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ICAS Mathematics study materials
ICAS Mathematics study resources

ICAS Mathematics practice questions

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Question 1Number and Arithmetic

In the number 47 382, what is the value of the digit 7?

  1. 70
  2. 70 000
  3. 7000 Correct answer.
  4. 700

Explanation

The digit 7 sits in the thousands column of 47 382, so its value is 7 × 1000 = 7000. The columns to its right are hundreds, tens and ones.

Question 2Number and Arithmetic

Round the number 6 847 to the nearest hundred.

  1. 6 850
  2. 6 800 Correct answer.
  3. 6 900
  4. 7 000

Explanation

The hundreds digit is 8 and the digit deciding the rounding is the 4 in the tens column. Since 4 is less than 5, the number rounds down to 6 800.

Question 3Number and Arithmetic

What is the value of 8 + 3 × 4?

  1. 24
  2. 32
  3. 44
  4. 20 Correct answer.

Explanation

Multiplication is carried out before addition, so first work out 3 × 4 = 12. Then 8 + 12 = 20.

Question 4Number and Arithmetic

Which of these fractions has the same value as three fifths?

  1. 6/15
  2. 9/20
  3. 5/3
  4. 9/15 Correct answer.

Explanation

Multiplying the numerator and denominator of 3/5 by the same number 3 gives 9/15, so the two fractions are equivalent.

Question 5Number and Arithmetic

Work out the value of -7 + 12.

  1. 19
  2. 5 Correct answer.
  3. -19
  4. -5

Explanation

Adding a positive number to a negative number moves to the right along the number line. Starting at -7 and moving 12 places right lands on 5.

Question 6Number and Arithmetic

What is 10% of 250?

  1. 50
  2. 2.5
  3. 250
  4. 25 Correct answer.

Explanation

Ten per cent means one tenth of the amount. Dividing 250 by 10 gives 25.

Question 7Number and Arithmetic

Which of these numbers is a factor of 36?

  1. 9 Correct answer.
  2. 8
  3. 5
  4. 7

Explanation

A factor divides the number exactly with no remainder. Since 36 ÷ 9 = 4, the number 9 is a factor of 36.

Question 8Number and Arithmetic

Which of these decimals has the greatest value?

  1. 0.509
  2. 0.6 Correct answer.
  3. 0.59
  4. 0.5

Explanation

Writing every decimal to three decimal places gives 0.509, 0.590, 0.500 and 0.600. Comparing the hundredths column shows 0.600 to be the largest.

Question 9Number and Arithmetic

What is the value of 20 - 3 × (4 + 2)?

  1. -2
  2. 2 Correct answer.
  3. 102
  4. 18

Explanation

Brackets are evaluated first: 4 + 2 = 6. Then the multiplication gives 3 × 6 = 18, and finally 20 - 18 = 2.

Question 10Number and Arithmetic

A jacket originally costing $80 is reduced by 15%. What is the sale price?

  1. 65
  2. 68 Correct answer.
  3. 72
  4. 92

Explanation

Fifteen per cent of $80 is 0.15 × 80 = $12. Subtracting the discount from the original price gives 80 - 12 = $68.

Question 11Number and Arithmetic

A sum of $45 is shared in the ratio 2 : 3. What is the larger share?

  1. 22.50
  2. 30
  3. 18
  4. 27 Correct answer.

Explanation

The ratio has 2 + 3 = 5 equal parts, so one part is $45 ÷ 5 = $9. The larger share is 3 parts, that is 3 × $9 = $27.

Question 12Number and Arithmetic

What is the lowest common multiple of 6 and 8?

  1. 14
  2. 24 Correct answer.
  3. 48
  4. 2

Explanation

Listing multiples of 6 gives 6, 12, 18, 24, and of 8 gives 8, 16, 24. The smallest number appearing in both lists is 24.

Question 13Number and Arithmetic

Write the number 0.00042 in standard form.

  1. 4.2 × 10⁻⁴ Correct answer.
  2. 42 × 10⁻⁶
  3. 4.2 × 10⁴
  4. 4.2 × 10⁻³

Explanation

Standard form is a × 10ⁿ where 1 ≤ a < 10. The decimal point must move 4 places to the right to turn 0.00042 into 4.2, so the index is -4.

Question 14Number and Arithmetic

Two fifths of a number is 18. What is the number?

