Number

What is the place value of the digit 7 in the number 27,543?

• A7,000✓
• B7
• C700
• D70

Answer: In 27,543, the digit 7 is in the thousands place. Reading from right to left: units, tens, hundreds, thousands. The 7 is worth 7 × 1,000 = 7,000.

Which of these numbers is closest to 5,000?

• A4,299
• B4,998✓
• C5,502
• D5,251

Answer: Distance from 5,000: 4,299 is 701 away, 5,502 is 502 away, 5,251 is 251 away, and 4,998 is only 2 away. So 4,998 is the closest.

Round 3,847 to the nearest 100.

• A3,800✓
• B3,850
• C3,900
• D4,000

Answer: To round 3,847 to the nearest 100, look at the tens digit (4). As 4 is less than 5, round down to 3,800.

Which of these is a prime number?

• A15
• B17✓
• C21
• D25

Answer: A prime number has exactly two factors: 1 and itself. 15 = 3 × 5, 21 = 3 × 7, 25 = 5 × 5. Only 17 has no factors other than 1 and 17, making it prime.

What are the factors of 12?

• A1, 2, 3, 4, 6, 12✓
• B1, 2, 3, 12
• C1, 12
• D2, 3, 4, 6

Answer: Factors of 12 are numbers that divide evenly into 12: 1 × 12, 2 × 6, 3 × 4. So the factors are 1, 2, 3, 4, 6, and 12.

Write 0.0057 in standard form.

• A5.7 × 10⁻³✓
• B57 × 10⁻⁴
• C5.7 × 10²
• D5.7 × 10⁻⁴

Answer: Standard form is A × 10ⁿ where 1 ≤ A < 10. The decimal point moves 3 places to the right to give 5.7, so 0.0057 = 5.7 × 10⁻³. The negative power shows the number is smaller than 1.

What is the highest common factor (HCF) of 18 and 24?

• A2
• B3
• C6✓
• D12

Answer: Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Common factors: 1, 2, 3, 6. The highest is 6.

What is the lowest common multiple (LCM) of 4 and 6?

• A2
• B24
• C10
• D12✓

Answer: Multiples of 4: 4, 8, 12, 16, 20... Multiples of 6: 6, 12, 18, 24... The smallest common multiple is 12.

Calculate 2³ × 3².

• A36
• B72✓
• C108
• D216

Answer: 2³ = 8 and 3² = 9. So 2³ × 3² = 8 × 9 = 72.

Simplify 5⁴ ÷ 5².

• A5⁶
• B5⁸
• C5
• D5²✓

Answer: Using the rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ: 5⁴ ÷ 5² = 5⁴⁻² = 5² which equals 25.

Order these numbers from smallest to largest: -3, 0, -7, 2, -1

• A-7, -3, -1, 0, 2✓
• B-1, -3, -7, 0, 2
• C2, 0, -1, -3, -7
• D0, -1, -3, -7, 2

Answer: On a number line, negative numbers are smaller than zero. -7 is furthest left, then -3, -1, then 0, then 2. Correct order: -7, -3, -1, 0, 2.

What is (-2) × (-3)?

• A-5
• B5
• C6✓
• D-6

Answer: Negative × negative = positive. (-2) × (-3) = 6.

Round 45.678 to 1 decimal place.

• A45.68
• B46
• C45.6
• D45.7✓

Answer: To round to 1 d.p., look at the second decimal place: 7. Since 7 ≥ 5, round up: 45.7.

What is the value of 10⁰?

• A10
• B1✓
• Cundefined
• D0

Answer: Any number to the power of 0 equals 1 (except 0⁰ which is undefined). So 10⁰ = 1.

What is 15 ÷ 0.5?

• A15.5
• B30✓
• C7.5
• D0.3

Answer: Dividing by 0.5 is the same as multiplying by 2. So 15 ÷ 0.5 = 15 × 2 = 30.

Write 48,000,000 in standard form.

• A4.8 × 10⁸
• B4.8 × 10⁶
• C4.8 × 10⁷✓
• D48 × 10⁶

Answer: Standard form: move decimal point to get 4.8, then count places: 48,000,000 = 4.8 × 10⁷. The exponent is 7 because we moved 7 places.

What is (-5) - (-2)?

• A-7
• B-3✓
• C3
• D7

Answer: Subtracting a negative is the same as adding a positive: (-5) - (-2) = (-5) + 2 = -3.

Which number is divisible by both 3 and 4?

• A14
• B18
• C24✓
• D26

Answer: Check each: 24 ÷ 3 = 8 ✓, 24 ÷ 4 = 6 ✓. For divisibility by 3: sum of digits is 2+4=6 (divisible by 3). For divisibility by 4: last two digits are 24 (divisible by 4).

Express 126 as a product of prime factors.

• A3 × 42
• B2 × 3² × 7✓
• C2 × 3 × 21
• D2 × 63

Answer: 126 = 2 × 63 = 2 × 9 × 7 = 2 × 3² × 7. All factors (2, 3, 7) are prime.

What is 0.002 in standard form?

• A2 × 10⁻³✓
• B2 × 10⁻²
• C2 × 10³
• D20 × 10⁻³

Answer: 0.002 = 2 × 10⁻³. Move decimal 3 places right to get 2, so the exponent is -3.

