Year 7 Mathematics Exam Booklet: Equations

Year 7 Mathematics Exam Booklet: Equations

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Year 7 Mathematics

Equations Name: …………………………

Easy:

1. If 𝑚 = 5 and 𝑛 = −3, write down the substitution and evaluate the following.

a) 𝑚𝑛

b) 𝑚+𝑛

4

c) 𝑚2 − 𝑛

2. Tom has 160 photos that he took while on a bus trip tour around Australia. He

puts his photos in an album, 6 on a page. How many whole pages will he need to

display all his photos?


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Year 7 Mathematics


3. Solve for 𝑥:

a) 𝑥 − 4 = 7

b) If 𝑣 = 𝑢 + 𝑎𝑡, find the value of 𝑣 when 𝑢 = −6, 𝑎 = 12, 𝑡 = −2.5.

4. Find the size of 𝑥 in the diagram below.

Give a reason for your answer.

Note: Part of Equations and Geometry

1300 𝑥0


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Year 7 Mathematics


5.

a) Is 𝑥 = 4.2 a solution to the equation 6𝑥

5=

126

25?

Answer YES or NO

b) If 𝑥 = 2 and 𝑦 = −3 determine the value of: 𝑥 − 𝑦 and 3𝑥3.

c) Circle the equation below.

2𝑥 + 1

2𝑥 + 1 = 3

d) Write a two-step equation that has the solution 𝑦 = −2.

e) Write an equation for the following statement. Let the number be 𝑥.

“The product of 7 and a number, is divided by 2 and the result is 15”.

Do not solve this equation.


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Year 7 Mathematics


6. Solve the following equations:

a) 𝑥

3= 10

b) 𝑥 + 8 = 23

7. The cost, C, in dollars, of an apartment on level 𝐿 of a building in Sydney is given

by:

𝐶 = 400 000 + 5000(𝐿 − 33)

Find the cost of an apartment on level 42.

8. Substitute 𝑎 = −4, 𝑏 = 3, and 𝑐 = 12 into each of the expressions and evaluate.

a) 2(𝑎 + 𝑏)

b) 4𝑎2 + 2𝑏

c) (4𝑎)2


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Year 7 Mathematics


9.

a) If 𝑎 = 3, 𝑏 = 4, and 𝑐 = −5,what is the value of 2𝑎𝑏 − 𝑐 ?

(a) 29

(b) 19

(c) 12

(d) 17

b) Solve the following equations

2𝑥 = 12

10.

a) Find the value of 𝑥, no reason is required.

Note: This is geometry and equations combined.

b) 𝑦 − 2 = 10

2𝑥

𝑥

𝑥

𝑥


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Year 7 Mathematics


c) Evaluate 𝑛2 − 𝑚 if 𝑚 = 5 and 𝑛 = −3

11.

a) Solve the following equations:

𝑥 − 4 = −5

b) Find the value of 2𝑎 + 𝑏2 if 𝑎 = 4 and 𝑏 = −3.


a) 𝑥 − 5 = 2

b) 𝑝

3= −4


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Year 7 Mathematics


13. Find the value of 𝑥𝑦 − 𝑦 when:

a) 𝑥 = 2 and 𝑦 = 3

b) 𝑥 = −2 and 𝑦 = −3


a) 𝑥 − 10 = 4

b) 6𝑎 = 18

c) 𝑏

9= 2


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Year 7 Mathematics


15.

a) Without trying to solve the equation, show that 𝑥 = 2 is a solution to the

equation

6

𝑥 + 1= 6 − 𝑥2

b) Evaluate the expression

𝑢𝑡 +1

2𝑎𝑡2

Where 𝑢 = 16.1, 𝑎 = −9.81, and 𝑡 = 2.

16. IF 𝑥 = 4 and 𝑦 = 3 find the value of 5(𝑥 + 𝑦)


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Year 7 Mathematics


17.

a) Evaluate: 3𝑚 − 2𝑛 if 𝑚 = 4 and 𝑛 = 9.

b) Evaluate: −3𝑎2 when 𝑎 = −4

18.

a) Find the value of (𝑎 + 𝑏)(𝑎 − 𝑏) if 𝑎 = −5 and 𝑏 = −2

b) Solve for 𝑥:

13 − 𝑥 = 18

19. Evaluate 4𝑥 − 𝑦3 when 𝑥 = 5 and 𝑦 = −2.


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Year 7 Mathematics


20.

a) Find 𝑟 if 6𝑟

11= 1.

b) Evaluate 7√𝑥, if 𝑥 = 25.

