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Exam questions · Maths

Number Without a Calculator

  • 35 exam questions
  • 85 marks
  • 48 quick checks

Place Value, Negatives and the Four Operations

Just this lesson
  1. 1 Write down [1 mark]

    Write down the value of the digit 6 in the number \(28.065\).

    Show answerHide answer

    Model answer

    The 6 is in the second column after the decimal point, so it is worth 6 hundredths, which is \(0.06\).

    Mark scheme

    • 0.06 or six hundredths — B1
  2. 2 Work out [2 marks]

    Work out \(7123 - 4856\).

    Show answerHide answer

    Model answer

    Subtracting column by column with exchanging gives \(7123 - 4856 = 2267\). Check: \(4856 + 2267 = 7123\).

    Mark scheme

    • A correct method for the subtraction, with an error in at most one column — M1
    • 2267 — A1
  3. 3 Work out [2 marks]

    Work out \(3.7 \times 0.6\).

    Show answerHide answer

    Model answer

    \(37 \times 6 = 222\). There are 2 decimal places in the question, so the answer is \(2.22\).

    Mark scheme

    • \(37 \times 6 = 222\) or \(3.7 \times 6 = 22.2\) — M1
    • 2.22 — A1
  4. 4 Work out [3 marks]

    Work out \(1764 \div 12\). You must show your working.

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    Model answer

    \(12 \times 100 = 1200\), and \(1764 - 1200 = 564\). \(12 \times 47 = 564\), so \(1764 \div 12 = 100 + 47 = 147\).

    Mark scheme

    • A correct first step, such as a chunking multiple like \(12 \times 100\) or a first bus-stop line — M1
    • A correct second stage — M1
    • 147 — A1
  5. 5 Work out [3 marks]

    Work out \((-6 + 2) \times (3 - 8) - 7\).

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    Model answer

    Brackets first: \((-6 + 2) = -4\) and \((3 - 8) = -5\). Then \((-4) \times (-5) = 20\) and \(20 - 7 = 13\).

    Mark scheme

    • One bracket evaluated correctly — M1
    • \((-4) \times (-5) = 20\) — M1
    • 13 — A1
  6. 6 Work out [3 marks]

    A tank holds 1200 litres of water when it is full. Some water is poured in using a bucket that holds 7.5 litres. The bucket is used 40 times. How many more litres are needed to fill the tank?

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    Model answer

    \(40 \times 7.5 = 300\) litres poured in, so \(1200 - 300 = 900\) litres are still needed.

    Mark scheme

    • \(40 \times 7.5\) or \(7.5 \times 4\) with a power of 10 correct — M1
    • \(1200 - 300\) — M1
    • 900 — A1
  7. 7 Work out [2 marks]

    Jack says, ‘The answer to \(0.2 \times 0.4\) is \(0.8\).’ Is Jack correct? You must show how you decide.

    Show answerHide answer

    Model answer

    No. \(2 \times 4 = 8\) and there are two decimal places in total, so \(0.2 \times 0.4 = 0.08\). Jack is wrong because he has put the decimal point in the wrong place.

    Mark scheme

    • Shows \(2 \times 4 = 8\) or \(0.08\) — M1
    • No, supported by \(0.08\) — Q1

Quick check

  1. 1

    Estimate \(38 \times 21\) by rounding each number to 1 significant figure.

    1. A600
    2. B80
    3. C1000
    4. D800
    Show answerHide answer

    D: 800

    \(38 \approx 40\) and \(21 \approx 20\), so the estimate is \(40 \times 20 = 800\).

  2. 2

    Tickets cost £8.50 each. How many tickets can be bought with £60?

    1. A7
    2. B8
    3. C6
    4. D70
    Show answerHide answer

    A: 7

    \(8.50 \times 7 = 59.50\), which fits, and \(8.50 \times 8 = 68\), which does not.

  3. 3

    What is the value of the digit 7 in the number 4.073?

    1. A70
    2. B7 hundredths
    3. C7 tenths
    4. D7 thousandths
    Show answerHide answer

    B: 7 hundredths

    The 7 is in the second column after the decimal point, so it is worth 7 hundredths.

