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LGCSE Mathematics (0178) study materials
LGCSE Mathematics (0178) study resources

LGCSE Mathematics (0178) practice questions

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Question 1Number

A shop reduces the price of a jacket from $80 to $68. By what percentage has the price been reduced?

  1. 17.6%
  2. 12%
  3. 15% Correct answer.
  4. 20%

Explanation

Percentage reduction = (reduction ÷ original) × 100 = (12 ÷ 80) × 100 = 15%. The reduction must be compared with the original price, not the new price.

Question 2Number

Which of these numbers is irrational?

  1. 22/7
  2. 0.75
  3. √49
  4. √2 Correct answer.

Explanation

An irrational number cannot be written as a fraction of two integers. √2 has a non-terminating, non-repeating decimal expansion and cannot be expressed as a ratio of integers.

Question 3Number

Write 0.000472 in standard form.

  1. 4.72 × 10⁻⁴ Correct answer.
  2. 4.72 × 10⁴
  3. 47.2 × 10⁻⁵
  4. 4.72 × 10⁻³

Explanation

Standard form is a × 10ⁿ where 1 ≤ a < 10. Moving the decimal point 4 places to the right gives 4.72, so the exponent is −4.

Question 4Number

What is the highest common factor (HCF) of 36 and 60?

  1. 18
  2. 12 Correct answer.
  3. 180
  4. 6

Explanation

36 = 2² × 3² and 60 = 2² × 3 × 5. The HCF takes the lowest power of each common prime: 2² × 3 = 12.

Question 5Number

A recipe for 4 people needs 300 g of rice. How much rice is needed for 10 people at the same rate?

  1. 700 g
  2. 600 g
  3. 1200 g
  4. 750 g Correct answer.

Explanation

This is direct proportion. Rice per person = 300 ÷ 4 = 75 g, so for 10 people the amount is 75 × 10 = 750 g.

Question 6Number

Evaluate 5 + 3 × 4 − 2².

  1. 24
  2. 28
  3. 13 Correct answer.
  4. 20

Explanation

Using the order of operations, evaluate the power and multiplication before addition and subtraction: 5 + 12 − 4 = 13.

Question 7Number

A car travels 150 km in 2 hours. What is its average speed in km/h?

  1. 300 km/h
  2. 75 km/h Correct answer.
  3. 50 km/h
  4. 65 km/h

Explanation

Average speed = distance ÷ time = 150 ÷ 2 = 75 km/h. The units are already consistent, so no conversion is required.

Question 8Number

Which list of numbers is written in ascending order?

  1. 0, −2, 3, −5
  2. 3, 0, −2, −5
  3. −5, −2, 0, 3 Correct answer.
  4. −2, −5, 0, 3

Explanation

Ascending order means from smallest to largest. On a number line −5 is furthest left, then −2, then 0, then 3.

Question 9Number

A sum of $5000 is invested at 4% simple interest per year. What is the total value of the investment after 3 years?

  1. $5200
  2. $6000
  3. $5624.32
  4. $5600 Correct answer.

Explanation

Simple interest = P × r × t = 5000 × 0.04 × 3 = $600. Adding this to the principal gives 5000 + 600 = $5600.

Question 10Number

A shirt is priced at $45. In a sale the price is reduced by 20%, and then a further 10% is taken off the sale price. What is the final price?

  1. $34.20
  2. $32.40 Correct answer.
  3. $31.50
  4. $36.00

Explanation

The reductions apply in sequence to the current price, not to the original. After 20% off: 45 × 0.8 = 36. After a further 10%: 36 × 0.9 = $32.40.

Question 11Number

Express 0.4̇5̇ (the digits 45 repeating) as a fraction in its simplest form.

  1. 45/100
  2. 5/11 Correct answer.
  3. 9/20
  4. 45/99

Explanation

Let x = 0.4545…, then 100x = 45.4545…, so 99x = 45 and x = 45/99 = 5/11. The fraction must be simplified.

