and 0.000275m high. What is its volume?

[Sol: 9.856×10-17m3]

21. The speed of light is 3·108 meters/second. If the sun is 1.5·1011 meters from earth, how

many seconds does it take light to reach the earth? And minutes? (Hint: Use scientific notation and write an equation and to solve it)

8

[Sol: 500s=8m20s]

3º ESO Mathematics Exam (trial exam)

IES Extremadura

Bilingual section

NUMBERS. Prime factors (2 pts.): 1) (1 pt.) Write as a product of prime numbers: a) 2520=

b) 1296=

c) 3388=

d) 2340=

2) (1 pt.) Compute LCM and GCD: a) 2520 and 1296

b) 3388, 2520 and 1100.

Calculations (4 pts.) 3) (1 pt.)

=

4) (1 pt.)

5) (1 pt.)

=

=

6) (1 pt.)

=

Decimal numbers (4 pts.) 7) (1.5 pts.) Write as a decimal number or as a fraction in lowest terms a)

=

d) 6.25=

b) 3.36363636...= e)

c)

=

8) (1 pt.) Use scientific notation: a) 3897000000000000 =

b) 0.0000000009137 =

c) 4.6·104 · 5.1·107=

d) 5.2·104 + 3.51·106=

9) (0.5 pts.) Find the absolute and relative errors when we say 30 people instead of 28 people. 10) (1 pt.) Oil (petróleo) reserves in the United States are estimated to be 3.5·1010 barrels. Consumption amounts to (asciende a) 3.2·109 barrels per year. At this rate (a este ritmo), how long would US oil reserves last? What should be the consumption if we want the oil to last 30 years?)

9

POTENCIAS Y RAÍCES  Raíz (radical):

porque 53=125. Es una raíz de índice 3 (raíz cúbica). porque 24=16. Es una raíz de índice 4 (raíz cuarta).

- Si el índice es par, no hay raíz de números negativos. Si es impar, la raíz sí existe. Ej.:

NO EXISTE,

pero

.

Ejercicio: memoriza y escribe usando raíces 23=8, 33=27, 43=64, 53=125, 63=216, 73=343.  Propiedades de las potencias 81=8 , 80=1 , (-3)1=-3 , (-3)0=1.

Exponentes 0 y 1 Base negativa Producto

Exponente par: se quita el signo del paréntesis. (-5)2=52=25, pero -52= -25. Exponente impar: el signo sale fuera del paréntesis

(-5)3=-53=-125.

52·58=510 , 56·5=57. 43·53=203.

Misma base: se suman los exponentes. Mismos exponentes: se multiplican las bases Misma base: se restan los exponentes

57 57 3 ,  5  5 7 ( 4)  511 54 5 4

Mismos exponentes: se dividen las bases

154  15      34  81 . 54  5 

División

4

Se multiplican los exponentes (43)2=46.

Potencia de potencia Exponente negativo

Se da la vuelta a la fracción.

Exponente fraccionario

Se convierte en radical. El denominador es el índice de la raíz

,

.

(y se quita el signo)

4 7

1

2  2  16 7

4

7

,

16 16 4 .  16  2      25 25   25 5

 Propiedades de los radicales Índice y exponente se simplifican como fracciones. Ej.:

;

.

Extraer factores: se calcula la raíz de cada factor del radicando. Ej. Suma/resta: si son iguales, se suman/restan los coeficientes. Ej. (*) Escribir con un solo radical: Reducir a índice común. Ej.

. .

11

POWERS AND ROOTS  Root (radical):

because 53=125. Is a root of index 3 (cubic root). because 24=16. Is a root of index 4 (fourth root).

- If the index is even, there is no root for negative numbers. If it is odd, the root does exist. E.g.:

DO NOT EXIST,

but

.

Exercise: memorize and write using roots 23=8, 33=27, 43=64, 53=125, 63=216, 73=343.  Properties of Exponents 81=8 , 80=1 , (-3)1=-3 , (-3)0=1.

Exponents 0 and 1 Negative base Product

Exponent even: remove the sign in parenthesis. (-5)2=52=25, but -52= -25. (-5)3=-53=-125.

Exponent odd: put the sign outside the parenthesis

52·58=510 , 56·5=57. 43·53=203.

Same base: add the exponents. Same exponents: multiply the bases Same base: subtract exponents

57 57  53 ,  5 7 ( 4)  511 4 4 5 5

Same exponents: divide the bases

154  15      34  81 . 4 5 5

Division

4

multiply the exponents (43)2=46.

Power to a power Negative exponent

Flip the base upside down.

,

.

Fractional exponent

Turns into a radical. The denominator is the index of the root

(and remove the sign)

4 7

1

2  2  16 7

4

7

,

16 16 4 .  16  2      25  25  25 5

 Properties of radicals Exponent and index can be converted or simplified. E.g.

;

.

Take out factors: find the root of each factor in the radicand. E.g. Add/subtract: if they are equal, add/subtract the coefficients. E.g. (*) Write using one radical: Reduce to a common index. E.g.

12

. .

POWERS AND ROOTS – REVISION EXERCISES 1. Write using only one radical (or find the result) a)

b)

c)

d)

e)

f)

Solutions: a)

b)

c)

d)

e) 6

f) .

2. Take out all possible factors. a)

b)

c)

d)

e)

f)

Solutions: a)

b)

c) 10

d)

e)

f)

.

3. Move factors to the inside of the radical a)

b)

c)

d) 22·7

e)

f)

Solutions: a)

b)

c)

d)

e)

.

f)

4. Simplify (add and subtract radicals) a) d) Solutions: a)

= b)

c)

b)

c)

e)

f) d)

e)

f)

.

13

OPERATIONS INVOLVING POWERS AND ROOTS – PRACTICE 1. Calculate. Remember to simplify, whenever it is possible.

=

a)

=

c)

=

e)

b)

=

d)

110 =

=

f)

2. Work out a) c)

=

b)

=

=

d)

e)

+

=

=

f)

=

3. Work out: a)

=

c)

b) d) 5-1+

=

e)

=

+ 2=

=

f)

=

Solutions:

14

1. [a] -2

[b]

[c] -5.

[d]

[e] -

[f]

2. [a]

[b]

[c]

[d]

[e] 7

[f]

3. [a]

[b] 3

[c] 7

[d] 0

[e]

[f] 5.

.

3º ESO Mathematics Exam (trial exam)

IES Extremadura

Bilingual section

POWERS AND ROOTS. Powers (5.5 pts): 1) (1.5 pts.) Write using only one power: a) 158 ·15-2 ·156 =

b)

e)

f)

c) 104·(-3)4·(-2)4=

=

d)

=

g) 52·82·40-6=

2) (1 pt.)

=

3) (1 pt.)

=

4) (1 pt.)

=

5) (1 pt.)

=

Radicals (5 pts.) 6) (1 pt.) Write using only one radical a)

=

b)

=

c)

=

7) (1.5 pts.) Take out all possible factors: a)

500 

b)

5

4 6 ·712·1010·4 

c)

5

20 4 

8) (1 pt.) Move factors to the inside of the radical: a) 3 5 

3 b) 4 10 

9) (1 pt.) Simplify: a) 14 13  16 13  42 13  b)  5 45  3 80  8 405 

15

SUCESIONES Y PROGRESIONES Sucesión: es un conjunto de números (términos) ordenado (hay primero, segundo, tercero,…) Suelen representarse como a1, a2, a3,… Ej. Para (2, 4, 6,…) a1=2, a2=4, a3=6,… Término general an: fórmula que representa el valor del término en el lugar “n”.  Progresión aritmética: cuando cada término se obtiene sumando siempre el mismo número al anterior. El número se llama diferencia (d). Ej. (1, 3, 5, 7, …), se obtiene sumando 2. Término general: Suma de los n primeros términos:

 Progresión geométrica: cuando cada término se obtiene multiplicando por el mismo número siempre. El número se llama razón (r). Ej. (2, 4, 8, 16,…), multiplicando por 2. Término general: Suma de los n primeros términos:

 Interés simple: cuando no lo acumulamos al capital (no sigue generando intereses).  Interés compuesto: cuando sí lo acumulamos al capital (sí sigue generando intereses). C=Capital (dinero) final, c=capital inicial, r=porcentaje (rédito), I=Interés, t = tiempo.

17

SEQUENCES AND PROGRESSIONS Sequence: is a set of ordered numbers (terms) (there is a first one, second one, third one,…) They are normally represented by a1, a2, a3,… E.g in (2, 4, 6,…) a1=2, a2=4, a3=6,… General Term an: formula that represents the value of the “nth ” term.  Arithmetic progression: when each successive term is obtained by adding always the same number to the previous term. This number is called difference (d). E.g. (1, 3, 5, 7…), is obtained by adding 2. General term: Sum of the n first terms:

 Geometric progression: when each successive term is obtained by multiplying the previous term always by the same factor. The number is called ratio (r). E.g.. (2, 4, 8, 16…), is obtained by multiplying by 2. General term: Sum of the n first terms

 Simple interest: when we do not cumulate it to the capital (it does not generate interest any more).  Compound interest: when we do cumulate it to the capital (it continues generating interest). B=Balance (final amount), p=principal (starting amount), r=percent (rate), I=Interest earned, t = time.

