OpenRevise

Flashcards · Maths

Number Without a Calculator

89 cards from 5 lessons

  1. What is the value of the digit 6 in 4.263?

    6 hundredths, which is 0.06.

  2. How do you order decimals such as 0.45, 0.405 and 0.54?

    Write them to the same number of decimal places (0.450, 0.405, 0.540) and compare digit by digit from the left.

  3. What happens to the digits when you multiply by 100?

    Each digit moves two columns to the left and the decimal point stays put, so \(3.2 \times 100 = 320\).

  4. How do you work out \(2.4 \times 0.35\) without a calculator?

    Multiply \(24 \times 35 = 840\), then give the answer three decimal places in total: 0.84.

  5. How do you work out \(8.4 \div 0.07\)?

    Multiply both numbers by 100 so the divisor is whole: \(840 \div 7 = 120\).

  6. How does the grid method work for \(34 \times 27\)?

    Split into \(30 + 4\) and \(20 + 7\), multiply each pair to get 600, 80, 210 and 28, then add them: 918.

  7. What is \(3 - (-5)\)?

    8, because subtracting a negative is the same as adding.

  8. What is \((-4) \times (-6)\)?

    24. The signs are the same, so the answer is positive.

  9. What is \((-20) \div 5\)?

    \(-4\). Different signs give a negative answer.

  10. What does BIDMAS stand for?

    Brackets, Indices, Division and Multiplication, Addition and Subtraction.

  11. What is \(24 \div 4 \times 3\)?

    18. Division and multiplication have equal priority, so work from left to right.

  12. What is \((-4)^2\), and how is it different from \(-4^2\)?

    \((-4)^2 = 16\) because the bracket squares the negative too, whereas \(-4^2 = -16\).

  13. How do you estimate \(38 \times 21\)?

    Round each number to 1 significant figure: \(40 \times 20 = 800\).

  14. 250 people travel on coaches with 48 seats. How many coaches are needed?

    6. \(250 \div 48 = 5\) remainder 10, and the last 10 people still need a coach.

  15. What is the difference between \(-12\) and 3?

    15, because \(3 - (-12) = 15\).

  16. Why is \(0.2 \times 0.4\) not 0.8?

    \(2 \times 4 = 8\), but there are two decimal places in total, so the answer is 0.08.

  17. How do you check a division such as \(4872 \div 12 = 406\)?

    Multiply the answer by the divisor: \(406 \times 12 = 4872\).

  18. What is a factor?

    A whole number that divides exactly into another number. The factors of 18 are 1, 2, 3, 6, 9 and 18.

  19. What is a multiple?

    A number in the times table of another number. The multiples of 7 are 7, 14, 21, 28 and so on.

  20. What is a prime number?

    A number with exactly two factors: 1 and itself.

  21. Is 1 a prime number? Is 2?

    1 is not prime because it has only one factor. 2 is prime, and it is the only even prime.

  22. What are the primes below 30?

    2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

  23. Why is 91 a trap in a prime question?

    It looks prime, but \(91 = 7 \times 13\).

  24. What is the divisibility test for 3 and 9?

    Add the digits. If the sum is a multiple of 3 (or 9), the number is divisible by 3 (or 9).

  25. How do you write 360 as a product of prime factors?

    Use a factor tree: \(360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5\).

  26. How do you find the HCF from prime factors?

    Multiply the primes they share, taking the lower power of each.

  27. How do you find the LCM from prime factors?

    Multiply every prime that appears, taking the higher power of each.

  28. What is the HCF of 24 and 36?

    12, because \(24 = 2^3 \times 3\) and \(36 = 2^2 \times 3^2\).

  29. What is the LCM of 24 and 36?

    72, because it uses \(2^3 \times 3^2\).

  30. Which words in a question point to the HCF?

    ‘Greatest’ or ‘largest’ number that divides exactly into both, or sharing things into identical groups.