  4. 4

    Which of these decimals is the largest?

    1. A0.75
    2. B0.809
    3. C0.098
    4. D0.8
    Show answerHide answer

    B: 0.809

    Writing each to 3 decimal places gives 0.800, 0.750, 0.809 and 0.098, so 0.809 is largest.

  5. 5

    What is 6.4 ÷ 100?

    1. A0.064
    2. B0.64
    3. C0.0064
    4. D640
    Show answerHide answer

    A: 0.064

    Dividing by 100 moves every digit two columns to the right, giving 0.064.

  6. 6

    Work out \(-3 - (-7)\).

    1. A\(10\)
    2. B\(4\)
    3. C\(-10\)
    4. D\(-4\)
    Show answerHide answer

    B: \(4\)

    Subtracting a negative is the same as adding: \(-3 + 7 = 4\).

  7. 7

    Work out \((-6) \times 3 \div (-2)\).

    1. A\(-36\)
    2. B\(-9\)
    3. C\(9\)
    4. D\(36\)
    Show answerHide answer

    C: \(9\)

    \((-6)\times 3 = -18\), then \(-18 \div (-2) = 9\) because the signs are the same.

  8. 8

    Work out \(2 + 3 \times 4^2\).

    1. A\(196\)
    2. B\(22\)
    3. C\(50\)
    4. D\(80\)
    Show answerHide answer

    C: \(50\)

    Indices first: \(4^2 = 16\). Then multiply: \(3 \times 16 = 48\). Then add: \(2 + 48 = 50\).

  9. 9

    Work out \(0.3 \times 0.2\).

    1. A0.06
    2. B0.006
    3. C0.5
    4. D0.6
    Show answerHide answer

    A: 0.06

    \(3 \times 2 = 6\) and there are 2 decimal places in total, so the answer is 0.06.

Factors, Multiples and Primes

Just this lesson
  1. 1 Write down [1 mark]

    Write down a multiple of 7 that is between 50 and 60.

    Show answerHide answer

    Model answer

    \(7 \times 8 = 56\), so 56 is the only multiple of 7 between 50 and 60.

    Mark scheme

    • 56 — B1
  2. 2 Write down [2 marks]

    Write down all the prime numbers between 40 and 50.

    Show answerHide answer

    Model answer

    \(41, 43, 47\). The other numbers are not prime: \(42\), \(44\), \(46\), \(48\) are even, \(45\) is a multiple of 5 and \(49 = 7 \times 7\).

    Mark scheme

    • Two correct primes and no more than one incorrect number — B1
    • 41, 43 and 47 only — B1
  3. 3 Write [3 marks]

    Write 252 as a product of its prime factors. Give your answer in index form.

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    Model answer

    \(252 = 4 \times 63 = 2 \times 2 \times 7 \times 9 = 2 \times 2 \times 3 \times 3 \times 7 = 2^2 \times 3^2 \times 7\).

    Mark scheme

    • A correct first step, such as a factor tree starting with a correct pair of factors — M1
    • \(2 \times 2 \times 3 \times 3 \times 7\) in any order — A1
    • \(2^2 \times 3^2 \times 7\) — A1
  4. 4 Work out [4 marks]

    (a) Find the highest common factor (HCF) of 36 and 60. [2 marks] (b) Find the lowest common multiple (LCM) of 36 and 60. [2 marks]

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    Model answer

    \(36 = 2^2 \times 3^2\) and \(60 = 2^2 \times 3 \times 5\). (a) HCF \(= 2^2 \times 3 = 12\). (b) LCM \(= 2^2 \times 3^2 \times 5 = 180\).

    Mark scheme

    • (a) Correct list of factors, or the prime factors of both numbers — M1
    • (a) 12 — A1
    • (b) Correct list of multiples, or a product that uses every prime once at its highest power — M1
    • (b) 180 — A1
  5. 5 Work out [2 marks]

    Is 91 a prime number? You must show your working.

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    Model answer

    No. \(91 = 7 \times 13\), so 91 has factors other than 1 and itself.

