Fix the mistake: Multiple Problems

Multiple-Ratio Problems ① Relationships Before and After a Ratio Changes

Multiple-ratio problems ask us to find two quantities from how their ratio changes before and after a change. The most important point is “how the ratio changed.”

Basics of Multiple-Ratio Problems

When one quantity increases or decreases, the ratio changes. Solve the problem by focusing on the “quantity that does not change” or the “difference that does not change.”

Example Problem
The ratio of A’s money to B’s money was 3：2, but after A gave B 100円, the ratio became 1：1. How much money did A have at first?

Solution:
① At first: A＝3k, B＝2k (k is one unit of the ratio)
② After giving the money: A＝3k－100, B＝2k＋100
③ Since the ratio is 1：1, A＝B
　3k－100＝2k＋100
　k＝200
④ A’s original amount＝3×200＝600円
Check: B’s original amount＝400円. Afterward, A＝500円 and B＝500円 → 1：1 ✓

Using “The Total Does Not Change”

When money is only transferred between two people, the total of A and B remains the same. We can also use this “unchanging quantity” effectively.

  • Original total: 3k＋2k＝5k
  • Since the ratio becomes 1：1, each amount is 5k÷2
  • A decreased from 3k to 5k÷2, so the amount given was 3k－5k÷2＝k/2 → k/2＝100 → k＝200
Key Points
  • Use the original ratio to write A＝3k and B＝2k.
  • Use the conditions after the change (the ratio or specific numbers) to set up an equation and find k.
  • Remember that “the total stays the same even when money is transferred,” and “the person receiving money increases while the person giving it decreases.”

Practice Problems

  1. The ratio of the numbers of cards X and Y was 5：3, but after X gave Y 20 cards, the ratio became 3：3. Find the original number of cards X had.
  2. The ratio of A’s money to B’s money was 7：3, but after A gave B 400円, the ratio became 3：2. Find A’s original amount of money.
  3. Containers P and Q contain water. The ratio of the amounts of water in P and Q was 5：3, but after 100ml was transferred from P, the ratio became 3：5. Find the original amount of water in P.

Answers and Explanations

  1. Answer: X originally had 100 cards
    Explanation: Before the change: X＝5k, Y＝3k. After X gave Y 20 cards: X＝5k－20, Y＝3k＋20. Since the ratio is 3：3 (＝1：1), 5k－20＝3k＋20. 2k＝40. k＝20. Original X＝5×20＝100 cards. Check: Original Y＝60 cards. Afterward, X＝80 cards and Y＝80 cards→1：1✓
  2. Answer: A originally had 2800円
    Explanation: Before the change: A＝7k, B＝3k. After A gave B 400円: A＝7k－400, B＝3k＋400. Since the ratio is 3：2, (7k－400)：(3k＋400)＝3：2. 2(7k－400)＝3(3k＋400). 14k－800＝9k＋1200. 5k＝2000. k＝400. Original A＝7×400＝2800円. Check: Original B＝1200円. Afterward, A＝2400円 and B＝1600円→2400：1600＝3：2✓
  3. Answer: P originally contained 250ml
    Explanation: Before the change: P＝5k, Q＝3k. After transferring 100ml from P: P＝5k－100, Q＝3k＋100. Since the ratio is 3：5, (5k－100)：(3k＋100)＝3：5. 5(5k－100)＝3(3k＋100). 25k－500＝9k＋300. 16k＝800. k＝50. Original P＝5×50＝250ml. Check: Original Q＝150ml. After the transfer, P＝150ml and Q＝250ml→150：250＝3：5✓

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