Question 12Number

The number 2⁴ × 3² × 5 is written as a product of primes. What is the smallest positive integer by which it must be multiplied to make a perfect square?

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

Explanation

In a perfect square every prime exponent is even. Here 2⁴ and 3² already have even exponents but 5 has exponent 1, so multiplying by 5 gives 2⁴ × 3² × 5².

Question 13Number

A map has a scale of 1 : 25 000. Two towns are 6 cm apart on the map. What is the real distance between them, in kilometres?

  1. 0.15 km
  2. 15 km
  3. 1.5 km Correct answer.
  4. 150 km

Explanation

Real distance = 6 × 25 000 = 150 000 cm. Dividing by 100 000 to convert to kilometres gives 1.5 km.

Question 14Number

A quantity decreases by 12% each year. After 2 years, by what single percentage has the original quantity decreased? Give your answer to 3 significant figures.

  1. 24.0%
  2. 25.4%
  3. 76.0%
  4. 22.6% Correct answer.

Explanation

After two years the value is 0.88² = 0.7744 of the original. The decrease is 1 − 0.7744 = 0.2256, i.e. 22.6% to 3 s.f.

Question 15Number

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

  1. 12 × 10²
  2. 1.2 × 10² Correct answer.
  3. 1.2 × 10⁹
  4. 1.2 × 10³

Explanation

Multiply the number parts and add the indices: 3 × 4 = 12 and 10⁵ × 10⁻³ = 10². Then 12 × 10² = 1.2 × 10³, so check the standard-form requirement carefully.

Question 16Number

A cyclist travels 18 km at an average speed of 24 km/h and then 12 km at an average speed of 16 km/h. What is the average speed for the whole journey?

  1. 19.2 km/h
  2. 20 km/h Correct answer.
  3. 21 km/h
  4. 20.4 km/h

Explanation

Total distance = 30 km. Times are 18 ÷ 24 = 0.75 h and 12 ÷ 16 = 0.75 h, total 1.5 h. Average speed = 30 ÷ 1.5 = 20 km/h.

Question 17Number

A recipe uses flour and sugar in the ratio 5 : 2. If 280 g of flour is used, how much sugar is needed?

  1. 700 g
  2. 70 g
  3. 140 g
  4. 112 g Correct answer.

Explanation

Each ratio part is 280 ÷ 5 = 56 g, so sugar = 2 × 56 = 112 g. The ratio must be scaled by the same factor for both quantities.

Question 18Number

A jacket costs $64 after a 20% discount. What was the original price before the discount?

  1. $80.00 Correct answer.
  2. $76.80
  3. $84.00
  4. $128.00

Explanation

The discounted price is 80% of the original. Original = 64 ÷ 0.8 = $80. Dividing by 0.8 reverses the percentage reduction.

Question 19Number

The number of bacteria in a culture doubles every 3 hours. If there are 500 bacteria initially, how many will there be after 12 hours?

  1. 2000
  2. 8000 Correct answer.
  3. 16000
  4. 4000

Explanation

In 12 hours there are 12 ÷ 3 = 4 doubling periods. The count is 500 × 2⁴ = 500 × 16 = 8000.

Question 20Number

Which of these is the correct order of operations result for 2 + 3 × (4² − 7) ÷ 3?

  1. 12
  2. 27
  3. 11 Correct answer.
  4. 18

Explanation

First evaluate the bracket: 4² − 7 = 16 − 7 = 9. Then 3 × 9 ÷ 3 = 9, and finally 2 + 9 = 11.

Question 21Number

A car worth $12 000 depreciates by 15% in the first year and by 10% of its value in the second year. What is its value after two years?

  1. $9720
  2. $9000
  3. $9180 Correct answer.
  4. $10 200

Explanation

After year 1: 12 000 × 0.85 = 10 200. After year 2: 10 200 × 0.90 = $9180. Each year's depreciation applies to the current value.

Question 22Number

Find the smallest positive integer n such that 2ⁿ × 3³ × 5² is a perfect cube.