    Mark scheme

    • Finds a factor of 91 other than 1 and 91, such as \(91 \div 7 = 13\) — M1
    • No, with \(7 \times 13\) — Q1
  6. 6 Work out [3 marks]

    Lighthouse A flashes every 15 seconds. Lighthouse B flashes every 20 seconds. They both flash at 8 pm. At what time do they next flash together?

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    Model answer

    The LCM of 15 and 20 is 60. Multiples of 15: 15, 30, 45, 60. Multiples of 20: 20, 40, 60. They next flash together after 60 seconds, which is at 8.01 pm.

    Mark scheme

    • A list of multiples of each, or a prime factor method — M1
    • 60 seconds — A1
    • 8.01 pm — B1
  7. 7 Work out [3 marks]

    The highest common factor (HCF) of two numbers is 12. Their lowest common multiple (LCM) is 180. One of the numbers is 36. Work out the other number.

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    Model answer

    For any two numbers, the product of the numbers equals the HCF multiplied by the LCM. So \(36 \times n = 12 \times 180 = 2160\), and \(n = 2160 \div 36 = 60\).

    Mark scheme

    • \(12 \times 180 = 2160\) — M1
    • \(2160 \div 36\) — M1
    • 60 — A1

Quick check

  1. 1

    What is the smallest whole number that 72 must be multiplied by to give a square number?

    1. A2
    2. B9
    3. C6
    4. D3
    Show answerHide answer

    A: 2

    \(72 = 2^3 \times 3^2\). The index of 2 is odd, so multiply by 2 to get \(144 = 12^2\).

  2. 2

    A teacher has 48 red pens and 72 blue pens. What is the greatest number of identical packs that she can make using all of the pens?

    1. A6
    2. B12
    3. C144
    4. D24
    Show answerHide answer

    D: 24

    The answer is the HCF of 48 and 72, which is 24.

  3. 3

    Which of these numbers is prime?

    1. A63
    2. B51
    3. C59
    4. D57
    Show answerHide answer

    C: 59

    59 has no factors other than 1 and 59. 51 = 3 × 17, 57 = 3 × 19 and 63 = 7 × 9.

  4. 4

    What is the highest common factor (HCF) of 18 and 30?

    1. A6
    2. B9
    3. C3
    4. D90
    Show answerHide answer

    A: 6

    The common factors are 1, 2, 3 and 6, and the highest of these is 6.

  5. 5

    What is the lowest common multiple (LCM) of 6 and 8?

    1. A14
    2. B48
    3. C24
    4. D2
    Show answerHide answer

    C: 24

    Multiples of 8 are 8, 16, 24 and 24 is the first one that is also a multiple of 6.

  6. 6

    Which of these is 72 written as a product of its prime factors?

    1. A\(2 \times 36\)
    2. B\(2^2 \times 3^3\)
    3. C\(2^3 \times 3^2\)
    4. D\(8 \times 9\)
    Show answerHide answer

    C: \(2^3 \times 3^2\)

    72 = 8 × 9 = 2 × 2 × 2 × 3 × 3 = \(2^3 \times 3^2\).

  7. 7

    How many factors does 20 have?

    1. A8
    2. B5
    3. C6
    4. D4
    Show answerHide answer

    C: 6

    The factors are 1, 2, 4, 5, 10 and 20, which makes six.

  8. 8

    Which of these numbers is divisible by 9?

    1. A4520
    2. B4531
    3. C4527
    4. D4528
    Show answerHide answer

    C: 4527

    The digits of 4527 add up to 18, which is a multiple of 9.

  9. 9

    Two lights flash every 8 seconds and every 12 seconds. They flash together now. After how many seconds will they next flash together?

    1. A20
    2. B96
    3. C4
    4. D24
    Show answerHide answer

    D: 24

    The LCM of 8 and 12 is 24, so they next flash together after 24 seconds.

Fractions

Just this lesson
  1. 1 Work out [1 mark]

    Work out \(\dfrac{3}{5}\) of 45.

    Show answerHide answer

    Model answer

    \(45 \div 5 = 9\) and \(9 \times 3 = 27\).