  1. 7.2
  2. 36
  3. 90
  4. 45 Correct answer.

Explanation

If 2/5 of the number is 18, then 1/5 of it is 18 ÷ 2 = 9. The whole number is 5 × 9 = 45.

Question 15Number and Arithmetic

Work out 0.4 × 0.03.

  1. 0.0012
  2. 1.2
  3. 0.012 Correct answer.
  4. 0.12

Explanation

First multiply 4 × 3 = 12. The two factors together have one decimal place plus two decimal places, so the answer needs three decimal places: 0.012.

Question 16Number and Arithmetic

Simplify 5⁶ ÷ 5², leaving your answer as a power of 5.

  1. 5⁸
  2. 5⁴ Correct answer.
  3. 25⁴
  4. 5³

Explanation

When dividing powers with the same base, subtract the indices. Here 6 - 2 = 4, so the answer is 5⁴.

Question 17Number and Arithmetic

How much simple interest is earned on $500 invested at 4% per year for 3 years?

  1. 560
  2. 120
  3. 20
  4. 60 Correct answer.

Explanation

Simple interest is found using P × r × t, so 500 × 0.04 × 3 = $60. The interest is the extra money earned, not the total amount in the account.

Question 18Number and Arithmetic

The temperature falls from 5 °C to -8 °C. By how many degrees does the temperature fall?

  1. 13 Correct answer.
  2. -3
  3. 3
  4. -13

Explanation

The fall is found by subtracting the final temperature from the starting temperature: 5 - (-8) = 5 + 8 = 13. A fall is a positive change of 13 degrees.

Question 19Number and Arithmetic

Which of these numbers is a prime number?

  1. 27
  2. 33
  3. 21
  4. 29 Correct answer.

Explanation

A prime number has exactly two factors, 1 and itself. The number 29 is divisible only by 1 and 29.

Question 20Number and Arithmetic

By rounding each number to the nearest whole number, estimate 39.6 × 4.9.

  1. 1 960
  2. 160
  3. 240
  4. 200 Correct answer.

Explanation

39.6 rounds up to 40 because the tenths digit is 6, and 4.9 rounds up to 5. The estimate is therefore 40 × 5 = 200.

Question 21Number and Arithmetic

After a 20% discount a bicycle costs $240. What was the original price?

  1. 288
  2. 320
  3. 300 Correct answer.
  4. 260

Explanation

A 20% discount means the sale price is 80% of the original price. Dividing 240 by 0.8 gives an original price of $300.

Question 22Number and Arithmetic

Work out (3 × 10⁵) × (4 × 10⁻²), giving your answer in standard form.

  1. 1.2 × 10⁴ Correct answer.
  2. 1.2 × 10⁷
  3. 1.2 × 10³
  4. 12 × 10⁴

Explanation

Multiply the number parts: 3 × 4 = 12. Add the indices: 5 + (-2) = 3, giving 12 × 10³, which is 1.2 × 10⁴ in standard form.

Question 23Number and Arithmetic

A car worth $12 000 loses 25% of its value each year. What is it worth after two years?

  1. 5 062.50
  2. 6 750 Correct answer.
  3. 9 000
  4. 6 000

Explanation

Keeping 75% of the value each year means multiplying by 0.75 twice. So 12 000 × 0.75 × 0.75 = 12 000 × 0.5625 = $6 750.

Question 24Number and Arithmetic

What is the highest common factor of 84 and 120?

  1. 24
  2. 12 Correct answer.
  3. 6
  4. 840

Explanation

Writing prime factors gives 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5. Taking the lowest power of each shared prime gives 2² × 3 = 12.

Question 25Number and Arithmetic

What is the value of 2⁻³?

  1. -8
  2. 0.125 Correct answer.
  3. -6
  4. 8

Explanation

A negative index means taking the reciprocal of the positive power. So 2⁻³ = 1/2³ = 1/8 = 0.125.

Question 26Number and Arithmetic

Work out 5 − (−8).

  1. -3
  2. 3
  3. -13
  4. 13 Correct answer.

Explanation

Subtracting a negative number is the same as adding its positive value, so 5 − (−8) = 5 + 8. Starting at 5 and moving 8 places to the right on the number line lands on 13.

Question 27Number and Arithmetic

Express 18/24 in its simplest form.