What is -5 + 3?

• A-8
• B-2✓
• C2
• D8

Answer: -5 + 3 = -2. When adding a positive number to a negative number, we move right on the number line.

What is -7 - (-2)?

• A-9
• B-5✓
• C5
• D9

Answer: -7 - (-2) = -7 + 2 = -5. Subtracting a negative is the same as adding a positive.

What is the product of -3 and -4?

• A-12
• B-7
• C7
• D12✓

Answer: A negative multiplied by a negative gives a positive. -3 × -4 = 12.

Estimate 4.8 × 7.2 by rounding to 1 significant figure.

• A28
• B30
• C35✓
• D40

Answer: To 1 significant figure, 4.8 ≈ 5 and 7.2 ≈ 7, so the estimate is 5 × 7 = 35.

In a sale a jacket is reduced by 20% and now costs £60. What was the original price?

• A£72
• B£48
• C£75✓
• D£80

Answer: After a 20% reduction the price is 80% of the original, so 0.8 × original = £60. The original price = 60 ÷ 0.8 = £75. Adding 20% back on to £60 gives £72, which is the classic mistake.

A measurement of 5 cm is recorded to the nearest centimetre. What is the lower bound?

• A4.5 cm✓
• B4 cm
• C5.5 cm
• D5 cm

Answer: When a measurement is recorded to the nearest cm, the lower bound is 0.5 cm below the recorded value: 5 - 0.5 = 4.5 cm.

A mass of 50 kg is recorded to the nearest 10 kg. What is the upper bound?

• A60 kg
• B54.9 kg
• C50 kg
• D55 kg✓

Answer: A mass recorded as 50 kg to the nearest 10 kg lies in the interval 45 kg ≤ mass < 55 kg. The upper bound is therefore 55 kg.

A distance of 23 m is given to the nearest metre. Which of these could be the actual distance?

• A23.6 m
• B23.8 m
• C22.4 m
• D23.2 m✓

Answer: Rounded to the nearest metre, 23 m means the actual distance x satisfies 22.5 ≤ x < 23.5. Only 23.2 m lies in this range: 22.4 rounds to 22, while 23.6 and 23.8 round to 24.

After a 5% pay rise, Sam earns £21000 a year. What was the salary before the rise?

• A£20000✓
• B£19950
• C£22050
• D£20500

Answer: After a 5% rise the new salary is 105% of the old one, so 1.05 × old salary = £21000. The old salary = 21000 ÷ 1.05 = £20000. Taking 5% off £21000 gives £19950, which is not the same thing.

Simplify √12.

• A3√2
• B2√3✓
• C√12
• D2√6

Answer: √12 = √(4 × 3) = √4 × √3 = 2√3.

Simplify √45.

• A5√3
• B√45
• C9√5
• D3√5✓

Answer: √45 = √(9 × 5) = √9 × √5 = 3√5.

Simplify 2√8 + √8.

• A3√16
• B3√8
• C8√2
• D6√2✓

Answer: 2√8 + √8 = 3√8 = 3 × 2√2 = 6√2. First simplify √8 = 2√2, then combine.

Write 45 000 in standard form.

• A4.5 × 10³
• B4.5 × 10⁴✓
• C45 × 10³
• D4.5 × 10⁵

Answer: Standard form is A × 10ⁿ where A is at least 1 and less than 10. Moving the decimal point in 45 000 four places to the left gives 4.5, so the answer is 4.5 × 10⁴. The form 45 × 10³ has the right value but A is 45, which is too large to be standard form, and 4.5 × 10⁵ is 450 000.

What is 3² + 2²?

• A5
• B11
• C13✓
• D25

Answer: 3² = 9 and 2² = 4, so 9 + 4 = 13.

Write 0.0053 in standard form.

• A5.3 × 10⁻²
• B5.3 × 10²
• C5.3 × 10⁻⁴
• D5.3 × 10⁻³✓

Answer: 0.0053 = 5.3 × 10⁻³. The decimal point moves 3 places to the right, so the power is -3.

Express 7.8 × 10⁴ as an ordinary number.

• A780
• B7800
• C78000✓
• D78

Answer: 7.8 × 10⁴ = 78000. We move the decimal point 4 places to the right.

What is (2 × 10³) × (3 × 10²) in standard form?

• A6 × 10⁶
• B6 × 10⁴
• C5 × 10⁵
• D6 × 10⁵✓

Answer: (2 × 10³) × (3 × 10²) = (2 × 3) × (10³ × 10²) = 6 × 10⁵.

What is the lowest common multiple (LCM) of 12 and 18?

• A6
• B18
• C36✓
• D216

Answer: Multiples of 12: 12, 24, 36... Multiples of 18: 18, 36... The LCM is 36.

What is the highest common factor (HCF) of 24 and 36?

• A6
• B8
• C12✓
• D24

Answer: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The HCF is 12.

A length of 8.7 cm is given to 1 decimal place. What is the upper bound?

• A8.70 cm
• B8.65 cm
• C8.75 cm✓
• D8.8 cm

Answer: When given to 1 decimal place, the upper bound is 0.05 above the recorded value: 8.7 + 0.05 = 8.75 cm.