    Mark scheme

    • 27 — B1
  2. 2 Write [2 marks]

    Write \(\dfrac{28}{42}\) in its simplest form.

    Show answerHide answer

    Model answer

    The HCF of 28 and 42 is 14. \(\dfrac{28 \div 14}{42 \div 14} = \dfrac{2}{3}\).

    Mark scheme

    • Divides the top and bottom by a common factor, such as \(\dfrac{14}{21}\) — M1
    • \(\dfrac{2}{3}\) — A1
  3. 3 Work out [3 marks]

    Work out \(\dfrac{2}{3} + \dfrac{4}{5}\). Give your answer as a mixed number.

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    Model answer

    The common denominator is 15: \(\dfrac{10}{15} + \dfrac{12}{15} = \dfrac{22}{15} = 1\dfrac{7}{15}\).

    Mark scheme

    • A common denominator of 15 — M1
    • \(\dfrac{22}{15}\) — A1
    • \(1\dfrac{7}{15}\) — A1
  4. 4 Work out [3 marks]

    Work out \(5\dfrac{1}{2} - 2\dfrac{2}{3}\). Give your answer as a mixed number.

    Show answerHide answer

    Model answer

    \(5\dfrac{1}{2} = \dfrac{11}{2} = \dfrac{33}{6}\) and \(2\dfrac{2}{3} = \dfrac{8}{3} = \dfrac{16}{6}\). Then \(\dfrac{33}{6} - \dfrac{16}{6} = \dfrac{17}{6} = 2\dfrac{5}{6}\).

    Mark scheme

    • Improper fractions or a common denominator of 6 — M1
    • \(\dfrac{33}{6} - \dfrac{16}{6}\) — M1
    • \(2\dfrac{5}{6}\) — A1
  5. 5 Work out [3 marks]

    Work out \(1\dfrac{3}{5} \times 2\dfrac{1}{2}\).

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    Model answer

    \(1\dfrac{3}{5} = \dfrac{8}{5}\) and \(2\dfrac{1}{2} = \dfrac{5}{2}\). Then \(\dfrac{8}{5} \times \dfrac{5}{2} = \dfrac{40}{10} = 4\).

    Mark scheme

    • Both mixed numbers converted to improper fractions — M1
    • \(\dfrac{8}{5} \times \dfrac{5}{2}\) or \(\dfrac{40}{10}\) — M1
    • 4 — A1
  6. 6 Work out [3 marks]

    Work out \(\dfrac{4}{9} \div \dfrac{2}{3}\). Give your answer in its simplest form.

    Show answerHide answer

    Model answer

    \(\dfrac{4}{9} \div \dfrac{2}{3} = \dfrac{4}{9} \times \dfrac{3}{2} = \dfrac{12}{18} = \dfrac{2}{3}\).

    Mark scheme

    • \(\dfrac{4}{9} \times \dfrac{3}{2}\) — M1
    • \(\dfrac{12}{18}\) — A1
    • \(\dfrac{2}{3}\) — A1
  7. 7 Work out [3 marks]

    A tank is \(\dfrac{3}{5}\) full of water. \(\dfrac{1}{3}\) of the water is then used. (a) What fraction of the tank is now full of water? [2 marks] The tank holds 300 litres when it is full. (b) How many litres of water are in the tank now? [1 mark]

    Show answerHide answer

    Model answer

    (a) \(\dfrac{1}{3}\) is used, so \(\dfrac{2}{3}\) of the water is left: \(\dfrac{2}{3} \times \dfrac{3}{5} = \dfrac{6}{15} = \dfrac{2}{5}\). (b) \(\dfrac{2}{5} \times 300 = 120\) litres.

    Mark scheme

    • (a) \(\dfrac{2}{3} \times \dfrac{3}{5}\) or \(\dfrac{3}{5} - \dfrac{1}{3} \times \dfrac{3}{5}\) — M1
    • (a) \(\dfrac{2}{5}\) — A1
    • (b) 120 — B1 (follow through)

Quick check

  1. 1

    \(\dfrac{3}{5}\) of a number is 36. What is the number?