18

SEQUENCES – REVISION EXERCISES 1. Fill the chart using progressions:

Terms

Description

General Term

{30,25,20,15,…}.

Some terms… a10=

Multiples of 2, plus 15.

a100= an= -3n+1

{-10,-7,-4,…}

a6= a12=

2. Compute the general term for the following sequences (0.75 pts. each section):

15 13 11 9  a)  , , , ,... .  2 4 8 16 

an=

c) {10 000, 5000, 2500, 1250, …}.

an=

. Compute a6=

3. Compute, and write the general term: a) For an arithmetic progression, we know that a3=19 and a7=39. Compute S15. b) Given the geometric progression {3, 6, 12, 24,…}, compute S12= c) We want to compute (using the correct formula), 500+ 500·1.8+500·1.82+...+500·1.820 = - What type of succession are we adding? - How many terms are we adding? - What is the 10th term of the succession? 4. Laura is going to deposit €15,000 in a bank for 10 years. She wants to know how much will she have after those 10 years, and how much she will have earned. a) If we use simple interest of 3 %: b) If we use compound interest of 3 % Solutions: [2] a)

, b)

[3] a) d=

. an=5n+4. a1=9, a15=79 S15=

. a6=312.5 . b) r =2. an=3·2n-1. a12=6144. S12=

c) It is a geometric progression. We are adding 1+20=21 terms. r=1.8; an=500·1.8n-1. a10=500·1.8999179.65 S21=



[4] a) Arithmetic progression: d=3% of 15000=450. {15 000,15 450, …} an=450n+14 550. a10=19 050. She will have €19 000, so she will have earned 19 050-15 000=4 050 (It is €450 nine times) b) Geometric progression. r=1.03. {15 000,15 000·1.03,…} an=15 000·1.03n-1. a10=15 000·1.03919 571.60 She will have €19571.60, so she will have earned 1 9751.60-15 000=4 571.60€

19

5. Find the general term for the following sequences, and find the missing terms: a)

,

b)

d)

,

g)

,

e)

c) ,

;

f)

;

h)

6. Find the general term and sums for the following arithmetic progressions: a) a4=19, a6=31; S30?

b) a1=26, a5=10; S10?

c) a3=-71, a53=79; S100?

d) a1=16, a5=56; a5+…+a40?

e) 3+ 60 … +239 = ?

f) a4=-8, a10=10; a10+…+a30?

g) a3=44, a13=24; a30+…+a50?

h) a1=1, a21=7; a40+…+a87?

7. Find the general term and sums for the following geometric progressions: a) {2, 6, 18,…}; S10?

b) {5, 10, 20, …}; S20?

c) 0.1+0.4+1.6+…+1638.4

d) 30+30·1.2+…+30·1.220

e) 10+30·0.4+…+10·0.415

f) 60+60·1.05+…+60·1.0511

Solutions: [5] a) d) g)

; ; ;

b)

;

c)

e)

;

f)

h)

[6] a) an=6n-5. S30=2640 d) an=10n+6. 56+ +406=8316

; ;

;

b) an=30-4n. S10=80

c) an=3n-80. S100=7150

e) an=4n-1. 3+…+239=7260

f) an=3n-20. 10+ +70=840

g) an=50-2n. (-10)+ +(-50) = - 630 h) an=0.3n+0.7 12.7+ +26.8 =948. [7] a) an= d) an=

20

. S10=59048

b) an=

. S21=

e) an=

. S20= . S16=

c) an= f) an=

. 0.1+…+1638.4= . S12=

SEQUENCES AND PROGRESSIONS - WORD PROBLEMS (*) Use the progressions formulas to compute the totals in the problems.

1. Pedro did 40 sit-ups (abdominales) on Tuesday, 50 sit-ups on Wednesday, 60 sit-ups on Thursday, 70 sit-ups on Friday, and 80 sit-ups on Saturday. If this pattern (pauta) continues, how many sit-ups will Pedro do on Sunday? 2. The teacher gave 18 gold stickers (pegatinas doradas) to the first student, 24 gold stickers to the second student, 30 gold stickers to the third student, and 36 gold stickers to the fourth student. If this pattern (pauta) continues, how many gold stickers will the teacher give to the fifth student? How many stickers will he give in total? 3. A new cookbook is becoming popular. The local bookstore ordered 1 copy in May, 5 copies in June, 25 copies in July, and 125 copies in August. If this pattern (pauta) continues, how many copies will the bookstore order in September? How many copies will it order in total? 4. Ana put 2 beads (cuentas-abalorios) in the first jar (tarro), 4 beads in the second jar, 8 beads in the third jar, 16 beads in the fourth jar, and 32 beads in the fifth jar. If this pattern continues, How many beads will Ana put in the sixth jar? How many in total? 5. Luisa picked 2 flowers from the first bush (arbusto), 4 flowers from the second bush, 8 flowers from the third bush, and 16 flowers from the fourth bush. If this pattern (pauta) continues, how many flowers will Luisa pick from the fifth bush? How many will be in total? [ (*) Use a sequences formulas] [Sol: 32; 62 in total] 6. In a theatre, there are 28 chairs in the first row, 32 chairs in the second row, 36 chairs in the third row and 40 chairs in the fourth row. If this pattern continues: how many chairs will there be in the twelfth row? If there are 15 rows, what is the capacity of the theatre? [Sol: 840 chairs] 7. We have a 40m deep well (tenemos un pozo de 40m de profundidad). We have paid €7.5 for the first metre and for each successive metre €2.3 more than for the previous. How much does the well cost? [Sol: €2094] 8. Juan has €90 in a savings account. The interest rate is 5% per year and is not compounded. How much interest will he earn in 5 years? How much will he have?

21

9. Certain businessman earns per year a 6% more than the previous year. If the first year he earned €25,000, how much will he earn the 10 th year? How much in those 10 years? 10. We are paying a debt. The first week we pay €5, the next week €9; then €13, €17, and so on. If we pay in 30 weeks, how much do we owe (debemos)? [Sol: €2010] 11. Leonardo deposited €100 in a savings account earning 5% interest, compounded annually. How much will he have in 6 years? [Sol: €134] 12. Monica has €80 in a savings account that earns 10% interest, compounded annually. How much will she have in 3 years? [Sol: €106.48] 13. Angeles has €90 in a savings account that earns 5% annually. The interest is not compounded. How much will she have in 2 years? [Sol: €99] 14. Compute the principal we have to pay in an account that pays a 5% of compounded

interest if we want to get €1,526.50 in twelve years. And how much if the interest is simple? [Sol: €460.63; and €954.06 if it is simple] 15. The ending balances in Carissa’s savings account for each of the past four years form the sequence {$1,000, $1,100, $1,210, $1,331,...}. Is the sequence arithmetic, geometric, or neither? Find the next two terms of the sequence. [Sol: Geom. a5=1464.1, a6=1610.51] 16. A large pizza at Joe’s Pizza Shack costs $7 plus $0.80 per topping. Write a sequence of pizza prices consisting of pizzas with no toppings, pizzas with one topping, pizzas with two toppings, and pizzas with three toppings. Is the sequence arithmetic, geometric, or neither? How do you know? [Sol: Arithmetic. an=6.20+0.8n] 17. A family purchased furniture on an interest-free payment plan with a fixed monthly payment. Their balances after each of the first four payments were $1,925, $1,750, $1,575, and $1,400. a) Is the sequence of the balances arithmetic, geometric, or neither? Explain how you know. If it is arithmetic or geometric, state the common difference or common ratio. b) Continue to find the terms of the sequence of balances until you get a term of 0. After how many payments will the balance be $0?

22

3º ESO Mathematics Exam (trial exam)

IES Extremadura

Bilingual section

SEQUENCES AND PROGRESSIONS. 1. (3 pts.) Fill the chart using progressions (0.25 pts. each cell): Terms

Description

General Term

{2,4,6,8,…}.

Some terms… a6=

Multiples of 3, plus 5.

a15= an= 4n-7

{20,15,10,…}

a20= a8=

2. (1.5 pts.) Compute the general term of the following sequences (0.5 pts. each section): 15 13 11 9 a)  , , , ,... . an= 3

7 11 15



b) 3, 6,12, 24,.... an=

. Compute also, a8=

c) {3000,300,30}. an=

. Compute also, a6=

3. (2.5 pts.) Calculations: a) For certain arithmetic progression, a1=2 and a3=12. Compute the sum of its 20 first terms. b) Compute 2+ 2·1.05+ 2·1.052+ 2·1.053+…+2·1.0515= c) Given the geometric progression {3, 6, 12, 24…}, compute S12= d) Use the formula for Sn to compute: (2+7+12+17+...+102)·(1+4+7+...+46)= 4. (1 pt.) During December, a shop manager has been taking note of his electricity consumption. The first day, it was 2Kwh; the second, 5Kwh and so on, increasing in arithmetic progression. a) What was the consumption the day 31st? b) What is the consumption of December in total? 5. (2 pts.) Luis is going to deposit €18,000 in a bank for 20 years. a) How much will he earn if we use simple interest of 5 %? b) How much will he earn if we use compound interest of 5 %?

23

PROPORCIONALIDAD - Proporción: igualdad entre dos razones (cociente de magnitudes). El número obtenido se llama constante de proporcionalidad. Ejemplo:

es una proporción y la constante es 2.5.

