Mathematical Reasoning

NSW Selective High School Placement Test · 35 questions · 40 minutes · no calculator

0 / 35 answered

Question 1

Which of the following decimals is the largest?

Show worked solution

Correct: E

Line up the decimal places: 0.450, 0.405, 0.400, 0.360, 0.350. The largest is 0.45.

Question 2

Sarah eats

\tfrac{2}{5}

of a pizza and Tom eats

\tfrac{1}{4}

of the same pizza. What fraction is left?

Show worked solution

Correct: B

Change both to twentieths: $\tfrac{2}{5}=\tfrac{8}{20}$ and $\tfrac{1}{4}=\tfrac{5}{20}$ . Together they eat $\tfrac{13}{20}$ , so $\tfrac{7}{20}$ is left.

Question 3

A jumper that normally costs $80 is on sale at 25% off. What is the sale price?

Show worked solution

Correct: C

25% of $80 = $20 discount. Sale price = $80 − $20 = $60.

Question 4

In the triangle below, two angles are 55° and 75°. What is the third angle?

Show worked solution

Correct: C

Angles in a triangle add up to 180°. Third angle = 180 − 55 − 75 = 50°.

Question 5

Three rays meet at a point on a straight line, as shown. What is the value of

x

Show worked solution

Correct: C

Angles on a straight line add up to 180°. So x = 180 − 35 − 75 = 70°.

Question 6

Three vertices of a square are shown on the grid. What are the coordinates of the fourth vertex?

Show worked solution

Correct: B

The vertices (1, 1), (5, 1) and (5, 5) are three corners of a square of side 4. The fourth is directly above (1, 1): (1, 5).

Question 7

Marbles are shared between Maya and Noah in the ratio 3 : 5. Noah receives 30. How many does Maya receive?

Show worked solution

Correct: D

Noah’s 5 parts = 30, so 1 part = 6. Maya gets 3 × 6 = 18.

Question 8

A bag has 4 red, 3 blue and 5 green marbles. One is drawn at random. What is the probability it is not blue?

Show worked solution

Correct: E

Total = 12, not blue = 9. $P=\tfrac{9}{12}=\tfrac{3}{4}$ .

Question 9

A student's scores are 75, 82, 91, 68 and 84. What is the mean?

Show worked solution

Correct: C

Sum = 400. Mean = 400 ÷ 5 = 80.

Question 10

A train leaves at 10:45 am and travels for 2 hours 35 minutes. At what time does it arrive?

Show worked solution

Correct: D

10:45 am + 2 h = 12:45 pm, + 35 min = 1:20 pm.

Question 11

Liam buys 3 notebooks at $4.50 and 2 pens at $1.75. He pays with $20. How much change?

Show worked solution

Correct: C

Cost = $13.50 + $3.50 = $17.00. Change = $20 − $17 = $3.00.

Question 12

What is the next number in the sequence

2,\,6,\,12,\,20,\,30,\,\ldots

Show worked solution

Correct: D

Differences are 4, 6, 8, 10, … (rising by 2). Next difference 12, so next term = 30 + 12 = 42.

Question 13

4 \times \square + 12 = 60

, what is the value of

\square

Show worked solution

Correct: C

$4\times\square = 48$ , so $\square = 48\div 4 = 12$ .

Question 14

A cyclist rides 18 km in 45 minutes at a steady speed. How long to ride 30 km at the same speed?

Show worked solution

Correct: C

18 km in 45 min = 6 km per 15 min. 30 km = 5 × 15 min = 75 min = 1 h 15 min.

Question 15

What is the perimeter of the shape below?

Show worked solution

Correct: D

Add up all the sides going around the shape: 10 + 3 + 4 + 3 + 6 + 6 = 32 cm.

Question 16

What is the area of the shape below?

Show worked solution

Correct: D

A 12 × 8 = 96 cm² rectangle with a 5 × 4 = 20 cm² corner removed. Area = 96 − 20 = 76 cm².

Question 17

A pool is 25 m long, 10 m wide and 2 m deep. How many litres when full? (1 m³ = 1000 L)

Show worked solution

Correct: D

Volume = 25 × 10 × 2 = 500 m³ = 500 000 L.

Question 18

The bar chart shows ice-cream cones sold by flavour. How many more chocolate than strawberry?

Show worked solution

Correct: C

Chocolate = 20, strawberry = 10. Difference = 10.

Question 19

The pie chart shows the favourite subjects of 120 students. How many chose Maths?

Show worked solution

Correct: C

30% of 120 = 36 students.

Question 20

A function machine multiplies a number by 3, then subtracts 7. If the output is 26, what was the input?

Show worked solution

Correct: C

Work backwards from 26: 26 + 7 = 33, then 33 ÷ 3 = 11.

Question 21

Emma doubles a number, adds 5, then divides by 3. The result is 7. What was her number?