21.

a) Solve the equation 18

𝑝=

3

5

b) Show that 𝑥 = 10 is a solution to the equation 3𝑥 − 7 = 2𝑥 + 3.

22. Solve 8

𝑥= 32


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Year 7 Mathematics


23. If 𝑝 = −1, 𝑞 = 2 and 𝑟 = 8, then evaluate √𝑟 + 𝑝𝑞2

24. The formula for the cost per kilogram $𝐶 of an alloy is given by 𝐶 =𝑛𝑥+𝑚𝑦

𝑛+𝑚. IF

𝐶 = $10, 𝑚 = 6, 𝑥 = 8, and 𝑦 = 14, find the value of 𝑛.

25. Find the value of −4𝑏2 when 𝑏 = −1


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Year 7 Mathematics


Medium

26.

a) Find the perimeter of a rectangle whose width is 7.5 𝑐𝑚 and area is

90 𝑐𝑚2.

b) A number is decreased by 6 then multiplied by 8. The result is 72. Find the

number.

27.

a) A stone dropped from the top of a cliff 50 𝑚 above the water and falls to

the bottom of the sea. The depth of the water is 73 𝑚. How far has the

stone dropped?

b) 3 + 2𝑥 = 13


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Year 7 Mathematics


28. Find three consecutive numbers such that the sum of the first and third number

is equal to −10.

29.

a) 𝑥

4− 5 = 7

b) 2𝑥

3= 6

30.

a) By substituting 𝑥 = 5, show that 6𝑥 + 4 − 2𝑥 is equivalent to 4(𝑥 + 1).

b) Is it possible for 3𝑥 + 7𝑦 and 10𝑥 to have the same value? Give a reason

for your answer.


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Year 7 Mathematics


31. Solve for 𝑏: 𝑏−5

2= 8

32.

a) By substituting 𝑚 = 3, show that 5𝑚 − 7 is equivalent to 2(𝑚 + 1).

b) Write an expression to describe how you would calculate the total cost of

2 hamburgers (ℎ) and a one drink (𝑑). If a drink costs $3.75 and the total

cost of 2 hamburgers and one drink is $7.25, find the cost of hamburger.

33.

a) If 𝑎 ∗ 𝑏 = 1 −𝑎

𝑏, what is the value of 2 ∗ (3 ∗ 4)?


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Year 7 Mathematics


b) Form a two-step equation whose solution is 𝑥 = 3.

c) 𝑦+2

2= −1

34. Solve the following equations and show all necessary working.

a) 5𝑥 + 4 = 24

b) 𝑥+1

2= 14

c) 2(5 − 𝑥) = 7


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Year 7 Mathematics


35. Solve the following equations

a) 3𝑥 − 4 = 14

b) 2𝑥

5= 10

c) 15 − 2𝑥 = −7

36.

a) The sum of the heights of Gerard and Rachel is 194 cm. Rachel is 8 cm

taller than Gerard. How tall is Gerard?


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Year 7 Mathematics


b) Note: This question is geometry and equations combined

In the diagram below, find the value of 𝑥 that will makes AB parallel to CD.

37. A box contains an unknown number of pencils ‘6𝑥’. Ansalee states that there are

60 pencils in the box and Allegra states that there are 30 pencils in the box.

Is this possible? Explain your reasoning.

A B

C D

𝑥 + 250


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Year 7 Mathematics



a) 2 − 3𝑥 = 0

b) 5𝑥 + 2 = 4𝑥

c) 8 − 3𝑘 = 4𝑘 + 2

d) If 5𝑥 + 2 = 5𝑦, find the value of 10(𝑥 − 𝑦).