    1. A60
    2. B21.6
    3. C108
    4. D12
    Show answerHide answer

    A: 60

    \(36 \div 3 = 12\) is one fifth, so the whole number is \(12 \times 5 = 60\).

  2. 2

    What fraction of 2 hours is 35 minutes? Give your answer in its simplest form.

    1. A\(\dfrac{35}{2}\)
    2. B\(\dfrac{7}{4}\)
    3. C\(\dfrac{7}{12}\)
    4. D\(\dfrac{7}{24}\)
    Show answerHide answer

    D: \(\dfrac{7}{24}\)

    Convert to minutes: \(\dfrac{35}{120} = \dfrac{7}{24}\).

  3. 3

    Work out \(\dfrac{1}{2} + \dfrac{1}{3}\).

    1. A\(\dfrac{5}{6}\)
    2. B\(\dfrac{2}{6}\)
    3. C\(\dfrac{1}{6}\)
    4. D\(\dfrac{2}{5}\)
    Show answerHide answer

    A: \(\dfrac{5}{6}\)

    Use a common denominator of 6: \(\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\).

  4. 4

    What is \(\dfrac{3}{5}\) of 40?

    1. A8
    2. B24
    3. C16
    4. D120
    Show answerHide answer

    B: 24

    \(40 \div 5 = 8\), then \(8 \times 3 = 24\).

  5. 5

    Work out \(\dfrac{2}{3} \div 4\).

    1. A\(\dfrac{8}{3}\)
    2. B6
    3. C\(\dfrac{1}{6}\)
    4. D\(\dfrac{1}{3}\)
    Show answerHide answer

    C: \(\dfrac{1}{6}\)

    Dividing by 4 is multiplying by \(\dfrac{1}{4}\): \(\dfrac{2}{3} \times \dfrac{1}{4} = \dfrac{2}{12} = \dfrac{1}{6}\).

  6. 6

    Write \(3\dfrac{2}{5}\) as an improper fraction.

    1. A\(\dfrac{11}{5}\)
    2. B\(\dfrac{6}{5}\)
    3. C\(\dfrac{17}{5}\)
    4. D\(\dfrac{32}{5}\)
    Show answerHide answer

    C: \(\dfrac{17}{5}\)

    \(3 \times 5 + 2 = 17\), so the fraction is \(\dfrac{17}{5}\).

  7. 7

    Which fraction is equal to \(\dfrac{18}{24}\)?

    1. A\(\dfrac{4}{5}\)
    2. B\(\dfrac{2}{3}\)
    3. C\(\dfrac{9}{10}\)
    4. D\(\dfrac{3}{4}\)
    Show answerHide answer

    D: \(\dfrac{3}{4}\)

    The HCF of 18 and 24 is 6, and dividing both by 6 gives \(\dfrac{3}{4}\).

  8. 8

    Work out \(\dfrac{3}{4} \times \dfrac{2}{9}\).

    1. A\(\dfrac{27}{8}\)
    2. B\(\dfrac{5}{13}\)
    3. C\(\dfrac{2}{3}\)
    4. D\(\dfrac{1}{6}\)
    Show answerHide answer

    D: \(\dfrac{1}{6}\)

    Multiply across and simplify: \(\dfrac{6}{36} = \dfrac{1}{6}\).

  9. 9

    What is the reciprocal of \(\dfrac{5}{7}\)?

    1. A\(\dfrac{5}{7}\)
    2. B\(\dfrac{7}{5}\)
    3. C\(-\dfrac{5}{7}\)
    4. D35
    Show answerHide answer

    B: \(\dfrac{7}{5}\)

    The reciprocal is the fraction turned upside down, which is \(\dfrac{7}{5}\).

  10. 10

    Which of these fractions is the largest?

    1. A\(\dfrac{2}{3}\)
    2. B\(\dfrac{5}{8}\)
    3. C\(\dfrac{3}{5}\)
    4. D\(\dfrac{7}{12}\)
    Show answerHide answer

    A: \(\dfrac{2}{3}\)

    As decimals they are 0.625, 0.667, 0.583 and 0.6, so \(\dfrac{2}{3}\) is the largest.

