Mathematical Reasoning

NSW Opportunity Class Placement Test · 35 questions · 40 minutes · no calculator

0 / 35 answered

Question 1

What number is missing from this equation?

54 - ? = 19 + 8

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Correct: A

First work out the right side: 19 + 8 = 27. So 54 − the missing number = 27, which means the missing number = 54 − 27 = 27.

Question 3

A film starts at 3:50 pm and finishes at 6:15 pm. How long is the film?

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Correct: E

3:50 pm to 4:50 pm is 1 hour. 4:50 pm to 6:15 pm is 1 hour 25 minutes. Total = 2 hours 25 minutes.

Question 5

Look at the number pattern

1,\,4,\,9,\,16,\,25,\,\ldots

What is the 7th term?

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Correct: C

These are the square numbers: 1×1, 2×2, 3×3, 4×4, 5×5. So the 7th term is 7×7 = 49.

Question 6

The bar chart shows the favourite fruit of 60 students. How many more students chose bananas than oranges?

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Correct: B

Bananas = 18, oranges = 9. Difference = 18 − 9 = 9.

Question 7

Liam buys 3 books at $7.50 each and 2 pencils at $1.20 each. What is the total cost?

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Correct: D

Books: 3 × $7.50 = $22.50. Pencils: 2 × $1.20 = $2.40. Total = $22.50 + $2.40 = $24.90.

Question 8

A jar holds exactly $4.65 in coins. There are six 50-cent coins, eight 20-cent coins, and the rest are 5-cent coins. How many 5-cent coins are there?

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Correct: A

Six 50c = $3.00 and eight 20c = $1.60, together $4.60. What is left is $4.65 − $4.60 = $0.05, which is exactly 1 five-cent coin.

Question 9

What number comes next in the pattern

3,\,8,\,15,\,24,\,35,\,\ldots

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Correct: D

The gaps are 5, 7, 9, 11 (the odd numbers). The next gap is 13, so the next number is 35 + 13 = 48.

Question 10

Which of the following statements is true?

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Correct: A

Every square is also a rectangle (four right angles, opposite sides equal). The other statements are all false.

Question 11

How many millilitres (mL) are there in 2.5 litres (L)?

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Correct: A

1 L = 1,000 mL, so 2.5 × 1,000 = 2,500 mL.

Question 12

A bag has 12 counters numbered 1 to 12. Lily takes one at random. Statement 1: she is more likely to take an even number than an odd number.Statement 2: she is more likely to take a number greater than 6 than less than 6.Statement 3: she is equally likely to take the number 3 or the number 11.Which statements are correct?

Show worked solution

Correct: E

Check each one.

Statement 1: 6 even and 6 odd, so they are equally likely. Not correct.
Statement 2: greater than 6 is {7,8,9,10,11,12} = 6 numbers; less than 6 is {1,2,3,4,5} = 5 numbers. Greater is more likely. Correct.
Statement 3: each number appears once, so 3 and 11 are equally likely. Correct.

So Statements 2 and 3.

Question 13

A rectangular floor is 6 metres long and 4 metres wide. What is its area?

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Correct: B

Area = length × width = 6 × 4 = 24 m².

Question 14

At 6:00 am the temperature was 4 °C. By midday it had risen to 24 °C. By how many degrees did it increase?

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Correct: C

Increase = 24 − 4 = 20 °C.

Question 15

What is the area of the shape below?

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Correct: C

Treat it as an 8 × 5 = 40 cm² rectangle with a 3 × 2 = 6 cm² corner removed. Area = 40 − 6 = 34 cm².

Question 16

In the pyramid below, each number is the sum of the two directly below it. What is

Q \times R

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Correct: B

Q + 5 = 13, so Q = 8. And 5 + R = 17, so R = 12. So $Q \times R = 8 \times 12 =$ 96.

Question 17

Which capital letter has exactly two lines of symmetry?

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Correct: E

H has two lines of symmetry: one down the middle and one across the middle. E has one, F and Z have none, and M has one.

Question 18

On a map where each step is 1 metre, Mia walks 3 steps north, then 2 steps east, then 5 steps south, then 2 steps west. How far is she from her starting point?

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Correct: A

The 2 east and 2 west cancel out, so she ends up directly below the start. She went 3 north then 5 south, which is 2 south overall. So she is 2 metres away.

Question 19

Today is Tuesday, 1st March. What day of the week will it be on 20th March?

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Correct: E

From the 1st to the 20th is 19 days. 19 = 2 weeks (14 days) + 5 days. Five days after Tuesday is Sunday (Wed, Thu, Fri, Sat, Sun).

Question 20

Lily thinks of a number, doubles it, then adds 7. The result is 25. What was her original number?

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Correct: C

Work backwards from 25: take off 7 to get 18, then halve it to get 9.

Question 21

The pie chart shows how a class spent one school day. Which statement is true?

AMore time was spent on Lunch than on Maths.BReading, Maths and Sport took equal time.CLess time was spent on Reading than on Other.DSport took up more than half of the day.EOther took up the largest share of the day.