  1. 3/4 Correct answer.
  2. 2/3
  3. 6/8
  4. 9/12

Explanation

The highest common factor of 18 and 24 is 6. Dividing the numerator and the denominator by 6 gives 18/24 = 3/4, and 3 and 4 share no common factor other than 1.

Question 28Number and Arithmetic

Work out 3.6 × 100.

  1. 36
  2. 0.036
  3. 360 Correct answer.
  4. 3.6

Explanation

Multiplying by 100 shifts every digit two places in place value, which moves the decimal point two places to the right. So 3.6 becomes 36 after one place and 360 after two places.

Question 29Number and Arithmetic

Write 7/20 as a decimal.

  1. 0.72
  2. 0.27
  3. 0.14
  4. 0.35 Correct answer.

Explanation

Multiplying the numerator and denominator by 5 gives 7/20 = 35/100. A denominator of 100 means the numerator counts hundredths, so the decimal is 0.35.

Question 30Number and Arithmetic

Write 2^3 × 2^4 as a single power of 2.

  1. 2^1
  2. 4^7
  3. 2^7 Correct answer.
  4. 2^12

Explanation

When powers with the same base are multiplied, the indices are added: 3 + 4 = 7. The base stays as 2, so the result is 2^7.

Question 31Number and Arithmetic

Round 6.847 to one decimal place.

  1. 7.0
  2. 6.84
  3. 6.9
  4. 6.8 Correct answer.

Explanation

The first decimal place is the tenths digit 8. The next digit is 4, which is less than 5, so the 8 stays as it is and the digits after it are dropped, giving 6.8.

Question 32Number and Arithmetic

Which of these numbers is the smallest?

  1. -1
  2. -9 Correct answer.
  3. -4
  4. 2

Explanation

On a number line, values further to the left are smaller. Of the four numbers given, −9 lies furthest to the left, so it is the smallest.

Question 33Number and Arithmetic

Work out 1/4 + 1/6.

  1. 1/24
  2. 2/10
  3. 5/24
  4. 5/12 Correct answer.

Explanation

The lowest common denominator of 4 and 6 is 12, so rewrite each fraction as twelfths: 3/12 + 2/12 = 5/12. The numerator and denominator of 5/12 share no common factor.

Question 34Number and Arithmetic

Work out 5/8 ÷ 1/4.

  1. 10
  2. 5/32
  3. 2/5
  4. 5/2 Correct answer.

Explanation

Dividing by a fraction is the same as multiplying by its reciprocal, so 5/8 ÷ 1/4 = 5/8 × 4/1. This gives 20/8, which simplifies to 5/2.

Question 35Number and Arithmetic

Increase 60 by 15%.

  1. 69 Correct answer.
  2. 72
  3. 66
  4. 75

Explanation

First find 15% of 60 by working out 0.15 × 60 = 9. Adding this increase to the original amount gives 60 + 9 = 69.

Question 36Number and Arithmetic

Write the ratio 18 : 24 : 30 in its simplest form.

  1. 2:3:4
  2. 3:4:5 Correct answer.
  3. 6:8:10
  4. 9:12:15

Explanation

The highest common factor of 18, 24 and 30 is 6. Dividing all three parts by 6 gives 3 : 4 : 5, and these three numbers share no common factor other than 1.

Question 37Number and Arithmetic

A shop sells rice in 5 kg bags for $11.50. What is the cost per kilogram?

  1. $2.30 Correct answer.
  2. $2.10
  3. $2.50
  4. $2.40

Explanation

The unit price is the total cost divided by the quantity, so work out 11.50 ÷ 5. Dividing 11.5 by 5 gives 2.3, so the rice costs $2.30 per kilogram.

Question 38Number and Arithmetic

Write 6 400 000 in standard form.

  1. 0.64 × 10^7
  2. 64 × 10^5
  3. 6.4 × 10^5
  4. 6.4 × 10^6 Correct answer.

Explanation

Standard form is written as A × 10^n where 1 ≤ A < 10. Moving the decimal point six places to the left in 6 400 000 gives 6.4, so the number is 6.4 × 10^6.

Question 39Number and Arithmetic

Work out 5^0 × 2^3.

  1. 0
  2. 8 Correct answer.
  3. 40
  4. 1

Explanation

Any non-zero number raised to the power 0 equals 1, so 5^0 = 1. Since 2^3 = 8, the product is 1 × 8 = 8.