Fractions, Decimals and Percentages

Just this lesson
  1. 1 Write [1 mark]

    Write \(0.45\) as a percentage.

    Show answerHide answer

    Model answer

    Multiply by 100: \(0.45 \times 100 = 45\%\).

    Mark scheme

    • 45% — B1
  2. 2 Write [2 marks]

    Write \(\dfrac{3}{20}\) as a percentage.

    Show answerHide answer

    Model answer

    \(\dfrac{3}{20} = \dfrac{15}{100} = 15\%\).

    Mark scheme

    • \(\dfrac{15}{100}\) or \(0.15\) — M1
    • 15% — A1
  3. 3 Work out [3 marks]

    Work out \(35\%\) of 480. You must show your working.

    Show answerHide answer

    Model answer

    \(10\% = 48\), so \(30\% = 144\). \(5\% = 24\). Then \(35\% = 144 + 24 = 168\).

    Mark scheme

    • Finds 10% (48) or 5% (24) — M1
    • Adds correct parts, such as 144 + 24 — M1
    • 168 — A1
  4. 4 Write [2 marks]

    Write these in order of size. Start with the smallest. \(\dfrac{2}{5}\) \(0.38\) \(41\%\)

    Show answerHide answer

    Model answer

    \(\dfrac{2}{5} = 0.4\) and \(41\% = 0.41\). The order is \(0.38,\ \dfrac{2}{5},\ 41\%\).

    Mark scheme

    • At least two converted to a common form — M1
    • \(0.38,\ \dfrac{2}{5},\ 41\%\) — A1
  5. 5 Work out [3 marks]

    A phone costs \(\pounds 250\). In a sale the price is reduced by \(18\%\). Work out the sale price.

    Show answerHide answer

    Model answer

    \(10\% = 25\) and \(8\% = 20\), so \(18\% = 45\). The sale price is \(250 - 45 = \pounds 205\).

    Mark scheme

    • Finds 10% or 1% correctly — M1
    • \(18\% = 45\) or \(250 \times 0.82\) — M1
    • \(\pounds 205\) — A1
  6. 6 Work out [3 marks]

    The price of a ticket rises from \(\pounds 25\) to \(\pounds 28\). Work out the percentage increase.

    Show answerHide answer

    Model answer

    The increase is \(28 - 25 = \pounds 3\). \(\dfrac{3}{25} \times 100 = 12\%\).

    Mark scheme

    • \(28 - 25 = 3\) — M1
    • \(\dfrac{3}{25} \times 100\) or \(\dfrac{3}{25} = 0.12\) — M1
    • 12% — A1
  7. 7 Work out [3 marks]

    After a \(20\%\) increase, a water bill is \(\pounds 144\). Work out the bill before the increase.

    Show answerHide answer

    Model answer

    \(144\) is \(120\%\) of the original bill. \(120\% = 144\), so \(10\% = 12\) and \(100\% = \pounds 120\).

    Mark scheme

    • Recognises that 144 is 120% — M1
    • \(144 \div 12 = 12\) (10%) or \(144 \div 1.2\) — M1
    • \(\pounds 120\) — A1

Quick check

  1. 1

    Which of these fractions is a recurring decimal?

    1. A\(\dfrac{7}{20}\)
    2. B\(\dfrac{1}{6}\)
    3. C\(\dfrac{9}{25}\)
    4. D\(\dfrac{3}{8}\)
    Show answerHide answer

    B: \(\dfrac{1}{6}\)

    6 has the prime factor 3, so \(\dfrac{1}{6} = 0.1\dot{6}\) recurs. The others have denominators with only the prime factors 2 and 5.

  2. 2

    £500 is invested for 4 years at 3% simple interest per year. How much interest is earned in total?

    1. A£62.75
    2. B£15
    3. C£60
    4. D£560
    Show answerHide answer

    C: £60

    3% of £500 is £15 a year, and \(15 \times 4 = £60\).

  3. 3

    Write 0.035 as a percentage.

    1. A350%
    2. B3.5%
    3. C0.35%
    4. D35%
    Show answerHide answer

    B: 3.5%

    Multiply by 100: \(0.035 \times 100 = 3.5\%\).

