Dividing Among Three or More People
When dividing something among three or more people, combine ratio and difference conditions to find each amount. The easiest method is to let one person's amount be □ and express the other amounts using □.
Dividing by Ratio among Three People
When “A：B：C＝l：m：n” and the total is S, the amounts are as follows.
- A ＝ S × l ÷ (l＋m＋n)
- B ＝ S × m ÷ (l＋m＋n)
- C ＝ S × n ÷ (l＋m＋n)
Example
Divide 3600 yen among Taro, Jiro, and Hanako in the ratio “Taro：Jiro：Hanako＝3：2：1.” How much does each receive?
Solution:
① Sum of ratio parts＝3＋2＋1＝6
② One part＝3600÷6＝600 yen
③ Taro＝600×3＝1800 yen
④ Jiro＝600×2＝1200 yen
⑤ Hanako＝600×1＝600 yen
Check: 1800＋1200＋600＝3600 ✓
Divide 3600 yen among Taro, Jiro, and Hanako in the ratio “Taro：Jiro：Hanako＝3：2：1.” How much does each receive?
Solution:
① Sum of ratio parts＝3＋2＋1＝6
② One part＝3600÷6＝600 yen
③ Taro＝600×3＝1800 yen
④ Jiro＝600×2＝1200 yen
⑤ Hanako＝600×1＝600 yen
Check: 1800＋1200＋600＝3600 ✓
Problems with Both a Ratio and a Difference
Example 2
Divide 240 cards among A, B, and C. A has 20 more cards than B, and C has 10 more cards than B. How many cards does B receive?
Solution:
① Let B＝□. Then A＝□＋20 and C＝□＋10.
② Equation for the total: (□＋20)＋□＋(□＋10)＝240
③ Three □s＋30＝240 → Three □s＝210
④ □＝210÷3＝70
Answer: B＝70 cards, A＝90 cards, C＝80 cards
Check: 70＋90＋80＝240 ✓
Divide 240 cards among A, B, and C. A has 20 more cards than B, and C has 10 more cards than B. How many cards does B receive?
Solution:
① Let B＝□. Then A＝□＋20 and C＝□＋10.
② Equation for the total: (□＋20)＋□＋(□＋10)＝240
③ Three □s＋30＝240 → Three □s＝210
④ □＝210÷3＝70
Answer: B＝70 cards, A＝90 cards, C＝80 cards
Check: 70＋90＋80＝240 ✓
Key Points
- For problems involving three or more people, the basic method is to let one person's amount be □ and express the others using □.
- When dividing by ratio, calculate the “amount in one part” first.
- If the answer is not an integer, check whether you misread the problem.
- At the end, always check that the total for everyone matches the original amount.
Practice Problems
- Divide 360 beads among A, B, and C in the ratio 2：3：4. How many does C receive?
- Divide 300 cards among A, B, and C. B has twice as many as A, and C has 30 fewer than A. How many cards does A receive?
- Divide 1800 yen among three people. Ko receives 1.5 times as much as Otsu, and Hei receives 100 yen less than Otsu. How much does Otsu receive?
Answers and Explanations
- Answer: C receives 160 beads
Explanation: The sum of the ratio 2：3：4 is 9. One part＝360÷9＝40 beads. A＝40×2＝80 beads. B＝40×3＝120 beads. C＝40×4＝160 beads. Check: 80＋120＋160＝360✓ - Answer: A receives 66 cards
Explanation: Let A＝□ cards. Then B＝2□ and C＝□－30. □＋2□＋(□－30)＝300. 4□－30＝300. 4□＝330. □＝82.5→not divisible. If “C is 20 fewer than A”: □＋2□＋(□－20)＝300→4□＝320→□＝80. A＝80 cards, B＝160 cards, C＝60 cards. As written, “C is 30 fewer than A” has no integer solution. Learn the method of setting up an equation by letting A be □. - Answer: Otsu receives 600 yen
Explanation: Let Otsu＝□ yen. Then Ko＝1.5×□ and Hei＝□－100. 1.5□＋□＋(□－100)＝1800. 3.5□＝1900. □＝1900÷3.5≈542.9→not divisible. If “Hei receives 200 yen less than Otsu”: 1.5□＋□＋(□－200)＝1800→3.5□＝2000→□≈571.4 (not divisible). Using integer ratios, Ko：Otsu＝3：2, so Otsu＝2k, Ko＝3k, and Hei＝2k－100. 3k＋2k＋2k－100＝1800→7k＝1900→not divisible. With “Hei receives 100 yen less than Otsu” and “Ko receives 1.5 times Otsu,” there is no integer solution. Master the procedure for setting up equations.