Show worked solution

Correct: C

Work backwards from 7: 7 × 3 = 21, 21 − 5 = 16, 16 ÷ 2 = 8.

Question 22

The mean of five numbers is 14. When one more number is added, the mean becomes 15. What was added?

Show worked solution

Correct: C

Old total = 70. New total = 6 × 15 = 90. Added = 90 − 70 = 20.

Question 23

A rectangle is 3 times as long as it is wide. Its perimeter is 48 cm. What is its area?

Show worked solution

Correct: B

The length plus the width is half the perimeter: 48 ÷ 2 = 24 cm. The width is 1 part and the length is 3 parts, so 4 parts = 24 cm and 1 part = 6 cm. Width = 6, length = 18, so area = 6 × 18 = 108 cm².

Question 24

The diagram shows the net of a closed rectangular box. What is its volume when folded?

Show worked solution

Correct: D

The net folds into a 5 × 4 × 3 box. Volume = 60 cm³.

Question 25

A solid is made by joining identical small cubes, as shown. How many small cubes were used?

Show worked solution

Correct: B

A 2 × 2 base = 4 cubes, plus 1 on top = 5 cubes.

Question 26

A school has 100 students; 60 play a sport. Of those, 35 are boys. There are 55 boys in total. How many girls do not play a sport?

Show worked solution

Correct: C

Boys not playing = 55 − 35 = 20. Girls = 45; girls playing = 60 − 35 = 25; girls not playing = 45 − 25 = 20.

Question 27

Four of a student's five scores are 75, 82, 91 and 68. The mean of all five is 80. What is the fifth?

Show worked solution

Correct: C

Total = 5 × 80 = 400. Known = 316. Fifth = 400 − 316 = 84.

Question 28

The numbers

2,\,6,\,12,\,20,\,30,\,\ldots

follow a pattern. What is the 10th number?

Show worked solution

Correct: C

Each number is a counting number times the next one: 1×2 = 2, 2×3 = 6, 3×4 = 12, … So the 10th number is 10 × 11 = 110.

Question 29

A class of 32 has girls : boys = 5 : 3. 25% of the girls and

\tfrac{1}{3}

of the boys play netball. How many play netball?

Show worked solution

Correct: C

20 girls, 12 boys. 25% of 20 = 5; $\tfrac{1}{3}$ of 12 = 4. Total = 9.

Question 30

Liam buys 3 notebooks at $4.50 and 2 pens at $1.75. There's 10% off the whole bill when you spend $15+. What does he pay?

Show worked solution

Correct: B

Subtotal = $17.00. Since $17 ≥ $15, 10% off: $17 × 0.9 = $15.30.

Question 31

A function machine multiplies by 4, subtracts 3, then divides by 5. If the output is 9, what was the input?

Show worked solution

Correct: C

Work backwards from 9: 9 × 5 = 45; 45 + 3 = 48; 48 ÷ 4 = 12.

Question 32

Cubes are stacked as a pyramid: a 4×4 layer, then 3×3 centred, then 2×2, then a single cube. How many cubes in total?

Show worked solution

Correct: D

Count each layer and add them up: 4×4 = 16, 3×3 = 9, 2×2 = 4, and 1 cube on top. 16 + 9 + 4 + 1 = 30 cubes.

Question 33

A book costs $50. It's increased by 20%, then by a further 10%, then decreased by 25%. What is the final price?

Show worked solution

Correct: A

Do one change at a time:

+20% of $50 is $10, giving $60.
+10% of $60 is $6, giving $66.
−25% of $66 is $16.50, giving $49.50.

Question 34

Sam reads

\tfrac{1}{3}

of a book on Monday, then

\tfrac{1}{4}

of the remainder on Tuesday. He has 120 pages left. How many pages in total?

Show worked solution

Correct: D

Think of the book in sixths. Monday: Sam reads $\tfrac{1}{3}=\tfrac{2}{6}$ , leaving $\tfrac{4}{6}$ . Tuesday: he reads $\tfrac{1}{4}$ of those, which is $\tfrac{1}{6}$ , leaving $\tfrac{3}{6}$ .

The 120 pages left are $\tfrac{3}{6}$ of the book, which is half. So the whole book is 240 pages.

Question 35

A piggy bank has only 5c, 10c, 20c and 50c coins. In total there are 25 coins worth $4.70. There are 3 times as many 10c coins as 5c coins, and the same number of 50c coins as 5c coins. How many 20c coins are there?

Show worked solution

Correct: D

The 5c, 10c and 50c coins all depend on how many 5c coins there are. Try 2 five-cent coins: then there are 6 ten-cent coins (3 times as many) and 2 fifty-cent coins (the same number).

That’s 2 + 6 + 2 = 10 coins, so the other 25 − 10 = 15 must be 20c coins.
Check the money: 2×5 + 6×10 + 15×20 + 2×50 = 10 + 60 + 300 + 100 = 470c = $4.70.