Higher June 2024 Paper 1 Q21
21 Prove algebraically that \(\quad 1.0\dot{1}\dot{8} = \dfrac{56}{55}\) [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: using \(x = 1.018\ldots\) | ||
| Multiplication by power of 10 | M1 | eg \(10x = 10.18\ldots\) or \(100x = 101.81\ldots\) or \(1000x = 1018.18\ldots\) any or no letter |
| Correct equation formed from subtraction of two equations to eliminate recurring digits | M1dep | eg \(99x = 100.8\) or \(990x = 1008\) or \(x = \dfrac{1008}{990}\) |
| \((x =)\ \dfrac{1008}{990}\) and \(\dfrac{56}{55}\) with no incorrect working | A1 | oe from using different powers of 10 |
| Alternative method 2: using \(x = 0.018\ldots\) | ||
| Multiplication by power of 10 | M1 | eg \(10x = 0.1818\ldots\) or \(100x = 1.818\ldots\) or \(1000x = 18.18\ldots\) any or no letter |
| Correct equation formed from subtraction of two equations to eliminate recurring digits | M1dep | eg \(99x = 1.8\) or \(990x = 18\) or \(x = \dfrac{1.8}{99}\) |
| \((x =)\ \dfrac{1.8}{99}\) or \(\dfrac{18}{990}\) and \((x =)\ \dfrac{1}{55}\) and \(\dfrac{56}{55}\) or \(\dfrac{100.8}{99}\) or \(\dfrac{1008}{990}\) and \(\dfrac{56}{55}\) with no incorrect working | A1 | oe from using different powers of 10 |
| Alternative method 3: using \(x = 1.018\ldots\) and addition | ||
| Multiplication by power of 10 | M1 | eg \(10x = 10.18\ldots\) or \(100x = 101.81\ldots\) or \(1000x = 1018.18\ldots\) any or no letter |
| Correct addition of two correct equations leading to 0.9 recurring | M1dep | eg \(110x = 111.99\ldots\) or \(1100x = 1119.99\ldots\) |
| \((x =)\ \dfrac{112}{110}\) and \(\dfrac{56}{55}\) with no incorrect working | A1 | oe from using different powers of 10 |
Additional guidance
Up to M2 may be awarded for correct work with no answer or incorrect answer if this is seen amongst multiple attempts
For all marks, numbers must be correct
Working with 1.018018018… scores 0
Recurring decimals should be denoted by correct notation or at least two of the recurring digits followed by at least two dots
In alt1 and alt2 condone incorrect recurring notation if the result of the subtraction is a correct equation