Además, 5 y 10 se llaman extremos, y 2 y 25 medios.  Relación entre magnitudes: Directa: si una aumenta, la otra también. Proporcional cuando al doble de una le corresponde el doble de la otra; al triple le corresponde el triple,… Ejemplos. Geometría: Teorema de Thales, figuras semejantes y mapas a escala. Porcentajes. (*) Para aumento/disminución encadenados es mejor hacerlos multiplicando. Inversa: si una aumenta, la otra disminuye. Proporcional cuando al doble de una le corresponde la mitad de la otra; al triple le corresponde un tercio,… (*) Compuesta: si está relacionada con más de una magnitud al mismo tiempo. Ejemplo: el espacio recorrido con la velocidad y la duración de un viaje.  Cálculo del cuarto proporcional y el medio proporcional Propiedad fundamental: “El producto de los medios es igual al producto de los extremos”. Ej. para

, 3x=1·25. Y en

, x·x=1·25; x2=25. “x” es el medio proporcional.

 Resolución de problemas. Se pueden hacer de dos formas: 1ª: calcular la constante de proporcionalidad (usando una función) “reducción a la unidad”. 2ª: hacer el planteamiento usando la regla de tres, que puede ser compuesta. - Reparto Proporcional: la cantidad a repartir se corresponde con el total. (*) Reparto con proporcionalidad compuesta: primero hay que juntar las magnitudes multiplicándolas (o dividiendo si el reparto es inversamente proporcional).

25

PROPORTIONALITY - Proportion: statement that two ratios (quotient of magnitudes) are equal. The number obtained is called proportionality constant. Example:

is a proportion and the constant

is 2.5. Moreover, 5 and 10 are called extremes, and 2 and 25 means.  Relationship between magnitudes (types of variation): Direct: the greater the 1st, the greater the 2nd. Proportional i If one doubles, the other will also double, etc,… Examples. Geometry: Thales Theorem, similar figures and scaled maps. Percentages. (*) For consecutive percentage increase/decrease it is better to compute them multiplying. Inverse: the greater the 1st, the smaller the 2nd. Proportional if one doubles, the other will become half as large, etc.,… (*) Compound: when two or more magnitudes are involved in the relationship Example: the distance travelled with speed and duration of a travel.  Finding the fourth proportional and the mean proportional) Propiedad fundamental: “El producto de los medios es igual al producto de los extremos”. E.g. for

, 3x=1·25

And in

, x·x=1·25; x2=25. “x” is the mean proportional.

 Problem solving. There are two ways: 1st: find the constant of variation (using a function) “unitary method”. 2nd: set the problem up using the rule of three, which can be compound (double). - For Proportional Distribution, the amount to share out corresponds to the total. (*) Distribution with compound proportion: first you have to put the magnitudes together by multiplying them (or dividing if it is inverse proportion).

26

PROPORTIONALITY - WORD PROBLEMS 1. In a shipment of 400 parts, 14 are found to be defective. How many defective parts

should be expected in a shipment of 1000?

[35 parts]

2. A piece of cable 8.5 cm long weighs 52 grams. What will a 10-cm length of the same cable

weigh?

[61.18 grams]

3. A snowstorm dumped (depositar) 18 inches of snow in a 12-hour period. How many

inches were falling per hour?

[1.5 inches]

4. 2 gardeners fence in a garden in 9 hours. How long would it have taken for 6 equally

productive gardeners?

[3 hours]

5. Mary can read 25 pages in 30 minutes. How long would it take her to read a 100 page

book?

[120 minutes]

6. An employee working at an electronics store earned $3582 for working 3 months during

the summer. What did the employee earn for the first two months?

[$2388]

7. A farmer has enough grain to feed 60 cattle (ganado) for 25 days. He sells 10 cattle. For

how many days will the grain last now?

[100 days]

8. A company’s quality control department found and average of 5 defective models for

every 1000 models that were checked. If the company produced 60,000 models in a year, how many of them would be expected to be defective? [300 defective models] 9. To determine the number of deer in a forest, a forest ranger tags 280 and releases them

back into the forest. Later, 405 deer are caught, out of which 45 of them are tagged. Estimate how many deer are in the forest. [2520 deer] 10. The ratio of men to women at a class is 6 to 5. How many women students are there if

there are 3600 men?

[3000 women]

11. A town has 800 inhabitants. Twelve percent of them have never seen the sea. How many

of the inhabitants have never seen the sea?

[96 inhabitants]

12. It takes 20 hours for a tap (grifo) with a flow of 15 litres per minute to fill a tank (depósito)

with water. How long will it take if its flow is reduced to 12 litres per minute?

[25 hours]

13. A farmer harvests 25,000 kg. of corn and sells 85% to an animal feed factory. How many

kilos did the feed factory buy?

[21,250kg.]

14. A 1 kilo cake contains 150 grams of sugar. What percentage of the cake is sugar?

[15%]

27

15. In a football team, 12 players have missed practice. This is 40% of all the players. How

many players are on the team?

[Sol: 30 players]

16. Ana lost 12kilos, which is 15% of what she weighed one year ago. How much did she

weigh one year ago?

[Sol: 80kg.]

17. A farmer has got food enough for 1200 rabbits for 180 days. If he sells 300 rabbits, how

long will the food last?

[240 days.]

18. The price of a bus ticket used to be €2, but today it goes up 5%. How much will a ticket

cost from now on?

[€2.10]

19. In a given population, 2,480 people last year had the flu. This year the number is 30%

lower. How many people had the flu this year?

[1736 people]

20. A company with 1,675 employees cuts its staff by 8%. How many employees does it have

after the cut?

[1206 employees]

21. Three partners make €12,900 in a business. Juan put €18,000, Antonio €15,000 and Marta

€10,000. How much should each of them receive?

[€5,400,: €4,500 and €3,000.]

22. Two workers earn 660€ for a job. The first one worked for 4 days and the second one

worked for 7 days. How much should they receive?

[Sol: €240 and €420]

Geometric proportionality 23. Use similarity of figures to work out the missing side lengths. Round to tenths.

[1]

[2]

Sol: 4.25 and 7.2

[5]

Sol: 2.3 and 3.2

[6]

[8]

Sol: 4 and 5.6

[7]

[9]

Sol: 4.3 and 6.5

[4]

Sol:6.4 and 7.6

Sol: 3.9 and 4.8

28

[3]

Sol:7.8 and 6.3

[10]

Sol: 2 and 4.1

Sol: 7.3 and 11.1

Sol: 2.8 and 2.8

3

[11]

[12]

[13]

8 Sol: 4.5 and 2.6

10

Sol: 2.6 and 6.5

Sol: 7.8, 6 and 7.7 and 3.7

24. Find the ratio of similarity (r). Then use it to find the surface of the larger base (A) and the

volume of the big pyramid/cone (V). Finally, work out the frustum’s volume (F). [Calcula la razón de semejanza (r). Luego úsala para encontrar el área de la base mayor (A) y el volumen de la pirámide/cono grande (V). Por último, deduce el volumen del tronco (F)]. a)

b) 3

10.7cm

4cm

c)

12cm

1.2cm3 1.8cm2

4cm

2cm

2

8cm

8cm

8cm 9.3cm3 7cm2

d)

3

e)

2.5dm

9.17dm

f)

4.5cm2

3cm2

6.25dm

2cm

3cm3

2cm3

5.5dm2

5cm

2.5cm 5cm

g)

h)

9m

1m 2

0.28m

7m2

i) 1cm

3m

0.44cm3 2 1.75cm

3.5cm

3

7m

0.4m

3

0.04m Solutions: a) r=2,

A=32cm2,

V=85.6cm3,

c) r=4,

A=28.8cm ,

V=76.8cm ,

F=75.6cm .

d) r=2.5, A=34.38dm , V=143.28dm , F=134.11dm .

e) r=2,

A=12cm2,

V=16cm3,

F=14cm3.

f) r=2.5, A=18cm2,

2

2

g) r=2.5, A=1.75m ,

3

3

V=0.63m ,

i) r=3.5, A=21.44cm2, V=18.87cm3,

F=74.9cm3. 3

3

F=0.59m .

b) r=3,

A=63cm2,

V=251.1cm3, F=241.8cm3. 2

h) r=3,

2

A=63m ,

3

3

V=24cm3,

F=21cm3.

3

F=182m .

V=189m ,

3

F=18.43cm3.

29

Compound proportion: 25. Two workers channel (canalizar) 100m pipes (tuberías) in 10 days. How long will take 5

workers to channel 350m pipes. 26. 10 men can lay a road 75 Km. long in 5 days. In how many days can 15 men lay a road 45

Km. Long? 27. Wheat (trigo) costing €480 is needed to feed 8 people over (durante) 20 days. What is the

cost of wheat required to feed 12 people over 15 days? 28. In a ship, they have food enough for 400 people and 64 days if they have portions of 1960g.

How much could they eat if there were 140 people but the trip lasted 80 days? 29. In a workshop (taller), spending 8 hours per day, it has taken them 5 days to make 1,000

pieces. How long will take them to make 3,000 pieces working 10 hours per day?