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

Explanation

For a perfect cube every prime exponent must be a multiple of 3. We need n ≡ 0 (mod 3), 3 is already a multiple of 3, and 2 needs to become 3, so the smallest n is 3.

Question 23Number

A sum of money is invested at a compound interest rate of 5% per year. After how many complete years will the investment first exceed 1.5 times its original value? (Take 1.05⁸ ≈ 1.477 and 1.05⁹ ≈ 1.551.)

  1. 11
  2. 10
  3. 9 Correct answer.
  4. 8

Explanation

We need 1.05ⁿ > 1.5. Since 1.05⁸ ≈ 1.477 is below 1.5 and 1.05⁹ ≈ 1.551 is above it, the first complete year is 9.

Question 24Number

A tank is filled by two pipes. Pipe A alone fills it in 6 hours and pipe B alone in 4 hours. If both pipes are opened together, how long will they take to fill the tank? Give your answer in hours and minutes.

  1. 3 hours
  2. 5 hours
  3. 2 hours 30 minutes
  4. 2 hours 24 minutes Correct answer.

Explanation

Rates add: 1/6 + 1/4 = 2/12 + 3/12 = 5/12 of the tank per hour. Time = 12/5 = 2.4 hours = 2 hours 24 minutes.

Question 25Number

A quantity is increased by 25% and the result is then decreased by 20%. What is the overall percentage change from the original quantity?

  1. A decrease of 5%
  2. No change Correct answer.
  3. An increase of 5%
  4. An increase of 45%

Explanation

Let the quantity be 100. After a 25% increase it is 125, and 20% of 125 is 25, so the decrease returns it to 100. The overall change is therefore no change.

Question 26Number

Which of these numbers is a rational number?

  1. π
  2. √7
  3. √2
  4. √9 Correct answer.

Explanation

A rational number can be written as a fraction of two integers. Since √9 = 3, it equals 3/1 and is therefore rational. The others are irrational because their decimal expansions never terminate or repeat.

Question 27Number

What is the value of the digit 7 in the number 4 372.5?

  1. 7 hundreds
  2. 7 tenths
  3. 7 tens Correct answer.
  4. 7 units

Explanation

Reading place value from the decimal point leftwards gives units, tens, hundreds. The digit 7 sits in the tens column of 4 372.5, so its value is 7 × 10 = 70.

Question 28Number

Work out the highest common factor (HCF) of 84 and 120.

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

Explanation

Writing 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5, the HCF takes the lowest power of each shared prime: 2² × 3 = 12. The HCF must divide both numbers exactly and be the largest such divisor.

Question 29Number

Which of these numbers is a prime number?

  1. 57
  2. 51
  3. 91
  4. 59 Correct answer.

Explanation

A prime has exactly two factors, 1 and itself. The number 59 is not divisible by 2, 3, 5 or 7, so it is prime. Each of the others has a factor pair beyond 1 and itself.

Question 30Number

A student's mark changes from 40 to 46. What is the percentage increase?

  1. 15% Correct answer.
  2. 60%
  3. 13%
  4. 6%

Explanation

Percentage increase is the change divided by the original value, then multiplied by 100. Here the change is 6 and the original is 40, so 6 ÷ 40 × 100 = 15%.

Question 31Number

Estimate 49.3 × 2.1 by rounding each number to 1 significant figure.

  1. 1 000
  2. 100 Correct answer.
  3. 50
  4. 105

Explanation

Rounding to 1 significant figure gives 50 × 2 = 100. Estimation uses rounded values, so the answer is an approximation rather than the exact product.

Question 32Number

A number is written as 3.6 × 10⁻⁴. What is it in ordinary decimal form?

  1. 36 000
  2. 0.000036
  3. 0.0036
  4. 0.00036 Correct answer.

Explanation

A negative index of −4 means the digits of 3.6 are divided by 10⁴, moving the decimal point four places left. This gives 0.00036.

