Using Ratios ② Problems Involving Sharing According to a Ratio
“Dividing a whole according to a specified ratio” is a common type of ratio problem.
The key is to think of it as “whole = sum of the ratio (how many units).”
What You Will Learn
Steps for sharing according to a ratio:
① First, add all the parts of the ratio to find “how many units in total.”
② Whole ÷ sum of the ratio = amount in 1 unit.
③ Multiply by how many units each person has to find each person’s amount.
Example
A, B, and C share 4200 yen in the ratio 2:3:1. How much does each person receive?
Solution:
① Find the sum of the ratio. 2 ＋ 3 ＋ 1 ＝ 6 (6 units)
② Calculate the amount in 1 unit. 4200 ÷ 6 ＝ 700 yen
③ Calculate each person’s amount.
A ＝ 700 × 2 ＝ 1400 yen
B ＝ 700 × 3 ＝ 2100 yen
C ＝ 700 × 1 ＝ 700 yen
A, B, and C share 4200 yen in the ratio 2:3:1. How much does each person receive?
Solution:
① Find the sum of the ratio. 2 ＋ 3 ＋ 1 ＝ 6 (6 units)
② Calculate the amount in 1 unit. 4200 ÷ 6 ＝ 700 yen
③ Calculate each person’s amount.
A ＝ 700 × 2 ＝ 1400 yen
B ＝ 700 × 3 ＝ 2100 yen
C ＝ 700 × 1 ＝ 700 yen
Key Points
Remember
- “Whole ÷ sum of the ratio” gives the amount in 1 unit.
- When you finish calculating, check whether the total for everyone is the same as the whole.
- When the difference, rather than the whole, is known, “difference ÷ difference in the ratio” gives the amount in 1 unit.
Practice Problems
- A piece of land with an area of 360 cm² is divided into two parts in the ratio 5:4. Find the area of each part.
- The ratio of the amounts of money A and B have is 3:5, and the difference is 400 yen. Find how much money A and B each have.
- An older brother, a younger brother, and a younger sister share 7200 yen in the ratio 4:2:3. How much does each person receive?
Answers and Explanations
- Answer: One part = 200 cm², the other part = 160 cm²
Explanation: Sum of the ratio = 5+4 = 9. 1 unit = 360÷9 = 40 cm². The 5-part = 40×5 = 200 cm², and the 4-part = 40×4 = 160 cm². - Answer: A = 600 yen, B = 1000 yen
Explanation: A:B=3:5, so the difference in the ratio is 5−3=2. Since the difference is 400 yen, 1 unit=400÷2=200 yen. A=200×3=600 yen, and B=200×5=1000 yen. - Answer: Older brother = 3200 yen, younger brother = 1600 yen, younger sister = 2400 yen
Explanation: Sum of the ratio = 4+2+3 = 9. 1 unit = 7200÷9 = 800 yen. Older brother=800×4=3200 yen, younger brother=800×2=1600 yen, younger sister=800×3=2400 yen. Total=3200+1600+2400=7200 yen ✓.