[12 days]

30. The dorm (residencia de estudiantes) charges $6300, for 35 students for 24 days, in how

many days will the dorm charges be $3375 for 25 students?

[Sol: 18 days]

31. 24 men working at 8 hours per day can do a piece of work in 15 days. In how many days

can 20 men working at 9 hours per days do the same work?

[Sol: 16 days]

32. Working 8 hours per day, a glass factory makes 6,000 bottles in 3 days. How long would it

take it to 10,000 bottles working 10 hours per day?

[Sol: 4 days]

33. In order to finish a building work in 360 days, 30 workers are needed working 8 hours per

day. How long will it take to finish to 45 workers working 6 hours per day?

[Sol: 320 days]

34. A transport company charges €80 for carrying 1500kg of goods a distance of 100km. How

much will charge for carrying 4500kg a distance of 250km?

[Sol: €600]

35. Working 8 hours per day a textile factory makes 15,000 pairs of socks in 12 days.

- How many pairs of socks will it produce over the next ten days if it doubles its working hours (to 16 hours)? [Sol: 25,000 pairs] - And how many days does it need to make 20,000 socks, working 16 hours?

[Sol: 8 days]

36. 20 cows consume 600kg of feed in three weeks. How many kg. of feed do 30 cows

consume in one month (4 weeks)?

30

[Sol: 1200kg]

Proportional distribution 37. In a competition, Pedro achieved a 10m shot with a 4kg weight and Oscar a 6m shot with a

3kg weight. Distribute a €174 prize directly to the weight and the distance. 38. A company is going to distribute €2,250 among three employees. Juan is 35 years old and

earns €1,400 monthly. Ana is 24 and earns 1,200, and Luis is 48 and earns €1,600. The distribution will be directly proportional to their ages and inversely to their salaries. 39. A company is giving €750 gratification to three of their typist. Ana typed 250 pages and but

had 5 typing errors per page, Juan 240 pages and had 4 errors per page, and María 280 but had 7 errors per page. Distribute the gratification proportionally among them (directly to the pages and inversely to the typing errors). 40. Distribute a €450 bonus (gratificación) between two employees, proportionally to the

number of hours worked and new clients made and inversely to the extra-money spent. Luis: 40hours, 20 clients and €100 extra. Julián: 35 hours, 18 clients and €90 extra. 41. Distribute 528€ between two brothers directly to their ages and final marks:

Juan is 15 and got 8 points. Virginia is 16 and got 9 points.

[Sol: Juan €240; Virginia €288.]

42. We have paid two teams 6888€ for clearing a forest. There are 12 people in the first team

and they worked for 8 days. In the second, there are 15 people and they worked for 10 days. How much should receive each team? [Sol: 1stteam: €2688; 2nd team €4200.] 43. Luis and Sonia got 430 points in a trivia competition. Luis answered correctly 23 questions

in 10 minutes, and Sonia 30 questions in 15 minutes. Distribute the points proportionally to the questions and inversely to the time spent. [Sol: Luis: 230 points; Sonia: 200 points] 44. A hospital is sending 28 patients to other hospitals. Distribute the patients proportionally

to the available (disponible) beds and inversely to the distance. Hospital A: 20 beds and is at 10km. Hospital B: 30 beds and 15km. Hospital C: 36 beds and 12 km. [Solution: Hospital A: 8 patients; Hospital B: 8 patients, Hospital C: 12 patients. ] Consecutive percent increases/decreases 45. Three years ago, Andrés rented his house for €400 per month. The R.P.I. “Retail Price

Index” (parecido al I.P.C.) has been: 3%, 4% and 3.5% these years. What should be the rental (precio de alquiler) now? [Sol: €443.48] 46. A factory has to pay a €15,000 invoice. They get a 15% discount for prompt payment

(pronto pago). V.A.T of the operation is 12 %. How much does the factory have to pay? 47. A fridge costs € 336 after a price increase of 40%. What was the original price?

[€240]

31

48. We have paid €24 for a skirt that had a 20% off. We have also paid a 21% VAT. What is the

price of the skirt without discount and without VAT?

[Sol: 24.79€]

49. Over the period of a year the price of an article first increases by 40%, then decreases by

10% before finally decreasing a further 20%. Calculate the percentage change over the whole year. [It increases a 0.8% (it is a 100.8% of the total)] 50. Over three decades an expanse of forest changes: 1st decade; it increases by 28%. 2nd: it

decreases by 40%. 3rd: 1970 to 1980 it increases by 15%.By what percentage did the forest change during all three decades? [Decreases by 11.68% (it is a 88.32% of 100%)] 51. Three years ago, the cost of living went up by 10% and by 8% two years ago. Then, last year

it went down by 5% (this data is not true). How much had the cost of living increased in these three years? [The amount has increased by 12.86% (and not 10+8-5=13%)] 52. In the sales we bought a painting for € 105, a bicycle for €50.40 and a book for €16.35. If all

the prices were reduced by 30% how much would we have spent on the same items before the sales? [Painting: €150; Bicycle: €72; Book: €23.36] 53. I have an investment that has returned a 20% increase in performance from the start of the

year until October, and then increased a further 3% in November, what has been the year to date? [23.6% (it is a 123.6% of the total)] 54. A coat is marked down by 40%, and you have a coupon good for an additional 25% off. If

VAT is 20% and you are charged €400, what is the original price (without VAT)?

[€740.74]

55. Your business is growing faster than you had ever imagined. Last year you had 100

employees statewide. This year, you opened several additional locations and increased the number of workers by 30%. With demand so high, next year you will be opening new stores nationwide and plan to increase your employee roll by an additional 50%. Determine the projected number of employees next year. [195 workers] 56. We are making a cone using a base having 8cm radius. Its slant height is 10cm,

and we need 4/5 of a circle for the top. [The net of a cone is a circle with part missing (and a smaller circle underneath). You use the slant height itself as the radius of the part-circle]. a) Write an equation relating “fractions of circle” with “radius” and “slant height”. Find the proportionally constant.

[Sol: Fraction =

. C=1]

b) What fraction of a circle do we need for a cone with 20cm slant height and base radius 10 cm? [Sol: One half of a circle] c) What is the base radius of the cone we can make using 3/4 of a circle having 9 cm radius (so the slant height is 9cm)? [Sol: 12 cm]

32

3º ESO Mathematics Exam (trial exam)

IES Extremadura

Bilingual section

PROPORTIONALITY. (*) All word problems must be answered using a sentence.

1. (1 pt.) Compute the proportional mean “x” of 75 and 12 (i.e. so that

)

2. (1.5 pts.) A farmer has got 300 animals and fodder (pienso) enough to feed them over 90 days. He decides to sell some of them so that he has fodder enough for 135 days. a) How many animals does he have now? b) How many has he sold? 3. (1.5 pts.) In a factory, 3 machines produce 480 pieces in 4 days, working 8 hours per day. a) How long will take them to produce 200 pieces, working 6 hours per day? b) How many hours per day do they have to work 5 machines to produce 500 pieces in 2 days? 4. (1.5 pts.) Three friends have a business that makes a profit of €3,500. Distribute it proportionally to the time and money invested by each of them. Juan: Luisa: Antonio:

3 hours and €20. 5 hours and €15. 4 hours and €10.

5. (1.5 pts.) Distribute a €360 incentive between two workers, inversely proportional to the number of days they have arrived late to work this year. Ana: 10 days and María: 8 days. 6. (1.5 pts.) A piece of cloth cost 30€ in December. In January they offer it with a 10% discount, and later they took another 30% off. If they add a 16% VAT to the final Price, a) Compute the price including VAT. b) Is the same as if (in total) they had made a 24% discount? What would have been the price in that case? 7. (1.5 pt.) The price of a house was marked up (subir el precio) a 10% two years ago and a 4% last year, but this year it was marked down (bajar el precio), a 4%. If now the price €200.000, how much was it two years ago?

33

POLINOMIOS - Monomio: producto de un número “coeficiente” por varias letras “parte literal” (se admiten exponentes en las letras pero no raíces de letras, o letras en el denominador, etc.). El grado del monomio es la suma de los exponentes.  Producto: juntamos las letras y sumamos los exponentes de las que son iguales. Ej: x 2y4·xy3=x3y7.  Suma y resta de monomios semejantes (con la misma parte literal): se suman o restan los coeficientes. Ej. 2x+5x=7x y 2x-5x=-3x. - Polinomio: suma y/o resta de varios monomios (términos). Si hay dos términos, es un binomio; si hay tres, trinomio, etc. Se escriben ordenados, del grado mayor al menor. El monomio de mayor grado da el grado del polinomio. Su coeficiente se llama principal. Término independiente: un número que aparece sin letras. Su grado es 0. Ej: 3x2-5x+1: trinomio de segundo grado, coeficiente principal 3 y término independiente 1. Valor numérico: resultado de sustituir las letras por un número. Raíz de un polinomio: cuando el valor numérico es 0.  Operaciones con polinomios: (el polinomio debe ir entre paréntesis) Signo “+”: se quita el paréntesis y se deja todo como está. Ej.: +(3x2-5x+1)= 3x2-5x+1. Signo “-“: se cambia el signo de cada término. Ej. -(3x2-5x+1)= -3x2+5x-1. Producto: se usa la propiedad distributiva. Ej. (x+3)·(x-2)=x2-2x+3x-6. Puede usarse para sacar factor común. Ej. x2+5x=x(x+5), 4x+6=2·(x+3) Igualdades notables: 2

(a+b) =a2+b2+2ab , (a-b)2=a2+b2-2ab (a+b)·(a-b)=a2-b2.