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Correct: B

Reading, Maths and Sport are each a quarter of the circle, so they are equal. The other statements are all false.

Question 22

A four-sided shape has all four sides the same length, but none of its angles are right angles. What is the best name for it?

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Correct: A

Four equal sides with no right angles is a rhombus. A square also has equal sides but its angles are right angles.

Question 23

Karim buys a notebook for $2.20 and pays with a $10 note. His change includes nine 50-cent coins, and the rest is made of 20-cent and 5-cent coins only. What is the greatest possible number of 20-cent coins?

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Correct: D

Change = $10.00 − $2.20 = $7.80. Nine 50c coins = $4.50, leaving $3.30. To use the most 20c coins, $3.30 holds at most 16 of them ($3.20), with the last 10c made of two 5c coins. So 16.

Question 24

Tom has 30 marbles.

\tfrac{1}{3}

are red and

\tfrac{1}{5}

are blue. The rest are green. How many are green?

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Correct: C

Red = $\tfrac{1}{3}$ of 30 = 10. Blue = $\tfrac{1}{5}$ of 30 = 6. Green = 30 − 10 − 6 = 14.

Question 25

The diagram shows the net of a cube. When it is folded, which face is directly opposite face A?

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Correct: E

In the row of four faces (B, A, C, F), folding makes A the front and F the back, so A is opposite F. D and E fold to the top and bottom, opposite each other.

Question 26

Three friends share 30 lollies. Anna has 4 more than Bea, and Cara has 2 more than Bea. How many does Bea have?

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Correct: B

Bea + (Bea + 4) + (Bea + 2) = 30. That is 3 lots of Bea plus 6 = 30, so 3 lots of Bea = 24 and Bea has 8.

Question 27

The scales below are not balanced. Anna is considering three methods:Method 1: add 5 kg to the right pan.Method 2: add 4 kg to the left pan and 9 kg to the right pan.Method 3: add 7 kg to the left pan and 11 kg to the right pan.Which methods would balance the scales?

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Correct: D

The left pan has 14 kg and the right has 9 kg. Check each method:

Method 1 (add 5 kg right): right = 9 + 5 = 14, so it balances.
Method 2 (add 4 kg left, 9 kg right): left = 18, right = 18, so it balances.
Method 3 (add 7 kg left, 11 kg right): left = 21, right = 20, so it does not balance.

So Methods 1 and 2.

Question 28

Two identical cubes are placed next to each other so they share one whole face. How many of the original cube faces can be seen from all sides combined?

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Correct: C

Two cubes have 6 + 6 = 12 faces in total. When they share a face, one face on each cube is hidden, so 2 faces are hidden. Visible = 12 − 2 = 10.

Question 29

A square sheet of paper is folded in half, then folded in half again at right angles. A single hole is punched through all the layers. When unfolded, how many holes are there?

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Correct: D

Each fold doubles the layers: 1 fold = 2 layers, 2 folds = 4 layers. One hole through 4 layers makes 4 holes.

Question 30

The table shows the number of cars sold each day last week.

Day	Mon	Tue	Wed	Thu	Fri	Sat	Sun
Cars sold	8	5	9	4	7	12	6

Claim 1: more than 50 cars were sold during the week.Claim 2: twice as many cars were sold on Saturday as on Thursday.Claim 3: Wednesday had the highest sales of any day.Which claims are correct?

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Correct: A

Check each claim.

Claim 1: total = 8 + 5 + 9 + 4 + 7 + 12 + 6 = 51, which is more than 50. Correct.
Claim 2: Saturday 12, Thursday 4. 12 is 3 times 4, not twice. Not correct.
Claim 3: Saturday (12) was the highest, not Wednesday (9). Not correct.

So Claim 1 only.

Question 31

In a sequence, each number is 3 less than triple the number before it. The third number is 24. What is the first number?

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Correct: B

Work backwards. Third = triple the second, minus 3 = 24, so triple the second = 27 and the second = 9. Second = triple the first, minus 3 = 9, so triple the first = 12 and the first = 4.

Question 32

Using the digits 3, 5 and 8 each exactly once, what is the largest possible three-digit number minus the smallest?

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Correct: D

Largest = 853, smallest = 358. 853 − 358 = 495.

Question 33

How many different three-digit numbers can be made using each of the digits 2, 5 and 7 exactly once?

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Correct: E

3 choices for the first digit, then 2 for the second, then 1 for the last: 3 × 2 × 1 = 6 (257, 275, 527, 572, 725, 752).

Question 34

In a class of 30, 18 play soccer and 14 play netball. 8 play both. How many play neither sport?

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Correct: B

Students playing at least one sport = 18 + 14 − 8 = 24 (the 8 who play both were counted twice). Neither = 30 − 24 = 6.

Question 35

A ruler has marks only at 0 cm, 2 cm, 5 cm and 11 cm. By joining two marks, which length CANNOT be drawn?

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Correct: E

The lengths you can make are the gaps between marks: 2, 5, 11, 3 (5−2), 9 (11−2) and 6 (11−5). So 2, 3, 5, 6, 9 and 11 are possible. 7 cm cannot be made.