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4MA1/1HR Jan 2021, Question 15: Surds

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 5 marks

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The question, in short

Convert a recurring decimal to a mixed fraction using algebra, and rationalise a surd fraction into a specified integer form.

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Part (a) · 2 marks

Using algebra, show that the recurring decimal 4.5757… equals 4 19/33.

  1. 1Let x = 4.5757…, so 100x = 457.5757…
  2. 2Subtract: 100x − x = 457.5757… − 4.5757…, so 99x = 453
  3. 3x = 453/99 = 151/33
  4. 4151 ÷ 33 = 4 remainder 19, so x = 4 19/33

Answer99x = 453, so x = 453/99 = 151/33 = 4 19/33

Part (b) · 3 marks

Show that 2/(6 − 3√2) can be written as (a + √a)/b with integers a and b, showing working.

  1. 1Multiply top and bottom by 6 + 3√2
  2. 2Top: 2(6 + 3√2) = 12 + 6√2
  3. 3Bottom: 6² − (3√2)² = 36 − 18 = 18
  4. 4(12 + 6√2)/18 = (2 + √2)/3, so a = 2 and b = 3

Answer(2 + √2)/3 or a = 2, b = 3

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