Pueden usarse para factorizar. Ej. x2–4=(x+2)(x-2), x2-5=

.

División: el grado del resto debe ser menor que el del divisor. Para dividir entre x+a puede aplicarse la regla de Ruffini. Teorema del resto: al dividir entre x-a, el resto es el valor numérico para x=a. Teorema del factor: un polinomio es divisible entre x-a cuando a es raíz suya.  Fracción algebraica: cociente de dos polinomios. Ej.

. Podemos operar con ellas o

simplificarlas como con las fracciones de números.

35

POLYNOMIALS - Monomial: algebraic expression where a number “coefficient” is multiplied by letters “literal part” (exponents are allowed on letters but not square roots, nor a letter in a denominator, etc.). The degree of a monomial is the sum of the exponents.  Product: join the letters and add the exponents of letters alike. Ex: x2y4·xy3=x3y7.  Addition and subtraction of like monomials (with the same literal part): add or subtract coefficients. For instance: 2x+5x=7x and 2x-5x=-3x. - Polynomial: addition and/or subtraction of several monomials (terms). If there are two terms, it is a binomial; if there are three, it is a trinomial, and so on. We write them ordered from the greatest degree to the least. The monomial with the greatest degree gives us the degree of the whole polynomial. It is called the leading term and its coefficient is called leading. Independent term: a number without letters. Its degree is 0. Ex: 3x2-5x+1: second degree trinomial, principal coefficient 3 and independent term 1. Numerical value: the result of replacing letters with numbers in an algebraic expression. Root of a polynomial: when the numerical value is 0.  Operations involving polynomials: (when the polynomial is in parentheses) “+” sign: remove the parentheses and leave everything as it is. Ex.: +(3x2-5x+1)= 3x2-5x+1. “-“ sign: change the sign of the terms. Ex.: -(3x2-5x+1)= -3x2+5x-1. Product: we use the distributive property. Ex.: (x+3)·(x-2)=x2-2x+3x-6. It can be used to find the common factor. Ex.: x2+5x=x(x+5), 4x+6=2·(x+3) Special binomial products: (a+b)2=a2+b2+2ab , (a-b)2=a2+b2-2ab (a+b)·(a-b)=a2-b2.

They can be used for factoring. Ex.: x2–4=(x+2)(x-2), x2-5=

.

Division: the degree of the remainder is less than the degree of the divisor. To divide by x+a we can use Ruffini’s rule. Remainder Theorem: the remainder of the division by x-a, is the numerical value for x=a. Factor Theorem: a polynomial is divisible by x-a when a is a root of the polynomial.  Algebraic fraction: quotient of two polynomials. Ex.: simplify them as well as we do with numerical fractions.

36

. We can operate with them or

POLYNOMIALS – REVISION EXERCISES 1. Use the special binomial products (either to multiply or to factorize) a)

b)

=

d)

=

=

e)

g)

=

h)

j)

=

k)

c)

=

=

f) =

=

i)

=

=

l)

=

2. Take out common factors. (*) Use them to simplify the algebraic fractions, if possible. a) 2x3-5x2+x=

b) (x-1)x2+5x(x-1)-3(x-1)=

d) 3x2(2x+5)+6x(2x+5)-9(2x+5)= e) g)

=

c) 6x8-4x5+2x3=

=

f)

h)

=

=

i)

=

3. Compute the following divisions (compute the quotient “q” and the remainder “r”): a)

b)

c)

d)

e)

f)

g)

h)

i)

Solutions: 1. [a] [d]

.

[b]

.

.

[c]

[e]

[g]

.

[h]

[j]

.

.

[f] [i]

.

[k]

[l]

.

[b] (x-1)(x2+5x-3).

[c] 2x3·(3x5-2x2+1)

[d] 3(2x+5)(x2+2x-3).

[e] 5x2-4x+1.

[f] x2-5x+2

[g]

[h]

2. [a] x·(2x2-5x+1).

.

3. [a] q=

; r=

[d] q=

r=

[g] q=

. .

.

.

[b] q=

; r=

[e] q= ; r= .

[i] 3x2+2x-5.

.

; r= 3

2

[h] q=x -2x +3x-1; r=

. .

[c] q=

; r=-3x2+2

[f] q= .

[i] q=

; r=0. ; r=

.

37

POLYNOMIALS - WORD PROBLEMS 1. Determine an expression representing the total income from selling roses at €6 each and

daffodils (narcisos) at €3 each. 2. Think of a number. Subtract 7. Multiply by 3. Add 30. Divide by 3. Subtract the original

number. The result is always 3. Use polynomials to illustrate this number trick. 3. Let an integer be represented by x. Find, in terms of x, the product of three consecutive

integers starting with x. 4. What expression represents “eight less than the product of five and a number”? 5. Find the area and the perimeter of these polygons. Simplify the answers.

a)

b)

c)

6. Write a variable expression for the area of a square whose side is x + 8. 7. The side of a cube is represented by x + 1. Find its volume in terms of x. 8. A circular courtyard has an area of 10 – 2x2. There are two rectangular

flower beds in the courtyard. Write an expression that represents the green lawn area. 9. Two weeks ago James bought 3 cans of tennis balls. Last week he bought

4 cans of tennis balls. This week he bought 2 cans of tennis balls. The tennis balls cost d dollars per can. Write an expression in simplest form that represents the total amount that James spent 10. For his birthday, Carlos’s parents give him €5 for each year of his age plus €50. His

grandmother gives him €10 for each year of his age. Let a represent Carlos’s age in years. Write a polynomial expression for the amount that Carlos receives from his parents. a) Write a polynomial expression for the amount that he receives from his grandmother. b) Write a polynomial expression for the total amount that Carlos receives from his parents and grandmother. c) How much will Carlos receive when he is 15 years old?

38

11. Lydia took a taxi from her home to school that charged $2 plus $0.50 per mile. Her brother

Luke took a taxi the same distance that charged $3 plus $0.30 per mile. Let d represent the distance in miles. a) Write a polynomial expression for the cost of Lydia’s taxi. Then write a polynomial expression for the cost of Luke’s taxi b) Find an expression representing the total cost of Lydia and Luke’s taxi rides c) What is the total cost if the distance is 20 miles? 12. Maria bought 7 CDs at x dollars each and used a coupon for $20 off her purchase of more

than 5 CDs. Ricky bought 4 CDs at x dollars each and redeemed (canjear) a coupon for $10 off his purchase of more than 3 CDs. a) Write polynomial expressions representing how much each spent after the discount. b) Write a polynomial representing how much more Maria spent than Ricky. 13. The polynomial expression (300 + 0.4s) – (500 + 0.3s) represents the difference between

two salary options that Chuck has in his new position as a salesperson (vendedor). Write this difference in simplest form. 14. On a test worth 100 points, Jerome missed 3 questions worth p points each but answered a

bonus question correctly for an extra 5 points. Suni answered 4 questions incorrectly and did not get the bonus. a) Write polynomial expressions representing each student’s score on the test. b) write a polynomial representing how many more points Jerome scored than Suni. 15. Sal’s Pizza Place charges €8 for a large pizza plus €0.75 for each topping (ingredient), while

Greco’s Cafe charges €10 for the same size pizza plus €0.90 for each topping. Write a polynomial in simplest form that represents how much more a pizza with t toppings would cost at Greco’s than at Sal’s. 16. The Marshalls’ pool is 5 feet longer than twice its width w. Write two expressions for the

area of the pool. What is the area of the pool if it is 12 feet wide? 17. When the Science Club members charged p dollars to wash each car at their car wash, they

had 8p customers. When they doubled their price, they had 12 fewer customers. a) Write expressions representing the new price and the new number of customers. b) Write an expression representing the amount of money they made at the new price. c) How much money did they raise at the new price if the original price was $5 for each car?

39

3º ESO Mathematics Exam (trial exam)

IES Extremadura

Bilingual section

POLYNOMIALS. 1. (1.5 pts.) Given: P(x) = 10x2 +6x-2 , Q(x) = 1– 3x2 –4x3, R(x) = 2x2 – 3 , calculate P(x) – [Q(x)-2x·R(x)] = 2. (1 pt.) Marta bought 5 CDs at x dollars each and used a coupon for $20 off. Juan bought 4 CDs at x dollars each and redeemed (canjear) a coupon for $10 off. a) Write polynomial expressions representing how much each spent after the discount. Marta:

Juan:

b) Write a polynomial representing how much more Maria spent than Juan: 3. (1.5 pts.) Take out the common factors and simplify: a)

=

b)

=

4. (1.5 pt.) Use the special binomials products (either to compute or to factorize): a) 9x2 - 144=

b) (5x2 –6)2 =

5. (3 pts.) Divide: P(x)= a) (1 pts.)

Quotient:

-

-

by Q(x)=

c) (x3-5)(x3+5) = -

Remainder:

b) (0.5 pts.) Write division and the result using algebraic fractions: c) (1.5 pt.) Check the result. 6. (1.5 pts.) Use Ruffini’s rule.