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Year 7 Mathematics



a) 3𝑝 = 𝑝 − 5

b) 2(𝑘 − 1) = 6

c) 13𝑥 + 4 = 9𝑥 − 12

d) 7 − 5𝑥 + 2(7 − 𝑥) = 0


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Year 7 Mathematics


40.


(i) 9 + 3𝑘 = 21

(ii) 29−𝑎

6= 4

b) Given 𝑚 = 0.0058 and 𝑛 = −0.29, find:

(i) 𝑚 × 𝑛

(ii) 100 𝑚2

𝑛


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Year 7 Mathematics


41.


(i) 𝑥+3

4= 5

(ii) −2(𝑧 + 1) = 8

b) Solve the equation 2𝑝 + 1 = 15 − 5𝑝.

42. If 𝑣2 = 𝑢2 + 2𝑎𝑠 where 𝑣 = 8, 𝑢 = 3, and 𝑎 = 10, find the value of 𝑠 as a fraction

in its simplest form.


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Year 7 Mathematics


43.

a) Find the value of 2𝑥 −1

𝑥 if 𝑥 − 1

2

3

b) Solve for 𝑦: 2𝑦

3−

𝑦

2=

5

6

44. Solve for 𝑥: −4𝑥 = 22.

45. Solve for 𝑎:

2𝑎 − 5 = 4𝑎 + 13


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Year 7 Mathematics


46.

a) Solve 3𝑚 + 16 = 25.

b) Evaluate 𝑥 exactly given that

𝑥 =−𝑏±√𝑏2−4𝑎𝑐

2𝑎 and = 8, 𝑏 = 1, and 𝑐 = −5.

47.

a) Solve 15𝑥 − 16 = 15 − 16𝑥

b) Solve 3𝑥 − 7 = −25

c) Given that 𝑉 =ℎ

3(𝑎2 + 𝑎𝑏 + 𝑏2) is the formula for the volume of a

frustum, find the value of 𝑉 if 𝑎 = 20, 𝑏 = 10 and ℎ = 15.


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Year 7 Mathematics


48. Find the value of 𝑥 if 𝑥, 20, 16, 30 are in proportion.

49. Solve the equation for 𝑥. (Show working)

3𝑥 + 15 = 66

50. The value of √20+𝑥2

√20−𝑥2 when 𝑥 = 4, is?

(A) 9

4

(B) 3

(C) 9

2

(D) 9

51.

a) Solve for 𝑥: 5𝑥−2

3= 9

b) Show that 𝑥 = 1 is a solution to

3𝑥2 − 2 = 5 − 4𝑥


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Year 7 Mathematics


Hard

52. A tower of 30 identical blocks is 450 cm high. What is the distance between the

top of the 16𝑡ℎ block and the top of the 24𝑡ℎ block?

?


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Year 7 Mathematics


53. Two buses took Year 7 and Year 8 on an excursion to the Zoo. Half of the

students in Year 7 got on the first bus with 15 Year 8 students. One quarter of the

students in Year 7 got on the second bus with 50 Year 8 students. The remaining

Year 7 students stayed at school.

If both buses carried the same number of students, how many students are there

in Year 7?

54. In this problem 𝑚 and 𝑛 stand for two missing numbers. 𝑚

8+

𝑛

8= 1

𝑚

8−

𝑛

8=

1

4

Find the values of 𝑚 and 𝑛.


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Year 7 Mathematics


55. Find the values of 𝑀 and 𝑁 if 4

5=

𝑀

40=

40

𝑁

56.

a) Solve the equation

5(2𝑦 + 1) + 1 = 2(1 − 𝑦) − 3

b) Emily and her daughter share the same birthday. This year Emily is six

times her daughter’s age and in twelve years time Emily will be three

times her daughter’s age. How old was Emily when her daughter was

born?


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Year 7 Mathematics


57. A number 𝑛 is tripled, then 14 is subtracted. It is found that the result is the same

when 𝑛 is doubled, then 8 is added.

a) Form an equation to represent the information above.

b) Solve your equation to find the value of 𝑛.

58.

a) The sale price of a car is 𝑥 dollars. Rather than paying upfront, the buyers

are able to pay a deposit of 25% of the sale price followed by 25 monthly

repayments of $350.

