Fix the mistake: Distribution Problems

Distribution Problems ① Dividing Quantities Using Ratios

Distribution problems involve dividing something according to set rules, such as ratios or differences. When dividing by ratio, the key is to first consider the “sum of the ratio parts.”

Basics of Dividing by Ratio

When “A：B＝m：n” and the total amount is S, they are found as follows.

  • A ＝ S × m ÷ (m＋n)
  • B ＝ S × n ÷ (m＋n)

In other words, divide by the sum of the ratio parts, then multiply by each ratio.

Example
Divide 1200 yen between A and B in the ratio 3：2. How much does each receive?

Solution:
① Sum of ratio parts＝3＋2＝5
② Dividing 1200 yen into 5 parts gives 1200÷5＝240 yen per part
③ A＝240×3＝720 yen
④ B＝240×2＝480 yen
Check: 720＋480＝1200 ✓

What Is a “Ratio”?

3：2 means “the amount of A is 3/2 times the amount of B.” Also, since the sum of the ratio parts is 5, you can imagine dividing the whole into 5 equal parts, with A receiving 3 parts and B receiving 2 parts.

For ratio problems, think in the following order.

  1. ① Find the sum of the ratio parts.
  2. ② Find the amount in one part: total ÷ sum of ratio parts.
  3. ③ Find each amount: one-part amount × each ratio.
Key Points
  • When using a ratio, first find the “amount in one part” to make the calculation much easier.
  • After dividing, always check that adding all the parts gives the original amount.
  • Use ratios in integer form (numbers that are not decimals or fractions). For example, for 0.5：1, multiply both parts by 2 to make 1：2 before using it.

Practice Problems

  1. Divide 90 candies between Elder Sister and Younger Sister in the ratio 4：5. How many does each receive?
  2. Divide 3600 yen among Elder Brother, Younger Brother, and Younger Sister in the ratio 2：1：3. How much does Elder Brother receive?
  3. Divide land with an area of 540㎡ between A and B in the ratio 5：4. What is the area of A's land?

Answers and Explanations

  1. Answer: Elder Sister＝40 candies, Younger Sister＝50 candies
    Explanation: The sum of the ratio 4：5 is 9. One part＝90÷9＝10 candies. Elder Sister＝10×4＝40 candies. Younger Sister＝10×5＝50 candies. Check: 40＋50＝90✓, 40：50＝4：5✓
  2. Answer: Elder Brother＝1200 yen
    Explanation: The sum of the ratio 2：1：3 is 6. One part＝3600÷6＝600 yen. Elder Brother＝600×2＝1200 yen. Younger Brother＝600×1＝600 yen. Younger Sister＝600×3＝1800 yen. Check: 1200＋600＋1800＝3600✓
  3. Answer: A's land＝300㎡
    Explanation: The sum of the ratio 5：4 is 9. One part＝540÷9＝60㎡. A＝60×5＝300㎡. B＝60×4＝240㎡. Check: 300＋240＝540✓

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