Multiple-Ratio Problems ② Using the Fact That “The Difference Does Not Change”
In multiple-ratio problems, there are cases where “the difference between two people does not change” or “the total does not change.” Using these facts well can help us solve some problems quickly.
Which Does Not Change: the Difference or the Total?
When money is only transferred between two people (and no money comes from outside), the following applies.
- The total A＋B does not change.
- The difference A－B changes by twice the amount transferred (A decreases by N and B increases by N, so the difference changes by 2N).
Example Problem
The ratio of the older brother’s money to the younger brother’s money was 4：1. After the older brother gave the younger brother 600円, the ratio became 2：1. How much money does the older brother have now?
Solution:
① Before the change: Older brother＝4k, younger brother＝k
② After the change: Older brother＝4k－600, younger brother＝k＋600
③ Since the ratio is 2：1, (4k－600)：(k＋600)＝2：1
1×(4k－600)＝2×(k＋600)
4k－600＝2k＋1200
2k＝1800
k＝900
④ Older brother’s current amount＝4×900－600＝3600－600＝3000円
The ratio of the older brother’s money to the younger brother’s money was 4：1. After the older brother gave the younger brother 600円, the ratio became 2：1. How much money does the older brother have now?
Solution:
① Before the change: Older brother＝4k, younger brother＝k
② After the change: Older brother＝4k－600, younger brother＝k＋600
③ Since the ratio is 2：1, (4k－600)：(k＋600)＝2：1
1×(4k－600)＝2×(k＋600)
4k－600＝2k＋1200
2k＝1800
k＝900
④ Older brother’s current amount＝4×900－600＝3600－600＝3000円
Learn Cross-Multiplication of Ratios
When “a：b＝c：d,” the products of the outer terms and the inner terms are equal. Using this can sometimes make calculations easier.
(Larger quantity before the change)×(smaller ratio after the change)＝(larger quantity after the change)×(smaller ratio before the change)
Key Points
- Use the original ratio to write amounts such as “4k” and “k.”
- “Outer×outer＝inner×inner” (cross-multiplication of ratios) makes it easy to set up an equation.
- After finding the answer, check that both the ratio before and the ratio after the change work.
Practice Problems
- The ratio of the numbers of cards A and B was 3：1. After A gave B 30 cards, the ratio became 3：2. Find the current number of cards B has.
- The ratio of P’s money to Q’s money was 5：2. After P gave Q 1000円, the ratio became 3：2. Find Q’s current amount of money.
- Boxes A and B contain balls, and their ratio is 7：3. After 20 balls are moved from A to B, the ratio becomes 3：2. Find the original number of balls in A.
Answers and Explanations
- Answer: B currently has 60 cards
Explanation: Before the change: A＝3k, B＝k. After A gave B 30 cards: A＝3k－30, B＝k＋30. Since the ratio is 3：2, (3k－30)：(k＋30)＝3：2. 2(3k－30)＝3(k＋30). 6k－60＝3k＋90. 3k＝150. k＝50. Current B＝k＋30＝50＋30＝80 cards. Check: A before＝150 cards, B before＝50 cards. A after＝120 cards, B after＝80 cards→120：80＝3：2✓ - Answer: Q currently has 3000円
Explanation: Before the change: P＝5k, Q＝2k. After P gave Q 1000円: P＝5k－1000, Q＝2k＋1000. Since the ratio is 3：2, (5k－1000)：(2k＋1000)＝3：2. 2(5k－1000)＝3(2k＋1000). 10k－2000＝6k＋3000. 4k＝5000. k＝1250. Current Q＝2k＋1000＝2500＋1000＝3500円. Check: P after＝5×1250－1000＝5250円, Q after＝3500円→5250：3500＝3：2✓ - Answer: A originally had 140 balls
Explanation: Before the change: A＝7k, B＝3k. After 20 balls are moved from A to B: A＝7k－20, B＝3k＋20. Since the ratio is 3：2, (7k－20)：(3k＋20)＝3：2. 2(7k－20)＝3(3k＋20). 14k－40＝9k＋60. 5k＝100. k＝20. Original A＝7×20＝140 balls. Check: Original B＝60 balls. After the move, A＝120 balls and B＝80 balls→120：80＝3：2✓