(i) Luke decides to pay the deposit followed by the monthly

repayments. Express the total amount he pays for the car in terms of

𝑥.

(ii) If the amount Luke pays is double the sale price, calculate the sale

price of the car.


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Year 7 Mathematics


b) Solve the following equations:

(i) 3𝑥 + 2 = 4(𝑥 + 1)

(ii) 2(𝑥 − 2) + 3(5 − 𝑥) = 4(2𝑥 + 1)

(iii) 2𝑥+1

3−

1−𝑥

6= −2


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Year 7 Mathematics


59.

a) Sam is participating in a mathematics competition. There are two types of

questions:

Easy questions worth 3 points, and hard questions worth 5 points. Sam correctly

answers 22 questions and she is awarded a total of 80 points.

(i) Let the number of easy questions correctly answered by Sam be 𝑥.

Write an equation in terms of 𝑥 to describe this scenario.

(ii) Solve your equation to find out how many of each type of question

Sam has answered correctly.

b) Solve the equation 17(7𝑥 + 4) = 18 + 13(7𝑥 + 4)


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Year 7 Mathematics


60. Five distinct points A, B, C, D and E lie on a line but not necessarily in that order.

Use the following four pieces of information to find the distance between C and

D.

a) E is the midpoint of AB.

b) D is the midpoint of AE.

c) Both C and E are the same distance from B.

d) The distance from B to D is 9cm.

61. A man paddles his canoe from his home to his friend’s place which is upstream

and takes 3 hours for the journey. He returns that afternoon, paddling at the

same speed, in 2 hours. The river is flowing at 2 km/hour.

Let 𝑠 km/hour be his paddling speed in still water.

a) Write down an equation in 𝑠, and then solve it, to find the paddling speed in

still water.

b) Hence, or otherwise, find the distance to his friends place.


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Year 7 Mathematics


62.

a) When an unknown number 𝑥 is added to the numerator and denominator

of the fraction 2

11, the new fraction that is formed is 4 times the original

fraction. Find the value of 𝑥.

b) A group of boys and girls sat for a test. Exactly 2

3 of the boys and exactly

3

4

of the girls passed the test. If an equal number of boys and girls passed the

test, what fraction of the entire group passed the test?

c) Solve for 𝑚:

𝑚 + 1

3−

1

2=

2𝑚 + 1

4


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Year 7 Mathematics


63. Solve the equation for 𝑥

2(𝑥 + 1) − 3(2𝑥 − 3) = 31

64.

a) If 𝑥 is an odd number, write down the next two consecutive odd numbers

in terms of 𝑥. If the sum of the first and third exceeds the second by 31,

find the three odd numbers.

b) If I add 4 to a number and multiply the result by 3, I get 8 less than 5 times

the number 1 started with. Write down an equation to solve this problem

and then find the number.


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Year 7 Mathematics


65. Solve the following equation for 𝑤:

𝑤

4÷ (2

1

3) =

3

4

66.

a) If 𝑔(𝑥) = −2𝑥 + 3, find 𝑔(𝑚 − 1)

b) A collection of coins consisting of 10-cent, 20-cent and 50-cent pieces has

a value of $8.00. The number of 20-cent pieces is twice the number of 10-

cent pieces and the number of 50-cent pieces is 2 less than twice the

number of 10-cent pieces.

(i) Express this information as an equation.

(ii) Find how many coins of each kind are there by solving the equation?


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Year 7 Mathematics


67. The lengths of the sides of a triangle are 𝑔 + 1, 7 − 𝑔, and 4𝑔 − 2. How many

different values of 𝑔 are there such that the triangle is isosceles; justify your

answers?

68.

a) Solve for 𝑤: 𝑊

5÷ (2

3

4) =

6

11

b) Solve for 𝑥: 2(4𝑥 − 1) − (1 − 𝑥) = 3𝑥

c) The sum of two numbers is 16. Let the first number be 𝑥.

If 2

3 of the first number plus

3

4 of the second is equal to 11, write an

equation in terms of 𝑥.

Hence find the numbers.


