Higher June 2023 Paper 1 Q23
23 Write \(0.1\dot{3}\) as a fraction in its simplest form. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – subtracting powers of 10 algebraically | ||
| Denotes the given recurring decimal by a letter and multiplies by one of 10, 100, etc | M1 | eg \(10x = 1.33\ldots\) or \(100x = 13.3\ldots\) |
| Denotes the given recurring decimal by a letter and multiplies by one or two of 10, 100, etc and subtracts accordingly | M1dep | eg \(10x - x = 1.333\ldots - 0.1333\ldots\) or \(9x = 1.2\) or \(\dfrac{1.2}{9}\) or \(100x - x = 13.333\ldots - 0.1333\ldots\) or \(99x = 13.2\) or \(\dfrac{13.2}{99}\) or \(100x - 10x = 13.333\ldots - 1.333\ldots\) or \(90x = 12\) or \(\dfrac{12}{90}\) |
| \(\dfrac{2}{15}\) | A1 | |
| Alternative method 2 – subtracting powers of 10 numerically | ||
| Multiplies the given decimal by one of 10, 100, etc | M1 | eg \(0.1\dot{3} \times 10 = 1.\dot{3}\) |
| Multiplies the given decimal by one or two of 10, 100, etc and subtracts appropriately in fraction form | M1dep | eg \(0.1\dot{3} \times 100 = 13.\dot{3}\) and \(0.1\dot{3} \times 10 = 1.\dot{3}\) and \(\dfrac{13.3 - 1.3}{100 - 10}\) or \(\dfrac{12}{90}\) |
| \(\dfrac{2}{15}\) | A1 | |
| Alternative method 3 – splitting into a known fraction and a recurring decimal | ||
| Splits into 0.1 and \(0.0\dot{3}\) and uses a correct first step from alt 1 or alt 2 with \(0.0\dot{3}\) | M1 | eg \(10x = 0.33\ldots\) or \(0.0\dot{3} \times 100 = 3.33\ldots\) 0.1 does not need to be seen separately at this stage |
| Correct method to evaluate \(0.0\dot{3}\) as a fraction and addition to \(\dfrac{1}{10}\) or \(\dfrac{1}{30} + \dfrac{1}{10}\) or \(\dfrac{4}{30}\) | M1dep | oe fraction |
| \(\dfrac{2}{15}\) | A1 | |
Additional guidance
| Condone decimals within fractions up to M2 eg \(\dfrac{1.2}{9}\) | M2 |
| Equals signs may be implied throughout | |
| Subtraction signs must be seen or the results correct | |
| Recurring decimals should be denoted by correct notation or at least two of the recurring digits followed by at least two dots. However, condone missing dots if the result is, or would be, correct eg condone \(13.3 - 1.3 = 100x - 10x\) |