Higher June 2024 Paper 6 Q7
7 Two bags of fruit contain only apples and bananas.
In bag X, the ratio of apples to bananas is 5 : 7.
In bag Y, \(\frac{5}{12}\) of the fruit are apples.
(a) Finley says
| Bag X and bag Y contain the same number of apples. |
Tick the correct statement.
- ☐ Finley is definitely correct
- ☐ Finley might be correct, or might not be correct
- ☐ Finley is definitely not correct
Show how you decided. [3]
(b) Finley adds 4 apples to bag X.
The ratio of apples to bananas is now 11 : 14.
The ratio of apples to bananas is now 11 : 14.
How many bananas are in bag X? [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Both bags may have 5 apples and 7 bananas or both bags may have 5 apples and 12 fruit | 1 | Accept 5 : 7 and \(\frac{5}{12}\) are equivalent or 5 + 7 = 12 with \(\frac{5}{12}\) or \(\frac{5}{12}\), \(\frac{7}{12}\) with 5 : 7 | See Appendix Check for working at the top of the page |
| A numerical example with some explanation (\(n\) fruit where \(n\) is a multiple of 12, \(n \ne 12\)) | 1 | More than just numbers | |
| “Finley might be correct or might not be correct” | 1dep | Dep on at least one other mark If 0 scored SC1 for explanation along lines of: don’t know how many fruit in the bag and middle box ticked | Accept single tick, cross or other highlight |
Appendix: Question 7(a)
| Response | Mark | Reason |
|---|---|---|
| 5 apples : 7 bananas means the same as \(\frac{5}{12}\) of the fruit are apples. 10 apples :14 bananas also means \(\frac{5}{12}\) of the fruit are apples C&NC | 1 1 1 | An example with the same number of apples An example with a different number of apples with some words of explanation |
| \(5 : 7 = \frac{5}{12}\) \(10 : 14 = \frac{5}{12}\) C&NC | 1 0 1 | Sufficient for first mark. Middle column of mark scheme. Insufficient for second mark. Some words of explanation required. |
| \(5 : 7 = \frac{5}{12}\). 1 × 5 = 5 apples and 1 × 7 = 7 bananas. 3 × 5 = 15 apples and 3 × 7 = 21 bananas still has the same ratio but different numbers. C&NC | 1 1 1 | Sufficient. Middle column of mark scheme. |
| Bag Y must be \(5x : 7x\) as \(\frac{5x}{12x} = \frac{5}{12}\) for the number of apples. This ratio is 5 : 7. The number of apples will only be equal if \(x = 1\). In other case the number of apples will not be equal. C&NC. | 1 0 1 | Not true |
| 5 + 7 = 12. So, \(a : b = 5 : 7\) and \(a = \frac{5}{12}\) However, the amount of fruit in each bag is unknown, so it cannot be certain. C&NC. | 1 0 1 | Not numerical but would be SC if 0 scored |
| 5 + 7 = 12. So, \(a = \frac{5}{12}\). C&NC. | 1 NR 1 | Sufficient. Middle column of mark scheme |
| There may be \(\frac{5}{12}\) apples in one bag but \(\frac{10}{24}\) apples in the other. C&NC. | 0 1bod 1 | Implies different numbers of apples but same fraction |
| 5 : 7 means that there must be 5 apples so the fraction of apples must be \(\frac{5}{12}\). C. | 1 NR 0 | Not strictly true; condone use of “must” |
| 25 apples and 35 bananas is 25 : 35 which is 5 : 7 and the fraction of apples is \(\frac{25}{60} = \frac{5}{12}\). C. | 1 0 0 | 25 apples and 35 bananas give 5 : 7 and \(\frac{5}{12}\) 0, 1, 0 is also acceptable |
| If both bags have same number of fruits they will have the same proportion and the same number of apples. However, if A or B have more fruits than the other, they will have more apples than the other. C&NC. | SC1 | No numbers. Correct statement and tick. |
| \(\frac{5}{12}\) of the fruit are apples in both bags. Both bags could have the same number of apples but they could be different as we do not know how much fruit is in each bag. C&NC. | SC1 | First sentence does not refer to ratio |
| Bag Y is \(\frac{5}{12}\) apples so bag X is \(\frac{5}{12}\) apples. C&NC. | 0 NR 0 | Insufficient compared with middle column of scheme. Ticked box mark is dependent on at least one other mark |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 56 as answer nfww | 3 | By ratios: B2 for both 40 : 56 and 44 : 56 identified or for 10 : 14, 11 : 14 and 44 : 56 or B1 for 2 ratios equivalent to 5 : 7 and 11 : 14 with a common number of bananas | eg. 10 : 14 and 11 : 14 or 20 : 28 and 22 : 28 |
| By equation: B2 for a correct equation that would lead directly to the number of bananas or B1 for a correct equation that would lead directly to the number of apples or total fruit, either before or after the addition of 4 apples | eg: \(b\) = bananas, \(a\) = apples, \(t\) = original total B2 for \(\frac{5b}{7} + 4 = \frac{11b}{14}\) oe or better or B1 for \(\frac{5t}{12} + 4 = \frac{11(t + 4)}{25}\) oe or better or for \(\frac{5a + 4}{7a} = \frac{11}{14}\) oe or better | ||
| By fractions: B2 for \(\frac{40[+4]}{96[+4]}\) and \(\frac{44}{100}\) identified or B1 for 2 fractions of the form \(\frac{5k + 4}{12k + 4}\) , where \(k\) is a positive integer | eg \(\frac{9}{16}\), \(\frac{14}{28}\) oe, \(\frac{19}{40}\), \(\frac{24}{52}\) oe, \(\frac{29}{64}\), \(\frac{34}{76}\) oe \(\frac{39}{88}\), \(\frac{44}{100}\) oe | ||
| All methods: If 0 scored SC1 for answer 44 or 100 | |||