  4. 4

    Write \(\dfrac{3}{8}\) as a percentage.

    1. A3.8%
    2. B38%
    3. C0.375%
    4. D37.5%
    Show answerHide answer

    D: 37.5%

    \(3 \div 8 = 0.375\), and \(0.375 \times 100 = 37.5\%\).

  5. 5

    What is 15% of 60?

    1. A15
    2. B6
    3. C9
    4. D90
    Show answerHide answer

    C: 9

    10% is 6 and 5% is 3, so 15% is 9.

  6. 6

    What is the multiplier for a decrease of 8%?

    1. A8
    2. B0.92
    3. C1.08
    4. D0.08
    Show answerHide answer

    B: 0.92

    100% − 8% = 92%, which is 0.92.

  7. 7

    Increase £60 by 25%.

    1. A£15
    2. B£75
    3. C£240
    4. D£85
    Show answerHide answer

    B: £75

    25% of 60 is 15, and 60 + 15 = 75.

  8. 8

    After a 20% decrease, a price is £48. What was the original price?

    1. A£57.60
    2. B£60
    3. C£96
    4. D£38.40
    Show answerHide answer

    B: £60

    The sale price is 80% of the original, so the original is 48 ÷ 0.8 = £60.

  9. 9

    A price rises from 40 to 50. What is the percentage increase?

    1. A25%
    2. B20%
    3. C80%
    4. D10%
    Show answerHide answer

    A: 25%

    The change is 10, and \(\dfrac{10}{40} \times 100 = 25\%\). It is divided by the original, not the new price.

  10. 10

    Write 12.5% as a fraction in its simplest form.

    1. A\(\dfrac{1}{8}\)
    2. B\(\dfrac{1}{12}\)
    3. C\(\dfrac{1}{5}\)
    4. D\(\dfrac{5}{8}\)
    Show answerHide answer

    A: \(\dfrac{1}{8}\)

    \(12.5\% = \dfrac{12.5}{100} = \dfrac{1}{8}\).

Powers, Roots and Indices

Just this lesson
  1. 1 Work out [2 marks]

    Work out \(5^2 - \sqrt{49}\).

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    Model answer

    \(5^2 = 25\) and \(\sqrt{49} = 7\), so \(25 - 7 = 18\).

    Mark scheme

    • \(25\) or \(7\) seen — M1
    • 18 — A1
  2. 2 Write down [1 mark]

    Write down the value of \(2^5\).

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    Model answer

    \(2 \times 2 \times 2 \times 2 \times 2 = 32\).

    Mark scheme

    • 32 — B1
  3. 3 Simplify [2 marks]

    Write each of these as a single power. (a) \(5^3 \times 5^4\) [1 mark] (b) \((6^2)^5\) [1 mark]

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    Model answer

    (a) Add the indices: \(5^{3+4} = 5^7\). (b) Multiply the indices: \(6^{2 \times 5} = 6^{10}\).

    Mark scheme

    • (a) \(5^7\) — B1
    • (b) \(6^{10}\) — B1
  4. 4 Work out [2 marks]

    Work out the value of \(5^0 + 2^{-2}\). Give your answer as a mixed number.

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    Model answer

    \(5^0 = 1\) and \(2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4}\). So \(1 + \dfrac{1}{4} = 1\dfrac{1}{4}\).

    Mark scheme

    • \(5^0 = 1\) or \(2^{-2} = \dfrac{1}{4}\) — M1
    • \(1\dfrac{1}{4}\) or \(\dfrac{5}{4}\) or \(1.25\) — A1
  5. 5 Show that [2 marks]

    Show that \(2^6 \times 4^2 = 2^{10}\).

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    Model answer

    \(4 = 2^2\), so \(4^2 = (2^2)^2 = 2^4\). Then \(2^6 \times 2^4 = 2^{6+4} = 2^{10}\).