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Year 7 Mathematics


69.

a) The size of angle 𝐴 in ∆𝐴𝐵𝐶 is 20% more than the size of angle 𝐶. Angle 𝐵

is 40% more than angle 𝐶. Find the angle of 𝐴.

b) Solve the equation for 𝑥:

3(2𝑥 − 4) = 5(1 − 3𝑥)

c) Sue and Tom both work at a shop. Sue does not work every ninth day.

Tom does not work every fifth day. Sue is not working on the 1st

November and Tom does not work on 2nd November. What is the date of

the first day they have off work together? Justify your answer.


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Year 7 Mathematics


d) Box A contains 33 green and 4 red balls. Box B contains 12 green and 5

red balls. A number of green balls is taken out of Box A and placed in Box

B to make the ratio of green to red balls in Box A the same as that in Box

B. Using an algebraic equation, find the number of green balls transferred.

70.

a) Solve

−3(𝑥 + 4)

7−

5(1 − 3𝑥)

3= 3𝑥 − 4

b) A rectangle has dimensions (3𝑥 + 1)cm by (𝑥 − 2)cm, if its perimeter is

70cm find the area in cm^2.


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Year 7 Mathematics


71.

a) Solve for 𝑥:

5(3𝑥 − 2) − 3(3 − 4𝑥) = 2𝑥 + 1

b) There are two consecutive odd numbers. Five more than twice the larger

one is triple the smaller number. Find the numbers.

c) James is currently 3 less than twice Andrew’s age. 4 years from now,

Matthew will be 2 more than twice Andrew’s age. 5 years ago, Matthew

was three times Andrew’s age. Let Andrew's age be 𝑥.

(i) Write Matthew’s current age in terms of 𝑥.

(ii) Determine Andrew’s current age.

(iii) How old is James?


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Year 7 Mathematics


Very Hard

72. Two numbers are said to be congruent modulo 6 is they have the same

remainder when divided by 6. For example, 17 ≡ 11 (mod 6) read as ‘seventeen

is congruent to eleven, modulo 6’ and 11 ≡ 5 (mod 6) read as ‘eleven is

congruent to five, modulo 6’ because seventeen and eleven both leave a

remainder of 5 when divided by 6. So both 17 and 11 can be reduced to 5 in

modulo 6.

Solve the equation 𝑥 + 3 ≡ 2 (mod 5)

73. A standard 12 hour clock has an hour hand, a minute hand and a second hand.

The hands move continuously around the clock to represent the time.

a) How many degrees will the hour hand move between 12 o'clock and 2

o'clock?


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Year 7 Mathematics


b) At exactly 𝑥 minutes after 2 o'clock, where 𝑥 < 60, the hour hand and

minute hand are pointing in exactly the same direction.

(i) Using the information above, form an equation involving 𝑥 and hence

find the exact value of 𝑥. Leave your answer as a mixed numeral.

(ii) What is the size of the angle between the second hand and minute

hand at this time? Round your answer to the nearest degree.

74. A small tray which is 4

5 full of water is suspended directly above a larger tray

which is 9

10 full. More water is then poured into the small tray until it overflows

into the larger tray below. The larger tray has 6 times the capacity of the small

tray.

What fraction of the original amount of water in the small tray must be poured

into the small tray to exactly fill the larger tray?


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Year 7 Mathematics


75. Michael wants to build a deck for his house. The diagram below shows his plans

for how the deck will attach to his house and the expected floor area.

Michael is unsure what lengths to make 𝑎, 𝑏, and 𝑐, but wants to spend exactly

$3400 on the deck. Given that the cost of the deck is $25 per square metre, find

the correct value for 𝑏. You may assume that all angles will be right angles.

𝑏

12 m

2𝑎

𝑎

𝑐

Deck

House

12 m


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Year 7 Mathematics


76. The sum of each row is written at the end of the row. The sum of each column is

written at the bottom of the column. Find the value of 𝑥 in the diagram below:

𝑚 𝑛 𝑝 𝑝 53

𝑚 𝑝 𝑝 𝑝 55

𝑝 𝑛 𝑛 𝑛 50

𝑟 𝑚 𝑚 𝑟 56

𝑥 51 53 55

Documents

Year 7 Mathematics Exam Booklet: Equations