  31. Which words in a question point to the LCM?

    ‘Next time together’, ‘smallest number both go into’, or events that repeat at different intervals.

  32. How can you tell from prime factors that a number is a perfect square?

    Every index is even. \(36 = 2^2 \times 3^2\) is a square, but \(72 = 2^3 \times 3^2\) is not.

  33. What is the smallest number to multiply 72 by to get a square number?

    2, because \(72 \times 2 = 144 = 12^2\).

  34. How do you find \(\sqrt{2^2 \times 3^4}\)?

    Halve every index: \(2 \times 3^2 = 18\).

  35. How are the HCF, the LCM and the two numbers linked?

    HCF \(\times\) LCM equals the product of the two numbers. For 24 and 36: \(12 \times 72 = 24 \times 36 = 864\).

  36. How do you simplify a fraction?

    Divide the top and bottom by their highest common factor, so \(\dfrac{36}{48} = \dfrac{3}{4}\).

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

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

  38. How do you work out \(\dfrac{3}{4} + \dfrac{5}{6}\)?

    Use a common denominator of 12: \(\dfrac{9}{12} + \dfrac{10}{12} = \dfrac{19}{12} = 1\dfrac{7}{12}\).

  39. What happens to the denominator when you add fractions?

    It stays the same. Only the numerators are added, and the denominators are never added.

  40. How do you multiply fractions?

    Multiply the numerators, multiply the denominators, and cancel first if you can.

  41. What is \(\dfrac{3}{4} \times \dfrac{2}{3}\)?

    \(\dfrac{6}{12} = \dfrac{1}{2}\).

  42. How do you divide by a fraction?

    Keep, change, flip: \(\dfrac{2}{3} \div \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{5}{6}\).

  43. What must you do to mixed numbers before multiplying or dividing?

    Turn them into improper fractions.

  44. How do you find \(\dfrac{3}{5}\) of 80?

    Divide by the denominator (\(80 \div 5 = 16\)), then multiply by the numerator (\(16 \times 3 = 48\)).

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

    \(\dfrac{7}{5}\).

  46. How do you write 35 minutes as a fraction of 2 hours?

    Use the same units first: \(\dfrac{35}{120} = \dfrac{7}{24}\).

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

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

  48. What is \(6 \div \dfrac{2}{3}\)?

    9. Write 6 as \(\dfrac{6}{1}\) and flip: \(\dfrac{6}{1} \times \dfrac{3}{2} = 9\).

  49. How do you work out \(3\dfrac{1}{3} - 1\dfrac{3}{4}\)?

    Improper fractions \(\dfrac{10}{3} - \dfrac{7}{4} = \dfrac{40}{12} - \dfrac{21}{12} = \dfrac{19}{12} = 1\dfrac{7}{12}\).

  50. What is the common mistake with \(\dfrac{1}{2} + \dfrac{1}{3}\)?

    Adding tops and bottoms to get \(\dfrac{2}{5}\). The correct answer is \(\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\).

  51. Which fraction operations need a common denominator?

    Adding and subtracting only. Multiplying and dividing do not.

  52. How do you compare \(\dfrac{5}{8}\) and \(\dfrac{3}{5}\)?

    Use a common denominator of 40: \(\dfrac{25}{40}\) and \(\dfrac{24}{40}\), so \(\dfrac{5}{8}\) is larger.

  53. What does per cent mean?

    Out of every hundred.

  54. Write 35% as a fraction in its simplest form.

    \(\dfrac{35}{100} = \dfrac{7}{20}\).

  55. How do you change a fraction to a percentage?

    Divide the top by the bottom to get a decimal, then multiply by 100: \(\dfrac{3}{8} = 0.375 = 37.5\%\).

  56. What are \(\dfrac{1}{8}\) as a decimal and as a percentage?

    0.125 and 12.5%.

  57. What are \(\dfrac{1}{3}\) and \(\dfrac{2}{3}\) as percentages?

    \(33\dfrac{1}{3}\%\) and \(66\dfrac{2}{3}\%\).