40

a) 2x4-3x3+6x2-x+5 divided by x+1.

Quotient:

Remainder:

b)

Quotient:

Remainder:

ECUACIONES - Ecuación: Igualdad entre expresiones algebraicas que sólo se cumple para algunos valores de las letras, que se llaman solución. Resolver la ecuación es encontrar las soluciones. Las letras se llaman incógnitas. El grado es el mayor grado de los términos. La parte de la izquierda primer miembro, y la de la derecha, segundo miembro.  Resolución de ecuaciones de primer grado (ecuaciones lineales): 1. Quitar paréntesis. 2. Quitar denominadores. 3. Transponer y agrupar términos semejantes. 4. Despejar la incógnita.

(reduciendo primero a común denominador, o si es una igualdad de fracciones, se multiplica en cruz). (se les cambia el signo al cambiarlos de miembro). (Si un número multiplica a un miembro, pasa al otro dividiendo, y viceversa)

 Resolución de ecuaciones de segundo grado (ecuación cuadrática): CASOS POSIBLES: 1. La incógnita siempre aparece con el mismo grado: se hace como con las de primer grado, y luego se hace una raíz. En la raíz cuadrada hay que poner ±. Ejemplo: x2=9;

. Soluciones x=+3 y x=-3.

2. Producto igual a 0: se resuelve cada factor por separado. Ejemplo: para (x-1)·(x-2)=0, se resuelve x-1=0 y x-2=0. Las soluciones son x=1 y x=2. 3. Se puede factorizar: si no hay término independiente, se saca la incógnita factor común. Ejemplo: x2-3x=0. Factorizamos x·(x-3)=0. Las soluciones son x=0 y x=3. 4. Ecuación “completa”: ax2+bx+c=0. Soluciones: Se calcula más rápido si hacemos primero el discriminante

.

Fórmulas de Cardano: cuando a=1, la suma de las soluciones es –b, y su producto es c.  Ecuaciones de Tercer grado o mayor: se puede usar el método de Ruffini. Para las raíces enteras, sólo hay que probar con divisores del término independiente.

41

EQUATIONS - Equation: Equality of two algebraic expressions that is correct for only certain values of the letters, called solution. To solve an equation is to find its solutions. Letters are called unknowns. The degree is the greatest of the terms’ degrees. The left side is the first member, and the right side, the second member.  Solving first degree equations (linear equations): 1. Eliminate parentheses. 2. Eliminate fractions.

(reduce first to common denominator, or if it is an equality of fractions, cross multiply.).

3. Transpose and combine like terms.

(changing the sign when changing to the other side).

4. Get the unknown by itself.

(If a number multiplies on one member, goes to the other dividing it, and vice versa)

 Solving second degree equations (quadratic equations): POSSIBLE CASES: 1. The unknown always appears with the same degree: it is solved as the first degree ones and then, one takes a root. With square roots one has to write ±. Example: x2=9;

. Solutions: x=+3 and x=-3.

2. A product equals 0: solve each factor separately. Example: for (x-1)·(x-2)=0, one solves x-1=0 and x-2=0. Solutions are x=1 and x=2. 3. It is possible to factorise: if there is no independent term, pick out common factor the unknown. Example: x2-3x=0. We factorise like x·(x-3) =0. Solutions are x=0 and x=3. 4. “Complete” equation: ax2+bx+c=0. Solutions: It is computed faster if we first work out the discriminant

.

Cardano formulae: when a=1, the sum of the solutions is –b, and their product is c.  Third or higher degree equations: we can use Ruffini’s method. For integer roots, choose only divisors of the independent term.

42

EQUATIONS – REVISION EXERCISES 1. Solve the following equations (first degree techniques). a)

b)

c)

d)

e)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

=

f)

2. Solve the following equations (second degree techniques). a) (x+1)(x-1)-5(x+2)=x-20

b) 2x(x+3)+x(5-3x)=x2+5

c) (x-4)·(3x+1)=x·(x+1)-4

d)

e)

f)

g)

h)

j)

k)

l)

m)

n) (2x-6)2+5=(x+2)(x-2)

o)

p)

q)

r)

=

i)

Solutions: 1. [a] x=2. [k] x=-3

[b] x=1.

[c] x=5.

[d] x=-1. [e] x=-1. [f] x=3.

[g] x=3.

[h] x=5.

[l] x=1

[m] x=6

[n] x=3

[q] x= 2

[r] x=5.

2. [a] x=3.

[o] x=5

[p] x=1

[i] x=-2.

[j] x=4

[b] x=5, x=1/2.

[c] x=0, x=6.

[d] x=2, x=-12/5

[e] x=-2, x=4.

[f] x=3, x=-7/2.

[g] x=5, x=-3/8.

[h] x=4, x=-1/3

[i] x=5, x=-4.

[j] x=2

[k]x=5, x=10

[l] x=-5, x=2.

[m] x=-4, x=3.

[n] x=3, x=5

[o] x=-2, x=5

[p] x=2, x=-14/5

[q] x=-3, x=6.

[r] x=±2.

43

3. Solve the following equations (using first degree equations techniques). a) x2+3·(5-2x2)=6-4x2

b) 3(15-x3)+4x3-45=3(x3+5)-18 c) 4·(x2-3)-6·(x2-2)+3x2-102=21

d) 4x3+100=5(x3-50)+7

e) 3·(40-2x4)-8·(x4-10)=-8-x4

4. Solve the following equations (remember

NOT

f) x5-3(2x5+4)+6(2-2x5)=3+2(3x5+10)

to multiply before solving).

a) (x-4)(3x+6) = 0

b) (5-x)·(2x+3)·x=0

c) x·(x2-5x+6)=0

d) (x2+x-2)(x-2)=0

e) (3x2+14x-5)·x2=0

f) (2x2-x-1)·(x-5)=0

g) (x2-4)·(x2-3)=0

h) (x3-343)·(2x2+3x-5)=0

i) (2x3-250)·(x3+64)=0

5. Take out common factors before solving (remember NOT to multiply the polynomials). a) x3+2x2 - 8x= 0

b) x·(5x-1)+3·(5x-1)=0

d) x2(x-1)-5x(x-1)-6(x-1)=0

e) (x2-4)x2+2x(x2-4)-15(x2-4)=0 f) 3x2(2x2+5x-3)+9x(2x2+5x-3)=0

g) x2·(x3+27) – 5·(x3+27)=0

h) (2x3-54)·x2 =-4·(2x3-54)

i) (x4+16)·x2+2x(x4+16)=-5(x4+16)

a) 2x3+16x=10x2+8

b) x3+11x=6·(x2+1)

c) x4+x3-7x2-x+6=0

d) x·(x2+5x+6)-5x2=5(6+5x)

e) x3+x=10·(4x-7)

f) 27·(x-2)=x3

g) 3x-6 = 3x·(x-2)

h) x·(x2-6x+5)=6·(x2-6x+7)

i) x2·(x3-2x)=3x·(x2-1)-x

c) x·(3x-6)=5·(3x-6)

6. Solve using Ruffini’s method:

Solutions: 3. [a] x=±3. 4. [a] x=4, x=-2

[b] x=2.

[c] x=±11.

[b] x=5, x=-3/2,x=0. [c] x=0, x=2, x=3.

[f] x=-1/2, x=1,x=5. [g] x=±2, x=± 5. [a] x=0, x=2, x=-4

[f] x=-6, x=3.

44

.

[e] x=±2. [d] x=-2, x=1, x=2.

[f] x=-1 [e] x=0, x=1/3, x=-5.

[h] x=7, x=-5/2,x=1

[i] x=1, x=9/4.

[c] x=2, x=5.

[d] x=-1, x=1, x=6.

[h] x=3.

[i] No solution.

[b] x=1, x=2,x=3

[c]x=-1, x=1, x=-3,x=2

[d] x=5, x=-3, x=-2

[g] x=1, x=2

[h] x=2, x=3, x=7

[i] x=-2, x=-1, x=0, x=1, x=2.

[b] x=-3, x=1/5.

[f] x=1/2, x=-3,x=0. [g] x=-3, x=± 6. [a] x=1, x=2.

[d] x=7.

.

[e] x=±2, x=3, x=-5.

[e] x=2, x=5, x=-7.

7. Solve using the most suitable method: a)

x3 +2x 4x2

-

x2 -6x+4

b)

c) (2x-4)·(3x-5) 2(x+1)

d) 2x·(x2 -9)-10·(x2 -9) 0

e)

f)

g) 2x·(x-5) (x-1)·(x+1)+x+1

h)

i)

j) x3 =19x+30

k) 3

m)

n)

o) (x2 -2x)(2x-5)-6x (x+4)(x-4)

p) x3 (5x-1)-8(5x-1)=0

q) 4x(x3-2)+8(x3-2)=0

r)

s)

t)

u)

v)

w) x·(x+1)+1 0

x)

y) x(x2+4)-5(x2+4)=0

z) x5 -17x3 +12x2 +52x-48 0.

3

2

·(x2 -5)-12·(x2 -5) 0

x3 -3x2 -3x+9 5

+

x2 +3 2

-2x+2

7x-1 10

l)

Solutions: 7. [a] x=1, x=2, x=6.

