Foundation June 2019 Paper 3 Q25
25 Towns \(P\), \(Q\) and \(R\) are connected by roads \(PQ\), \(PR\) and \(QR\).
\(PR\) is 10 km longer than \(PQ\).
\(QR\) is twice as long as \(PR\).
The total length of the three roads is 170 km
Not drawn accurately
Work out the length of \(PQ\). [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – PQ as the unknown | ||
| \(x + 10\) or \(2(x + 10)\) | M1 | any unknown |
| \(x + x + 10 + 2(x + 10) = 170\) | M1dep | oe any consistent unknown \(x\) + their two expressions (with at least one correct) = 170 |
| \(4x + 30 = 170\) | M1dep | oe \(4x = 140\) must be correct |
| 35 | A1 | |
| Alternative method 2 – PR as the unknown | ||
| \(x - 10\) or \(2x\) | M1 | any unknown |
| \(x + x - 10 + 2x = 170\) | M1dep | oe any consistent unknown \(x\) + their two expressions (with at least one correct) = 170 |
| \(4x - 10 = 170\) or \(x = 45\) | M1dep | oe \(4x = 180\) must be correct |
| 35 | A1 | |
| Alternative method 3 – QR as the unknown | ||
| \(\dfrac{x}{2}\) or \(\dfrac{x}{2} - 10\) | M1 | any unknown |
| \(x + \dfrac{x}{2} + \dfrac{x}{2} - 10 = 170\) | M1dep | oe any consistent unknown \(x\) + their two expressions (with at least one correct) = 170 |
| \(2x - 10 = 170\) or \(x = 90\) | M1dep | oe \(2x = 180\) must be correct |
| 35 | A1 | |
| Alternative method 4 – trial and improvement with addition of three lengths | ||
| A correctly evaluated trial with a difference of 10 (km) between the two shorter lengths and the longest length twice the length of the middle length | M1 | may be seen as a subtraction of three numbers from 170 |
| A different correctly evaluated trial with a difference of 10 (km) between the two shorter lengths and the longest length twice the length of the middle length | M1dep | may be seen as a subtraction of three numbers from 170 |
| 35, 45 and 90 | A1 | |
| 35 | A1 | |
| Alternative method 5 – trial and improvement with subtraction from 170 | ||
| A correctly evaluated trial of two lengths subtracted from 170 with a difference of 10 (km) between the two lengths or one length twice the length of the other | M1 | |
| A different correctly evaluated trial of two lengths subtracted from 170 with a difference of 10 (km) between the two lengths or one length twice the length of the other | M1dep | |
| 35, 45 and 90 | A1 | |
| 35 | A1 | |
Additional guidance
| If the student attempts more than one method, mark each method and award the highest mark | |
| Alt 1 \(\quad PQ + PQ + 10 + 2(PQ + 10) = 170\) | M1M1 |
| Alt 1 \(\quad PQ + PQ + 10 + 2PR = 170\) | M1 |
| Alt 2 \(\quad x\), \(x + 10\) and \(2x\) seen on diagram, \(4x + 10 = 170\) | M1M1M0A0 |
| Alt 4 \(\quad 35 + 45 + 90\) with no choice made | M1M1A1A0 |
| Alt 4 \(\quad 170 - 30 - 40 - 80 = 20\) | M1 |
| Alt 4 \(\quad 170 - 30 - 40 - 60 = 40\) incorrect number is doubled | M0 |
| Alt 5 \(\quad 170 - 30 - 60 = 80\) | M1 |