Equivalent Quantity Problems ③ Solving Clearly with Line Segment Diagrams
In problems involving several fractions, such as “using a certain fraction of the whole, and then taking a certain fraction of what remains…,” drawing a line segment diagram makes the relationships immediately clear.
Especially when a problem seems difficult, try drawing a line segment diagram.
What You Will Learn
How to draw a line segment diagram:
① Represent the whole with one straight line.
② Divide the line at points corresponding to known fractions.
③ Write in known quantities (such as ○ yen or ○ people).
④ Calculate the parts that are not yet known.
Example
One-fourth of the purchase price of an item (the price the store paid) was the profit, and that profit was 450 yen. Find the selling price of the item (the price at which it was sold to the customer).
Solution:
① Since “1/4 of the purchase price = 450 yen,”
purchase price = 450 ÷ 1/4 = 450 × 4 = 1800 yen.
② Selling price = purchase price + profit = 1800 + 450 = 2250 yen.
[Line Segment Diagram]
|←── Purchase price (1800 yen) ──→|← Profit (450 yen) →|
|←────── Selling price (2250 yen) ─────────────→|
One-fourth of the purchase price of an item (the price the store paid) was the profit, and that profit was 450 yen. Find the selling price of the item (the price at which it was sold to the customer).
Solution:
① Since “1/4 of the purchase price = 450 yen,”
purchase price = 450 ÷ 1/4 = 450 × 4 = 1800 yen.
② Selling price = purchase price + profit = 1800 + 450 = 2250 yen.
[Line Segment Diagram]
|←── Purchase price (1800 yen) ──→|← Profit (450 yen) →|
|←────── Selling price (2250 yen) ─────────────→|
Key Points
Remember
- In a line segment diagram, let “the whole = 1” and write in each fraction.
- When a problem says “a certain fraction of the remainder is ○○,” first find the amount that remains.
- The more difficult the problem, the more you can reduce mistakes by drawing a diagram and organizing the information.
Practice Problems
- When a number that is 20 greater than one-third of a certain number is 80, find the original “certain number.”
- Three people, A, B, and C, have some money. A has 2/3 as much as B, and B has 3/4 as much as C. If C has 1200 yen, how much do A and B have, respectively?
- A person used 2/5 of the money they had at first, and then used 1/3 of the remaining money. After that, 280 yen remained. Find how much money they had at first.
Answers and Explanations
- Answer: Certain number = 180
Explanation: “A number 20 greater than one-third of a certain number is 80” → certain number × 1/3 + 20 = 80. Therefore, certain number × 1/3 = 60, so the certain number = 60 × 3 = 180. Check with a line segment diagram: 1/3 of the whole is 60, so the whole is 60 × 3 = 180. - Answer: B = 900 yen, A = 600 yen
Explanation: B = C × 3/4 = 1200 × 3/4 = 900 yen. A = B × 2/3 = 900 × 2/3 = 600 yen. Substitute the values in order to find the answers. - Answer: Initial amount of money = 700 yen
Explanation: Let the whole be 1. After using 2/5, the remaining fraction is 1 − 2/5 = 3/5. Since 1/3 of the remainder was then used, the fraction remaining is 3/5 × (1 − 1/3) = 3/5 × 2/3 = 2/5. Since whole × 2/5 = 280, whole = 280 ÷ (2/5) = 280 × 5/2 = 700 yen.