Question 33Number

Work out 2⁻³ ÷ 2⁻⁵, giving your answer as an integer.

  1. −4
  2. 1/4
  3. 1/256
  4. 4 Correct answer.

Explanation

When dividing powers with the same base, subtract the indices: 2⁻³ ÷ 2⁻⁵ = 2^(−3 − (−5)) = 2² = 4. Subtracting a negative index adds, so the result is positive.

Question 34Number

A number n satisfies 2 < n < 3. Which statement about n² is definitely true?

  1. n² < 4
  2. 4 < n² < 9 Correct answer.
  3. n² > 9
  4. n² is irrational

Explanation

For positive n, squaring preserves order, so squaring each part of 2 < n < 3 gives 4 < n² < 9. This holds for every n in that interval, whatever its exact value.

Question 35Number

The number 1 080 is written as a product of its prime factors. What is this product?

  1. 2³ × 3³ × 5 Correct answer.
  2. 2³ × 3² × 5
  3. 2⁴ × 3³ × 5
  4. 2⁴ × 3² × 5

Explanation

Dividing 1 080 repeatedly by primes gives 1 080 = 8 × 27 × 5 = 2³ × 3³ × 5. Checking: 2³ = 8, 3³ = 27 and 8 × 27 × 5 = 1 080.

Question 36Number

A bag contains only red and blue counters in the ratio 3 : 5. If there are 40 counters in total, how many are red?

  1. 8
  2. 15 Correct answer.
  3. 24
  4. 25

Explanation

The ratio 3 : 5 has 8 equal parts, and 40 ÷ 8 = 5 counters per part. Red counters are 3 parts, so 3 × 5 = 15.

Question 37Number

Round 0.004 678 to 2 significant figures.

  1. 0.004 68
  2. 0.00
  3. 0.005
  4. 0.0047 Correct answer.

Explanation

Significant figures start at the first non-zero digit, which is the 4 in the thousandths place. The next digit is 6, so the 4 rounds up to give 0.0047.

Question 38Number

Which symbol correctly completes the statement −3/4 ___ −0.7, using < or >?

  1. >
  2. < Correct answer.
  3. cannot be compared
  4. =

Explanation

Writing −3/4 as a decimal gives −0.75. On a number line −0.75 lies to the left of −0.7, so −0.75 < −0.7.

Question 39Number

A quantity is multiplied by 1.06 four times in succession. What single multiplier is equivalent to these four multiplications?

  1. 1.06⁴ × 4
  2. 1.2625 Correct answer.
  3. 1.24
  4. 4.24

Explanation

Repeated multiplication by the same factor is a power: 1.06 × 1.06 × 1.06 × 1.06 = 1.06⁴ ≈ 1.2625. This is the compound multiplier, not 4 × 1.06.

Question 40Number

Express the recurring decimal 0.1̇6̇ (the digits 16 repeating) as a fraction in its simplest form.

  1. 1/6
  2. 16/99 Correct answer.
  3. 8/45
  4. 16/100

Explanation

Let x = 0.161616…, then 100x = 16.161616… Subtracting gives 99x = 16, so x = 16/99. Since 16 and 99 share no common factor, this is already simplest form.

Question 41Number

The prime factorisations of two numbers are p = 2³ × 3 × 5² and q = 2² × 3² × 7. What is the lowest common multiple (LCM) of p and q?

  1. 2⁵ × 3³ × 5² × 7
  2. 2² × 3² × 5² × 7
  3. 2² × 3 × 5
  4. 2³ × 3² × 5² × 7 Correct answer.

Explanation

For the LCM, take the highest power of each prime that appears in either factorisation. This gives 2³, 3², 5² and 7, so the LCM is 2³ × 3² × 5² × 7.

Question 42Number

A sum of money grows by 20% in the first year and then falls by 20% in the second year. What is the overall percentage change from the start, and what does this show?

