Higher June 2022 Paper 1 Q22
22 Work out \(\quad 0.6\dot{8} - 0.4\dot{5}\)
Give your answer as a fraction in its simplest form. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(0.2\dot{3}\) or 0.23… | M1 | implied by \(10x = 2.33\ldots\) |
| \(10x = 2.33\ldots\) (and \(x = 0.23\ldots\)) | M1dep | oe multiplication by a power of 10 any letter |
| \(10x - x = 2.1\) or \(9x = 2.1\) | M1dep | oe subtraction to eliminate recurring digits eg \(100x - 10x = 23.3\ldots - 2.3\ldots\) or \(90x = 21\) |
| \(\dfrac{21}{90}\) | A1 | oe fraction eg \(\dfrac{23.1}{99}\) |
| \(\dfrac{7}{30}\) | A1ft | ft full simplification of their \(\dfrac{21}{90}\) with all M marks awarded |
| Alternative method 2 | ||
| \(10x = 6.88\ldots\) (and \(x = 0.68\ldots\)) or \(10y = 4.55\ldots\) (and \(y = 0.45\ldots\)) | M1 | oe multiplication by a power of 10 any letter |
| \(10x - x = 6.88\ldots - 0.68\ldots\) or \(9x = 6.2\) and \(10y - y = 4.55\ldots - 0.45\ldots\) or \(9y = 4.1\) | M1dep | oe subtractions to eliminate recurring digits eg \(100x - 10x = 68.8\ldots - 6.8\ldots\) or \(90x = 62\) and \(100y - 10y = 45.5\ldots - 4.5\ldots\) or \(90y = 41\) |
| \(\dfrac{62}{90}\) and \(\dfrac{41}{90}\) | M1dep | oe fractions the fractions do not need to have a common denominator |
| \(\dfrac{21}{90}\) | A1 | oe fraction eg \(\dfrac{23.1}{99}\) |
| \(\dfrac{7}{30}\) | A1ft | ft full simplification of their \(\dfrac{21}{90}\) with all M marks awarded |
| Alternative method 3 | ||
| \(0.2\dot{3}\) or 0.23… | M1 | implied by \((0.0\dot{3} =)\ \dfrac{3}{90}\) oe fraction |
| \((0.0\dot{3} =)\ \dfrac{3}{90}\) | M1dep | oe fraction |
| \(\dfrac{2}{10} + \dfrac{3}{90}\) | M1dep | oe fractions |
| \(\dfrac{21}{90}\) | A1 | oe fraction eg \(\dfrac{23.1}{99}\) |
| \(\dfrac{7}{30}\) | A1ft | ft full simplification of their \(\dfrac{21}{90}\) with all M marks awarded |
| Alternative method 4 | ||
| \((0.0\dot{8} =)\ \dfrac{8}{90}\) or \((0.0\dot{5} =)\ \dfrac{5}{90}\) | M1 | oe fraction |
| \((0.0\dot{8} =)\ \dfrac{8}{90}\) and \((0.0\dot{5} =)\ \dfrac{5}{90}\) | M1dep | oe fractions |
| \(\dfrac{6}{10} + \dfrac{8}{90} - \left(\dfrac{4}{10} + \dfrac{5}{90}\right)\) | M1dep | oe condone missing brackets |
| \(\dfrac{21}{90}\) | A1 | oe fraction eg \(\dfrac{23.1}{99}\) |
| \(\dfrac{7}{30}\) | A1ft | ft full simplification of their \(\dfrac{21}{90}\) with all M marks awarded |
Additional guidance
| For the second mark in alt 1 and the first mark in alt 2, accept multiplication by a power of 10 seen without algebra | |
| Accept fractions with non-recurring decimal numerator and/or denominator up to the first A1 eg \(\dfrac{2.1}{9}\) | M1M1M1A1 |
| \(\dfrac{7}{30}\) with no incorrect working | M1M1M1A1A1 |
| If their incorrect fraction cannot be simplified the final mark cannot be awarded |