  58. How do you find 35% of \(\pounds 240\) without a calculator?

    10% is 24, 30% is 72 and 5% is 12, so 35% is \(\pounds 84\).

  59. How do you find 17.5% of \(\pounds 640\)?

    10% is 64, 5% is 32 and 2.5% is 16, so the total is \(\pounds 112\).

  60. What is the multiplier for a 12% increase?

    1.12.

  61. What is the multiplier for an 8% decrease?

    0.92.

  62. What is the formula for percentage change?

    Change \(\div\) original \(\times\) 100.

  63. A price rises from \(\pounds 40\) to \(\pounds 47\). What is the percentage increase?

    \(\dfrac{7}{40} \times 100 = 17.5\%\).

  64. A price is \(\pounds 84\) after a 20% increase. How do you find the original price?

    \(\pounds 84\) is 120%, so divide by 1.2: \(\pounds 70\).

  65. A coat costs \(\pounds 63\) after a 25% reduction. What was the original price?

    \(\pounds 63\) is 75%, so 25% is \(\pounds 21\) and 100% is \(\pounds 84\).

  66. Why can you not just add 25% of the sale price back on to find the original?

    The 25% was taken off the original price, not off the sale price.

  67. Which fractions give terminating decimals?

    Those whose denominator, in simplest form, has only 2 and 5 as prime factors, such as \(\dfrac{7}{20}\).

  68. What is simple interest?

    Interest worked out on the original amount only, so it is the same every year.

  69. What is the total for \(\pounds 800\) at 4% simple interest for 3 years?

    4% is \(\pounds 32\) a year, so interest is \(\pounds 96\) and the total is \(\pounds 896\).

  70. What does \(0.\dot{3}\) mean?

    0.333... recurring, which is \(\dfrac{1}{3}\).

  71. What does \(a^n\) mean?

    \(a\) multiplied by itself \(n\) times, so \(2^5 = 32\).

  72. Which square numbers should you know up to \(15^2\)?

    1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 and 225.

  73. Which cube numbers should you know?

    1, 8, 27, 64, 125 and \(10^3 = 1000\).

  74. What is \(\sqrt{196}\)?

    14.

  75. Is \(4^3\) equal to 12?

    No. \(4^3 = 4 \times 4 \times 4 = 64\).

  76. What is the law for \(a^m \times a^n\)?

    Add the indices: \(a^{m+n}\).

  77. What is the law for \(a^m \div a^n\)?

    Subtract the indices: \(a^{m-n}\).

  78. What is the law for \((a^m)^n\)?

    Multiply the indices: \(a^{mn}\).

  79. Can the index laws simplify \(2^3 \times 5^2\)?

    No, because the bases are different. Work out each power instead: \(8 \times 25 = 200\).

  80. What is \(a^0\)?

    1, for any non-zero \(a\).

  81. What is \(5^{-2}\)?

    \(\dfrac{1}{25}\). A negative index means a reciprocal, not a negative number.

  82. What is \(\left(\dfrac{2}{3}\right)^{-2}\)?

    Flip the fraction, then square: \(\left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\).

  83. What is \(49^{\frac{1}{2}}\) (Higher tier)?

    The square root of 49, which is 7.

  84. What is \(8^{\frac{2}{3}}\) (Higher tier)?

    Take the cube root first, then square: \(2^2 = 4\).

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

    6 and 7, because \(36 < 40 < 49\).

  86. How do you write \(16 \times 8\) as a single power of 2?

    \(2^4 \times 2^3 = 2^7\).

  87. What is \(10^{-3}\) as a decimal?

    0.001.

  88. What is the mistake in \(3^2 \times 3^4 = 9^6\)?

    Add the indices and keep the base: \(3^2 \times 3^4 = 3^6\).

  89. What is \((2x)^2\)?

    \(4x^2\). The 2 is squared as well.