Question 40Number and Arithmetic

Work out 4.8 ÷ 0.6.

  1. 0.8
  2. 8 Correct answer.
  3. 80
  4. 0.08

Explanation

Multiply both numbers by 10 to remove the decimal from the divisor: 48 ÷ 6. Scaling the dividend and divisor by the same factor leaves the quotient unchanged, and 48 ÷ 6 = 8.

Question 41Number and Arithmetic

A car travels 180 km in 2.5 hours at a constant speed. What is its speed?

  1. 45 km/h
  2. 90 km/h
  3. 75 km/h
  4. 72 km/h Correct answer.

Explanation

Speed is distance divided by time, so work out 180 ÷ 2.5. Dividing both numbers by 2.5 gives 72, so the speed is 72 km/h.

Question 42Number and Arithmetic

Which number is equal to 2^2 × 3 × 5?

  1. 90
  2. 45
  3. 30
  4. 60 Correct answer.

Explanation

Evaluate the power first: 2^2 = 4. Then 4 × 3 = 12 and 12 × 5 = 60, so the expression equals 60.

Question 43Number and Arithmetic

Write 0.004562 correct to 2 significant figures.

  1. 0.005
  2. 0.0046 Correct answer.
  3. 0.00456
  4. 0.0045

Explanation

The leading zeros are not significant, so the first two significant figures are 4 and 5. The next digit is 6, which is 5 or more, so the 5 rounds up to 6 and the answer is 0.0046.

Question 44Number and Arithmetic

Work out (−5) × (−3) − 7.

  1. -22
  2. 22
  3. 8 Correct answer.
  4. -8

Explanation

The product of two negative numbers is positive, so (−5) × (−3) = 15. Subtracting 7 then gives 15 − 7 = 8.

Question 45Number and Arithmetic

Sam spends 1/4 of his $60 on a book and then 1/3 of the money that remains on lunch. How much money does he have left?

  1. $22.50
  2. $30 Correct answer.
  3. $45
  4. $25

Explanation

The book costs 1/4 × 60 = $15, which leaves $45. Lunch then costs 1/3 × 45 = $15, so he has 45 − 15 = $30 left.

Question 46Number and Arithmetic

Work out (5 × 10^3) + (3 × 10^2), giving your answer in standard form.

  1. 8 × 10^5
  2. 5.3 × 10^3 Correct answer.
  3. 5.3 × 10^5
  4. 8 × 10^3

Explanation

Write both terms as ordinary numbers: 5 × 10^3 = 5000 and 3 × 10^2 = 300. Their sum is 5300, which in standard form is 5.3 × 10^3.

Question 47Number and Arithmetic

After a 15% increase, a ticket costs $460. What was the price before the increase?

  1. $400 Correct answer.
  2. $414
  3. $445
  4. $391

Explanation

The new price represents 115% of the original price. Dividing by 1.15 reverses the increase: 460 ÷ 1.15 = 400, so the original price was $400.

Question 48Number and Arithmetic

Lena invests $2000 at 5% per year compound interest. How much interest does she earn in total after 2 years?

  1. $205 Correct answer.
  2. $210
  3. $220
  4. $200

Explanation

After one year the investment is worth 2000 × 1.05 = $2100. After the second year it is 2100 × 1.05 = $2205, so the interest earned is 2205 − 2000 = $205.

Question 49Number and Arithmetic

The recurring decimal 0.4444... is equal to which fraction?

  1. 2/5
  2. 1/2
  3. 4/9 Correct answer.
  4. 4/11

Explanation

Let x = 0.4444..., so 10x = 4.4444.... Subtracting the first equation from the second gives 9x = 4, so x = 4/9, and in general a single recurring digit equals that digit divided by 9.

Question 50Number and Arithmetic

A rectangular floor measures 84 cm by 60 cm. It is to be covered completely with identical square tiles, with no gaps or overlaps. What is the largest possible side length, in centimetres, of each tile?

  1. 6
  2. 12 Correct answer.
  3. 4
  4. 24

Explanation

The side length must divide both 84 and 60 exactly, so the largest possible value is the highest common factor. Writing 84 = 2² × 3 × 7 and 60 = 2² × 3 × 5, the shared prime factors give 2² × 3 = 12.