Equivalent Quantity Problems ②: The Type “△/〇 of the Whole Is 〇〇 Yen”
The most common type of equivalent quantity problem is: “We know what fraction of the whole something is, but we do not know the whole. Let’s find the whole.”
These problems appear in various situations, such as money, the number of pages, and the number of people.
What You Will Learn
For example, if you are told that “3/7 of the whole is 420 yen,”
The whole = 420 ÷ 3/7 = 420 × 7/3 = 980 yen
In this way, when dividing by a fraction, you can “turn it upside down and multiply.”
Example Problem
An older brother and a younger brother have pocket money. 5/8 of the older brother’s pocket money and 3/4 of the younger brother’s pocket money are the same amount, and both are 600 yen. How much pocket money does each brother have?
Solution:
① Since “5/8 of the older brother’s pocket money = 600 yen,”
The older brother’s pocket money = 600 ÷ 5/8 = 600 × 8/5 = 960 yen
② Since “3/4 of the younger brother’s pocket money = 600 yen,”
The younger brother’s pocket money = 600 ÷ 3/4 = 600 × 4/3 = 800 yen
An older brother and a younger brother have pocket money. 5/8 of the older brother’s pocket money and 3/4 of the younger brother’s pocket money are the same amount, and both are 600 yen. How much pocket money does each brother have?
Solution:
① Since “5/8 of the older brother’s pocket money = 600 yen,”
The older brother’s pocket money = 600 ÷ 5/8 = 600 × 8/5 = 960 yen
② Since “3/4 of the younger brother’s pocket money = 600 yen,”
The younger brother’s pocket money = 600 ÷ 3/4 = 600 × 4/3 = 800 yen
Key Points
Remember
- “□/△ of the whole is ○○” → The whole = ○○ ÷ □/△ = ○○ × △/□.
- First, read the problem carefully and make clear “what, what fraction of it, and how many yen (items).”
- After finding the answer, check whether “the whole × the fraction = the original amount.”
Practice Problems
- When 3/8 of the number of students at a school is 150, find the total number of students at the school.
- When 2/5 of a ribbon was cut off, the remainder was 72 cm. How many centimeters long was the ribbon originally?
- 0.6 times the list price of a product is 1,800 yen. What is the list price?
Answers and Explanations
- Answer: Total number of students = 400
Explanation: Since total number of students × 3/8 = 150, total number of students = 150 ÷ (3/8) = 150 × 8/3 = 400. - Answer: Total length of the ribbon = 120 cm
Explanation: Since 2/5 of the whole was cut off, the remainder is 1 − 2/5 = 3/5. Since the whole × 3/5 = 72, the whole = 72 ÷ (3/5) = 72 × 5/3 = 120 cm. - Answer: List price = 3,000 yen
Explanation: Since list price × 0.6 = 1,800, list price = 1,800 ÷ 0.6 = 3,000 yen. “0.6 is 1,800 yen” means “0.6 times the whole is 1,800,” so divide 1,800 by 0.6 to find the whole.