Foundation November 2017 Paper 3 Q16
16 A train has 1 first-class carriage and 6 standard carriages.
The first-class carriage has 64 seats.
\(\dfrac{3}{8}\) are being used.
Each standard carriage has 78 seats.
\(\dfrac{7}{13}\) in each carriage are being used.
Are more than half the seats on the train being used?
You must show your working. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 of 6 | ||
| \(64 \times \dfrac{3}{8}\) or 24 or \(78 \times \dfrac{7}{13}\) or 42 or \(6 \times 78 \times \dfrac{7}{13}\) or 252 | M1 | oe \(64 \times \dfrac{5}{8}\) or 40 or \(78 \times \dfrac{6}{13}\) or 36 or \(6 \times 78 \times \dfrac{6}{13}\) or 216 |
| \(64 \times \dfrac{3}{8} + 6 \times 78 \times \dfrac{7}{13}\) or their 24 + their 252 or 276 | M1dep | oe \(64 \times \dfrac{5}{8} + 6 \times 78 \times \dfrac{6}{13}\) or their 40 + their 216 or 256 |
| \(64 + 6 \times 78\) or 64 + 468 or 532 | M1 | |
| their 532 \(\div\) 2 or 266 | M1dep | dep on 3rd method mark only |
| 266 and 276 and Yes or 266 and 256 and Yes | A1 | |
| Alternative method 2 of 6 | ||
| \(64 \times \dfrac{3}{8}\) or 24 or \(78 \times \dfrac{7}{13}\) or 42 or \(6 \times 78 \times \dfrac{7}{13}\) or 252 | M1 | oe \(64 \times \dfrac{5}{8}\) or 40 or \(78 \times \dfrac{6}{13}\) or 36 or \(6 \times 78 \times \dfrac{6}{13}\) or 216 |
| \(64 \times \dfrac{3}{8} + 6 \times 78 \times \dfrac{7}{13}\) or their 24 + their 252 or 276 | M1dep | oe \(64 \times \dfrac{5}{8} + 6 \times 78 \times \dfrac{6}{13}\) or their 40 + their 216 or 256 |
| \(64 + 6 \times 78\) or 64 + 468 or 532 | M1 | |
| their 532 – their 276 | M1dep | dep on M1M1M1 their 532 – their 256 |
| 256 and 276 and Yes | A1 | |
| Alternative method 3 of 6 | ||
| \(64 \times \dfrac{3}{8}\) or 24 or \(78 \times \dfrac{7}{13}\) or 42 or \(6 \times 78 \times \dfrac{7}{13}\) or 252 | M1 | oe \(64 \times \dfrac{5}{8}\) or 40 or \(78 \times \dfrac{6}{13}\) or 36 or \(6 \times 78 \times \dfrac{6}{13}\) or 216 |
| \(64 \times \dfrac{3}{8} + 6 \times 78 \times \dfrac{7}{13}\) or their 24 + their 252 or 276 | M1dep | oe \(64 \times \dfrac{5}{8} + 6 \times 78 \times \dfrac{6}{13}\) or their 40 + their 216 or 256 |
| \(64 \div 2\) or 32 and \((6 \times 78) \div 2\) or \(468 \div 2\) or 234 | M1 | |
| their 32 + their 234 or 266 | M1dep | dep on 3rd method mark only |
| 266 and 276 and Yes or 266 and 256 and Yes | A1 | |
| Alternative method 4 of 6 | ||
| \(64 \times \dfrac{3}{8}\) or 24 or \(78 \times \dfrac{7}{13}\) or 42 or \(6 \times 78 \times \dfrac{7}{13}\) or 252 | M1 | oe |
| \(64 \times \dfrac{3}{8} + 6 \times 78 \times \dfrac{7}{13}\) or their 24 + their 252 or 276 | M1dep | oe |
| \(64 + 6 \times 78\) or 64 + 468 or 532 | M1 | |
| their 276 \(\div\) their 532 or 0.51… or 0.52 or their 532 \(\div\) their 276 or 1.9… or 1.93 | M1dep | oe dep on M1M1M1 |
| 532 and 276 and 0.51… or 0.52 and Yes or 532 and 276 and 1.9… or 1.93 and Yes | A1 | |
| Alternative method 5 of 6 | ||
| \(64 \times \dfrac{3}{8}\) or 24 or \(78 \times \dfrac{7}{13}\) or 42 or \(6 \times 78 \times \dfrac{7}{13}\) or 252 | M1 | oe \(64 \times \dfrac{5}{8}\) or 40 or \(78 \times \dfrac{6}{13}\) or 36 or \(6 \times 78 \times \dfrac{6}{13}\) or 216 |
| \(64 \times \dfrac{3}{8} + 6 \times 78 \times \dfrac{7}{13}\) or their 24 + their 252 or 276 | M1dep | oe \(64 \times \dfrac{5}{8} + 6 \times 78 \times \dfrac{6}{13}\) or their 40 + their 216 or 256 |
| their 276 \(\times\) 2 or 552 | M1dep | their 256 \(\times\) 2 or 512 |
| \(64 + 6 \times 78\) or 64 + 468 or 532 | M1 | |
| 532 and 552 and Yes or 532 and 512 and Yes | A1 | |
| Alternative method 6 of 6 | ||
| \(\dfrac{1}{2} - \dfrac{3}{8}\) or \(\dfrac{1}{8}\) or \(\dfrac{7}{13} - \dfrac{1}{2}\) or \(\dfrac{1}{26}\) | M1 | oe |
| \(64 \times\) their \(\dfrac{1}{8}\) or 8 (under) or \(78 \times\) their \(\dfrac{1}{26}\) or 3 (over) | M1dep | oe |
| \(78 \times\) their \(\dfrac{1}{26} \times 6\) or 18 (over) | M1dep | oe |
| \(64 \times\) their \(\dfrac{1}{8}\) or 8 (under) and \(78 \times\) their \(\dfrac{1}{26} \times 6\) or 18 (over) | M1dep | oe May be subtracted |
| 8 under (half) and 18 over (half) and Yes or 10 over (half) and Yes | A1 | |
Additional guidance
| Condone \(\dfrac{24}{64}\) for 24 or \(\dfrac{42}{468}\) for 42 or \(\dfrac{252}{468}\) for 252 for first method mark | |
| 276 and 10 over (266) and Yes implies 266 and 276 and Yes | M1M1M1M1A1 |
| In Alt 2 256 and 276 and Yes | M1M1M1M1A1 |
| In Alt 4 accept working with unused seats leading to their 256 \(\div\) their 532 or 0.4… or 0.49 or their 532 \(\div\) their 256 or 2.07… or 2.08 |