[c] x=1, x=3.

[d] x=±3, x=5.

[e] x=.

[f] x=2, x=3, x=

[g] x=0, x=11.

[h] x=±3, x=-1, x=2.

[i] x=-3, x=2.

[j] x=-3, x=-2, x=5

[k] x=±2, x=

[l] x=2

[m] x=1, x=

[n] x=0, x=2, x=

[o] x=-1, x=2, x=4

[p] x=2, x=

[q] x=-2, x=

[r] x=±1, x=7

[s] x=±1

[t] x=0, x=±2

[u] x=±1, x=±3

[v] x=-3, x=1, x=

[w] No solution.

[x]x=0, x=

[y] x=5

[z]x=1, x=±2, x=3, x=-4 .

[e] x=1, x=

.

[b] x=1, x=3, x=

.

45

EQUATIONS - WORD PROBLEMS 1. The length of a rectangular window is 5 dm more than its width, x. The area of the window

is 36 square decimetres. Write an equation to find the dimensions of the window. 2. Mike’s Fitness Centre charges €30 per month for a membership. All-Day Fitness Club

charges €22 per month plus an €80 initiation fee for a membership. After how many months will the total amount paid to the two fitness clubs be the same? How much will it be? 3. A Plumbing (fontanería) Service charges €35 per hour plus a €25 travel charge for a service

call. “P4U Plumbing Repair” charges €40 per hour for a service call with no travel charge. How long must a service call be for the two companies to charge the same amount? How much will they charge? 4. The Lone Star Shipping Company charges €14 plus €2 a kg. to ship an overnight (nocturno)

package. Discount Shipping Company charges €20 plus €1.50 a kg. to ship an overnight package. For what weight is the charge the same for the two companies? How much will the shipment cost? 5. Julia and Isa are playing games at the arcade. Julia started with $15, and the machine she is

playing costs $0.75 per game. Isa started with $13, and her machine costs $0.50 per game. After how many games will the two girls have the same amount of money remaining? How much will they have then? 6. The Wayside Hotel charges its guests €1 plus €0.80 per minute for long distance calls.

Across the street, the Blue Sky Hotel charges its guests €2 plus €0.75 per minute for long distance calls. Find the length of a call for which the two hotels charge the same amount. What would be the price? 7. Juan is a part-time student at Horizon Community College. He currently has 22 credits, and

he plans to take 6 credits per semester until he is finished. Juan’s friend Lidia is also a student at the college. She has 4 credits and plans to take 12 credits per semester. After how many semesters will Juan and Lidia have the same number of credits? How many credits will they have? 8. A car leaves Badajoz at 5:30pm, driving at 120km per hour. At the same time, a bus leaves

Mérida (60km further) driving at 100kmph. a) How long will take the car to catch up with the bus? What time will it be? b) How long will have travelled each of them?

46

9. Juan’s got 215€ and Pedro’s got 160€. How much should Juan borrow from Pedro to have

twice the money as Pedro? How much will each have?

[Solution: Juan should borrow €35].

10. Pedro is 34 years old and his son is 6. In how many years will Pedro be three times older

than his son? How old will each of them be?

[Solution: In 8 years. Pedro will be 42 and his son 14].

11. [Clocks] Juan is watching a clock, and realizes that the minute-hand moves at a speed of 12

numbers per hour, and the hour-hand at 1 number per hour. a. At 12:00a.m., both hands coincide, and not at 3:15p.m., but a little later. What time after 3:00p.m. do both hands coincide? [Sol: 3+3/11=3h16m21’6s] b. At 6:00pm. they form a straight angle (their difference is 6 numbers). At 7:05p.m. the angle is not straight; that happens a little later. What time after 7:05p.m. do they form a straight angle? [Sol: At 7+1/11=7h05m27s] c. At 9:00p.m. they are perpendicular (so their difference is 3 numbers), but not at 2:25p.m. What time after 2:00p.m. do they form a right angle? [Sol: At 2+5/112h27m16’36s] 12. Juan is 3 years old and his father is 32. In how many years will the product of their ages be

252? How old will they be then?

[Sol. In 4 years time. Juan will be 7 and his father 36]

13. Ana is 10 years old and her mother is 34. How long ago was the product of their ages equal

to 145? How old were them?

[Sol: 5 years ago. Ana was 5 and her mother 29].

14. A rectangular swimming pool is twice as long as it is wide. A small concrete (cemento,

hormigón) walkway (pasillo) surrounds the pool. The walkway is a constant 2 metres wide and has an area of 196 square metres. Find the dimensions of the pool. (Hint: solve (2x + 4)(x + 4) - (2x)(x) = 196) Solution: The pool is 15 meters wide and 30 meters long.

47

3º ESO Mathematics Exam (trial exam)

IES Extremadura

Bilingual section

EQUATIONS. Solutions:

1. (1.75 pts.) x2(3x-1)+2x(3x-1)=-5(3x-1) .. 2. (1.75 pts.) (2x2-50)(2x2-4x-6)=0. 3. (2 pts.)

4. (1.75 pts.) x3+16=12x. 5. (1 pt.) Complete, so the solutions are… a) x=1, x=3:

x2 _____·x _____=0

b) x=-1, x=2:

3x2 _____·x _____=0.

6. (1.75 pts.) A car leaves Mérida at 10:20am in direction to Madrid, driving at 120km per hour. At the same time, a van leaves Madrid (350km away) driving at 90km per hour in direction to Mérida. a) How long will it take them to meet on the road?

What time will it be?

b) How many km will have travelled each of them? [*] Set the problem up using a one unknown equation. Write answer (a) using hours and minutes.

48

SISTEMAS DE ECUACIONES  SISTEMA DE ECUACIONES: es cuando tenemos varias ecuaciones de las que buscamos una solución común. Si las ecuaciones son de primer grado, el sistema se llama lineal. - Para dar la solución del sistema, hay que dar un valor para cada incógnita.  RESOLUCIÓN DE SISTEMAS DE DOS ECUACIONES Y DOS INCÓGNITAS. MÉTODOS ALGEBRAICOS: Se elimina una incógnita y queda una ecuación de una incógnita. La resolvemos. Después, se sustituye el valor obtenido en una de las ecuaciones y se resuelve lo que quede. - Método de SUSTITUCIÓN: despejamos una incógnita en una ecuación y sustituimos en la otra. Lo usamos en sistemas no lineales cuando alguna incógnita aparece con varios exponentes. - Método de IGUALACIÓN: despejamos la misma incógnita en las dos ecuaciones. Hacemos una nueva ecuación igualando los resultados. - Método de REDUCCIÓN: sumamos las ecuaciones de manera que desaparezca una incógnita. Puede que primero haya que multiplicarlas por algún número. MÉTODO GEOMÉTRICO: Cada ecuación lineal se dibuja como una recta del plano (usando una tabla de valores). La solución es el punto donde se cortan las rectas.

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EQUATIONS SYSTEMS  EQUATIONS SYSTEM: it is when we have several equations and we are looking for a common solution. If the equations are of first degree, the system is called linear. - In order to find a solution for the system, we have to give a value for each unknown.  SOLVING SYSTEMS OF TWO EQUATIONS AND TWO UNKNOWNS. ALGEBRAIC METHODS: Eliminate one unknown in order to get a “one variable equation” that we solve. Then, we substitute the value obtained in one of the equations and solve the remaining equation. - SUBSTITUTION method: isolate one variable in one equation and substitute in the other. We use it in nonlinear systems when any of the unknowns appears with different exponents. - ALGEBRAIC EQUATION method: isolate the same variable on each equation. Make another equation equalling the results. - ELIMINATION (ADDITION) method: add the equations so that one variable disappears. You may first need to multiply them by some number. GRAPHING METHOD: Draw each linear equation as a line on a coordinate plane (using a table of values). The solution is the point where the lines meet.

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EQUATION SYSTEMS – PRACTICE 1. Solve the following equation systems. a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

2. Solve the following equation systems. a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

Solutions: 1. [a] x=1, y=3. [g] x=-2, y=1 2. [a] x=3, y=2. [g] x=3, y=6

[b] x=-3, y=1.

[c] x=2, y=5

[d] x=-3, y=10

[e] x=3, y=1

[f] x=5, y=3

[h] x=1, y=4.

[i] x=7, y=7

[j] x=1, y=-1

[k] x=5, y=10

[l] x=2, y=-3.

[b] x=4, y=-1.

[c] x=5, y=1

[d] x=4, y=2

[e] x=2, y=-1

[f] x=3, y=6

[h] x=2, y=5.

[i] x=1, y=-2

[j] x=-1, y=-3.

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3. Solve the following equation systems. a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

4. Solve the following equation systems. Use substitution a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

Solutions: 3. [a] x=±1, y=2. [g] x=

, y=3

[b] x=±3, y=2.

[c] x=1, y=-3

[d] x=±1, y=5

[e] x=7, y=±5

[f] x=±

[h] x=±4, y=3.

[i] x=3, y=2

[j] x=5, y=±4

[k] x=6, y=±4

[l] x=±2, y=±7 (4 sol.)

4. [a] {x=2, y=3}, {x=14/3, y=1/3}.

[b] {x=-1, y=-1}, {x=-2/7, y=-8/7}

[c] x=±3, y=4.