    Mark scheme

    • \(4^2 = 2^4\) or \(4 = 2^2\) — M1
    • \(2^6 \times 2^4 = 2^{10}\) with a clear conclusion — Q1
  6. 6 Work out [4 marks]

    Work out the value of each of these. (a) \(16^{\frac{3}{4}}\) [2 marks] (b) \(\left(\dfrac{1}{9}\right)^{-\frac{1}{2}}\) [2 marks]

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    Model answer

    (a) \(16^{\frac{1}{4}} = 2\), then \(2^3 = 8\). (b) The negative index flips the fraction: \(\left(\dfrac{1}{9}\right)^{-\frac{1}{2}} = 9^{\frac{1}{2}} = 3\).

    Mark scheme

    • (a) \(\sqrt[4]{16} = 2\) or \(16^{\frac{1}{4}} = 2\) — M1
    • (a) 8 — A1
    • (b) \(9^{\frac{1}{2}}\) — M1
    • (b) 3 — A1
  7. 7 Simplify [3 marks]

    Simplify \(\dfrac{(2x^3)^2}{4x^2}\).

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    Model answer

    \((2x^3)^2 = 4x^6\). Then \(\dfrac{4x^6}{4x^2} = x^4\).

    Mark scheme

    • \((2x^3)^2 = 4x^6\) — M1
    • Divides the numbers and the powers of \(x\) — M1
    • \(x^4\) — A1

Quick check

  1. 1

    Between which two whole numbers does \(\sqrt{40}\) lie?

    1. A6 and 7
    2. B5 and 6
    3. C4 and 5
    4. D7 and 8
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    A: 6 and 7

    \(6^2 = 36\) and \(7^2 = 49\), and 40 is between 36 and 49.

  2. 2

    Write \(16 \times 8\) as a single power of 2.

    1. A\(4^7\)
    2. B\(2^{24}\)
    3. C\(2^{12}\)
    4. D\(2^7\)
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    D: \(2^7\)

    \(16 = 2^4\) and \(8 = 2^3\), so \(2^4 \times 2^3 = 2^7\).

  3. 3

    What is the value of \(\sqrt{196}\)?

    1. A98
    2. B14
    3. C16
    4. D13
    Show answerHide answer

    B: 14

    14 × 14 = 196.

  4. 4

    What is the value of \(4^3\)?

    1. A12
    2. B81
    3. C64
    4. D256
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    C: 64

    \(4 \times 4 \times 4 = 64\), not \(4 \times 3 = 12\).

  5. 5

    Simplify \(a^5 \times a^3\).

    1. A\(a^2\)
    2. B\(a^{15}\)
    3. C\(a^8\)
    4. D\(2a^8\)
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    C: \(a^8\)

    Add the indices when multiplying: \(a^{5+3} = a^8\).

  6. 6

    Simplify \((3^2)^4\).

    1. A\(3^8\)
    2. B\(9^8\)
    3. C\(3^6\)
    4. D\(3^{16}\)
    Show answerHide answer

    A: \(3^8\)

    For a power of a power, multiply the indices: \(3^{2 \times 4} = 3^8\).

  7. 7

    What is the value of \(5^{-2}\)?

    1. A\(-10\)
    2. B\(\dfrac{1}{25}\)
    3. C\(-25\)
    4. D\(\dfrac{1}{10}\)
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    B: \(\dfrac{1}{25}\)

    A negative index means a reciprocal: \(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\).

  8. 8

    What is the value of \(7^0\)?

    1. A1
    2. BUndefined
    3. C0
    4. D7
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    A: 1

    Any non-zero number to the power 0 is 1.

  9. 9

    Simplify \(2^6 \div 2^{-2}\).

    1. A\(2^4\)
    2. B\(2^{-3}\)
    3. C\(2^8\)
    4. D\(2^{-12}\)
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    C: \(2^8\)

    Subtract the indices: \(6 - (-2) = 8\), so the answer is \(2^8\).

  10. 10

    What is the value of \(8^{\frac{1}{3}}\)?

    1. A24
    2. B\(\dfrac{8}{3}\)
    3. C\(\dfrac{1}{8}\)
    4. D2
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    D: 2

    A power of one third means the cube root, and \(2 \times 2 \times 2 = 8\).