  1. A 4% decrease, because the percentage is taken from a different base each year Correct answer.
  2. A 4% increase, because 20% growth comes first
  3. No change, because +20% and −20% cancel out
  4. A 40% decrease, because the percentages are subtracted

Explanation

Multiplying by 1.2 then 0.8 gives 0.96, so the value ends 4% below the start. The two percentages act on different amounts, so they do not cancel.

Question 43Number

A car's value is modelled by V = 18 000 × 0.85ⁿ, where n is the age in years. After how many complete years does the value first fall below $10 000?

  1. 5 years
  2. 3 years
  3. 4 years Correct answer.
  4. 6 years

Explanation

Evaluating gives n = 3: 18 000 × 0.85³ ≈ 11 054, still above 10 000. For n = 4: 18 000 × 0.85⁴ ≈ 9 396, which is below 10 000, so 4 is the first complete year.

Question 44Number

The number 2⁴ × 3² × 5³ is a perfect square after being multiplied by the smallest possible positive integer k. What is k?

  1. 15
  2. 10
  3. 2
  4. 5 Correct answer.

Explanation

A perfect square needs every prime index to be even. Here 2⁴ and 3² already have even indices, but 5³ has an odd index, so multiplying by one more 5 gives 5⁴. Hence the smallest k is 5.

Question 45Algebra and Graphs

Expand and simplify: 3(2x - 5) - 2(x - 4).

  1. 4x - 7 Correct answer.
  2. 8x - 7
  3. 4x - 23
  4. 4x + 3

Explanation

Multiply out each bracket: 3(2x - 5) = 6x - 15 and -2(x - 4) = -2x + 8. Combining like terms gives 6x - 2x = 4x and -15 + 8 = -7, so the result is 4x - 7. The sign of the final constant is the usual trap here.

Question 46Algebra and Graphs

Factorise fully: 12x²y - 18xy².

  1. 3xy(4x - 6y)
  2. 6xy(2x - 3)
  3. 6x(2xy - 3y²)
  4. 6xy(2x - 3y) Correct answer.

Explanation

The highest common factor of 12 and 18 is 6, and both terms contain x and y, so the common factor is 6xy. Dividing each term by 6xy leaves 2x and -3y, giving 6xy(2x - 3y). Fully factorised means no further common factor remains inside the bracket.

Question 47Algebra and Graphs

Solve the equation 5x - 7 = 2x + 11.

  1. x = 6 Correct answer.
  2. x = 4
  3. x = 18
  4. x = 3/7

Explanation

Subtract 2x from both sides to get 3x - 7 = 11, then add 7 to both sides to get 3x = 18. Dividing by 3 gives x = 6. Checking: 5(6) - 7 = 23 and 2(6) + 11 = 23, so the solution is confirmed.

Question 48Algebra and Graphs

A straight line passes through the points (0, 4) and (3, 10). What is the equation of the line in the form y = mx + c?

  1. y = 2x + 4 Correct answer.
  2. y = 6x + 4
  3. y = 2x - 4
  4. y = 3x + 4

Explanation

The gradient is the change in y divided by the change in x: (10 - 4)/(3 - 0) = 6/3 = 2. The point (0, 4) is on the y-axis, so the intercept c = 4. Hence the equation is y = 2x + 4.

Question 49Algebra and Graphs

Given f(x) = 4x - 1, what is the value of f(-2)?

  1. -7
  2. 9
  3. 7
  4. -9 Correct answer.

Explanation

Substitute x = -2 into the rule: f(-2) = 4(-2) - 1 = -8 - 1 = -9. The multiplication 4 × (-2) is negative, and subtracting 1 makes it more negative still.

Question 50Algebra and Graphs

Which of these is the correct solution of the inequality 3x + 5 < 20?

  1. x < 5 Correct answer.
  2. x < 15
  3. x < 25/3
  4. x > 5

Explanation

Subtract 5 from both sides to get 3x < 15, then divide both sides by the positive number 3, which keeps the inequality direction: x < 5. Dividing by a positive number never flips the inequality sign.