[d] {x=10, y=-2}, {x=14, y=0}.

[e] {x=-4, y=12}, {x=3, y=5}.

[f] {x=-1, y=0}, {x=2, y=3}

[g] {x=3, y=-1}, {x=41/9, y=4/3}.

[h] {x=-1, y=1}, {x=3, y=5}.

[i] {x=2, y=1}, {x=-8/5, y=11/5}.

[j]

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, y=±3 (4 sol.)

5. Solve the following equation systems. (Use algebraic equation method –igualación-) a)

b)

c)

d)

e)

f)

g)

h)

i)

6. Solve the following equation systems. Choose the most suitable method. a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

Solutions: 5. [a]

-

-

[d]

-

-

-

[g] 6. [a]

-

-

[e]

-

-

-

-

-

-

-

[g] -

-

[b]

-

[e]

-

[h]

[c] There is no solution. [f]

[h]

[d]

[j]

[b]

-

[i]

-

-

-

[c] -

[f] -

[i]

-

-

-

(four solutions)

-

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EQUATIONS SYSTEMS - WORD PROBLEMS 1.

Two years ago, Carlos’ age was triple the age of his son Luis, but in twelve years his age will only be double that of Luis. Calculate their ages. [Sol: 44 and 16].

2.

On a test with 30 answers, students receive 2 points for each correct answer and lose 1 point for each incorrect answer. A boy who answered all of the test’s questions got a score of 24 points. How many correct answers and how many incorrect answers did he give? [Sol: 18 correct and 12 incorrect]

3.

How old is my youngest son if the sum of triple his age and five times his older brother's age is 78, and his age is half the age of his older brother’s age? [Sol: youngest: 6, older 12]

4.

A trader splits (reparte) 315 litres of oil between 1.5 litre and 0.75 litre bottles. If he fills 300 bottles, how many bottles of each capacity does he use?

5.

I paid €45 for a shirt and a pair of trousers which would have cost me €52 had they not been on offer. The shirt had been discounted 20% and the trousers 10%. What were the original prices?

6.

Find the value of two numbers if twice the lesser number is one more than the greater number and three times the lesser number minus the greater number is five. [Sol: 4 and 7]

7.

In my piggy bank (hucha) I have three times 2€ coins than 1€ coins. In all, I have 84€. How many coins do I have of each type? [Sol: 36 coins of 2€ and 12 coins of 1€.]

8.

In a language school, 430 students studied either English or French last year. This year, students studying English have increased 18% and students studying French have increased 15%. There are now 502 students studying the two languages. Calculate how many English students and how many French students there were last year.

9.

I changed a lot (lote) of 20 cent coins for €1 coins, and now I have 12 coins less than I did before I changed them. How many 20 cent coins did I have?

10. Three years ago, Laura’s age was half of Ana’s age, and in seven years their ages will add up

to 50. How old are they?

[Sol: Laura is 13 years old and Ana is 23 years old.]

11. Ana and Raúl, the boy she was babysitting, were playing basketball together. Her score was

30 points, and his score was 10 points. Ana wanted to make the game fairer (más justo), so she called a time-out and modified the rules a bit. Ana explained that, for the rest of the game, she would get 3 points per basket, and Raúl would get 4 points per basket. Then they played a bit longer. After the time-out, they both made the same number of baskets and ended up with a tied score. How many points did each person have at the end?

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12. A merchant has two types of coffee: one of 6€ per kg, and another of 4€ per kg. He mixes

them, and the price turns out to be 4.5€ per kg. In all, he has 8 kg of coffee. How much is there of each type of coffee? (*) Set the problem up using an equations system. The first one adding the kilograms and the second one computing the price of 8kg of coffee.

13. We mix two types of flour, one that costs €0.75 per kg and another that costs €1.15 per kg,

and we obtain 50 kg of a mix which costs €1 per kg. How much of each type of flour is used in the mix? [Sol.: 18.75kg of the cheaper flour and 21.35kg. of the more expensive flour] 14. A producer’s union mixes two types of coffee, one that costs €5 per kg and another that

costs €7.5 per kg. They obtain 30kg of a mix which costs €6 per kg. How much coffee of each type does the mix contain? 15. I want to mix two types of wine that costs €5 per litre and €3 per litre to obtain 25 litres of

a mix that will cost €3.80 per litre. How many litres of each type of wine do I need to mix? 16. The freshman (estudiante de primer año) and sophomore (estudiante de segundo año)

classes at Norwood High School are decorating floats (carrozas) for homecoming (fiesta de vuelta a casa). The freshmen have already spent £90 on their float, plus they need to buy floral sheeting that costs £66 per roll. The sophomores, who have spent £61 so far on theirs, still need to purchase vinyl grass at £95 per roll. Both classes plan to buy the same number of rolls, since they have the same area to cover. By coincidence, the two floats will have the same total cost in the end. How many rolls will each class be buying? How much will each class spend in total? [Sol: 1 roll, £156] 17. The admission fee (entrada) at a small fair (feria) is 1.50€ for children and 4.00€ for adults.

On a certain day, 2200 people enter the fair and 5050€ is collected. How many children and how many adults attended? [Sol: 1500 children and 700 adults attended the fair.] 18. A boat travelled 210 miles downstream (río abajo) and back. The trip downstream took 10

hours. The trip back took 70 hours. What is the speed of the boat in still water (si el agua estuviese quieta)? What is the speed of the current? (Measure them in miles per hour mph). (*) In order to set the problem up, think if you have to add or subtract the speed of the boat and the speed of the current. [Sol: boat: 12 mph, current: 9 mph.] 19. A landscaping company placed two orders with a nursery (vivero). The first order was for

13 bushes (arbustos) and 4 trees, and totalled $487. The second order was for 6 bushes and 2 trees, and totalled $232. What were the costs of one bush and of one tree? (They do not know it because the bills do not list the per-item price). [Sol: bush $23; tree: $ 47.]

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20. Flying to Kampala with a tailwind (volando con viento de cola) a plane averaged 158 km/h.

On the return trip the plane only averaged 112 km/h while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. (*) In order to set the problem up, think if you have to add or subtract the speed of the plane and the speed of the wind. [Sol. Plane: 135 km/h, Wind: 23 km/h.] 21. Matt and Ming are selling fruit for a school fundraiser. Customers can buy small boxes of

oranges and large boxes of oranges. Matt sold 3 small boxes of oranges and 14 large boxes of oranges for a total of $203. Ming sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220. Find the cost each of one small box of oranges and one large box of oranges. [Sol: small box of oranges: $7, large box of oranges: $13.] 22. 3 kilos of coffee, which costs €16 per kg, are mixed with 5 kilos of a different type of coffee,

which costs €12 per kg. How much does each kg of the mixture cost? 23. 400 litres of wine costing €9 per litre is watered down with 50 litres of water. How much

does each litre of the watered down wine cost? 24. A passenger jet took three hours to fly 1800 miles in the direction of the jetstream

(corriente de aire). The return trip against the jetstream took four hours. What was the jet's speed in still air (con el aire quieto) and the jetstream's speed? (Write the answer in miles per hour mph). [Sol: The jet’s speed in still air was 525mph and the jetstream’s spead 75mph.] (*) In order to set the problem up, think if you have to add or subtract the speed of the plane and the speed of the wind before multiplying by the number of hours. 25. Find two positive numbers whose sum is 20 and their product is 96.

[Sol: 12 and 8].

26. Last Wednesday, two friends met up after school to read the book they were both assigned

in Literature class. Tony can read 1 page per minute, and he had already read 60 pages. Belle, who has a reading speed of 2 pages per minute, had read 20 pages. Eventually they had read the same number of pages. How many pages had each of them read? 27. I spent 16€ on potatoes yesterday. I only remember that the price per kg (which was less

than the kg.) plus the number of kilograms was 10. How many kg did I buy? What was the price per kg? [Sol: 8kg, and they cost 2€ per kg]. 28. The side of my house is 6m largest than the side of my swimming-pool. Their areas sum

90m2. If both are squared, find the measure of their sides.

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[Sol: House: 9m, Pool: 3m].

3º ESO Mathematics Exam (trial exam)

IES Extremadura

Bilingual section

EQUATION SYSTEMS.

1. (1 pt.) Fill so the solution to the system is x=3, y=2.

2 (1 pt.) Solve using the graphing method

3 (2 pts.) Solve using the elimination method

4 (2 pts.) Solve

5. (2 pts.) Solve using the substitution method

6 (2 pts.) Pedro and Juan are making presentations for a class project. Pedro's slideshow starts with a verbal introduction that is 19 seconds long, and then each slide (diapositiva) is left up for 5 seconds. Juan leaves each slide onscreen for 10 seconds, and his introduction lasts 9 seconds. Pedro and Juan notice that their presentations have both the same number of slides and the same duration. - How many slides are in each presentation? - How long is each presentation? (Solve the problem using the substitution method).

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RECTA, PARÁBOLA E HIPÉRBOLA  FUNCIÓN AFÍN (y=mx+n). Es una recta; m es la pendiente y n la ordenada en el origen. Pendiente (m): mide la inclinación de la recta. m>0: es creciente; m0, y convexa () si a0: it is increasing; m0, and convex () when a