How to Divide When There Is a Difference
In this lesson, we study distribution problems that include a difference condition, such as “A is ○ more than B” or “A is △ less than ○ times B,” rather than only a ratio.
How to Solve Problems with a Difference
When a difference condition is given, the basic method is to let the smaller amount be □ and create an equation.
Example 1
Divide 720 yen between A and B. If A receives 120 yen more than B, how much does each receive?
Solution: The sum-and-difference formulas can be used.
① A＝(720＋120)÷2＝840÷2＝420 yen
② B＝(720－120)÷2＝600÷2＝300 yen
Divide 720 yen between A and B. If A receives 120 yen more than B, how much does each receive?
Solution: The sum-and-difference formulas can be used.
① A＝(720＋120)÷2＝840÷2＝420 yen
② B＝(720－120)÷2＝600÷2＝300 yen
Example 2 (Both a Ratio and a Difference)
Divide 840 yen between A and B so that A receives 60 yen less than twice B's amount. How many yen more does B receive than A?
Solution:
① Let B＝□. Then A＝2×□－60.
② Equation for the total: □＋(2×□－60)＝840
③ Three □s－60＝840 → Three □s＝900
④ □＝900÷3＝300, so B＝300 yen
⑤ A＝2×300－60＝540 yen
Answer: B receives 540－300＝240 yen more than A.
Divide 840 yen between A and B so that A receives 60 yen less than twice B's amount. How many yen more does B receive than A?
Solution:
① Let B＝□. Then A＝2×□－60.
② Equation for the total: □＋(2×□－60)＝840
③ Three □s－60＝840 → Three □s＝900
④ □＝900÷3＝300, so B＝300 yen
⑤ A＝2×300－60＝540 yen
Answer: B receives 540－300＝240 yen more than A.
Tips for Organizing the Conditions
When there are several conditions, first write them down as a list in your notebook, then create an equation. This helps prevent mistakes.
Key Points
- For problems involving only a ratio, use the “sum of the ratio parts.” For problems involving only a difference, use the “sum-and-difference formulas.”
- When both a ratio and a difference are given, let the smaller amount be □ and create an equation.
- After finding the answer, check whether the total, difference, and ratio all satisfy the conditions.
Practice Problems
- Divide 500 sheets of origami paper between A and B. A receives 50 more sheets than B. How many sheets does A receive?
- Divide 2000 yen between X and Y. X is 200 yen less than three times Y. How much is X?
- Divide cookies between two people. Elder Brother receives 10 more than twice Younger Brother's amount, and the total is 100 cookies. How many cookies does Younger Brother receive?
Answers and Explanations
- Answer: A receives 275 sheets
Explanation: A has 50 more sheets than B. Use the sum-and-difference formula. A＝(500＋50)÷2＝550÷2＝275 sheets. B＝(500－50)÷2＝450÷2＝225 sheets. Check: 275＋225＝500✓, 275－225＝50✓ - Answer: X＝1300 yen
Explanation: Let Y＝□. Then X＝3×□－200. □＋3□－200＝2000. 4□＝2200. □＝550 yen (Y). X＝3×550－200＝1650－200＝1450 yen. Check: 1450＋550＝2000✓. Note: “200 yen less than three times Y” means X＝3Y－200. 1450＋550＝2000✓, 1450＝3×550－200＝1450✓ - Answer: Younger Brother receives 30 cookies
Explanation: Let Younger Brother's amount be □ cookies. Elder Brother＝2□＋10. □＋2□＋10＝100. 3□＝90. □＝30 cookies (Younger Brother). Elder Brother＝2×30＋10＝70 cookies. Check: 30＋70＝100✓, 70＝2×30＋10✓