Equivalent Quantity Problems ① Fractions, Decimals, and Percentages (%)
Equivalent quantity problems are problems in which you “consider the whole as 1 (or 100%) and find the whole or the remainder from the amount that represents a part of it.”
For this purpose, let’s learn to convert between fractions, decimals, and percentages (%).
What You Will Learn
Examples of expressing the same quantity in different ways:
1/4 = 0.25 = 25%
“□ wari of something” means □ × 0.1 times it.
For example, “3 wari” means 0.3 times, which has the same meaning as 30%.
Example
After reading 3/5 of all the pages in a book, there were 48 pages left. How many pages are there in the book altogether?
Solution:
① Let the whole book be 1.
② Since 3/5 was read, the remainder is 1 − 3/5 = 2/5.
③ Since the amount corresponding to “2/5” is 48 pages,
Whole = 48 ÷ 2/5 = 48 × 5/2 = 120 pages
After reading 3/5 of all the pages in a book, there were 48 pages left. How many pages are there in the book altogether?
Solution:
① Let the whole book be 1.
② Since 3/5 was read, the remainder is 1 − 3/5 = 2/5.
③ Since the amount corresponding to “2/5” is 48 pages,
Whole = 48 ÷ 2/5 = 48 × 5/2 = 120 pages
Key Points
Remember
- In equivalent quantity problems, the key is to first define the whole as 1 (or 100%).
- Practice until you can freely convert between fractions, decimals, and percentages.
- If you know that “□/△ of the whole is ○○,” you can calculate the whole as ○○ ÷ □/△.
Practice Problems
- Convert 40% to a fraction. Also, convert 7/20 to a percentage (%).
- If 2/3 of a number is 54, find the original number.
- When 80% of the capacity entered, there were still 30 people who had not entered. What is the capacity?
Answers and Explanations
- Answer:40% = 2/5. 7/20 = 35%
Explanation: 40% = 40/100 = 2/5. 7/20 = 7 ÷ 20 × 100 = 35%. To convert a fraction to a percentage, calculate numerator ÷ denominator × 100. - Answer:The number = 81
Explanation: Since the number × 2/3 = 54, the number = 54 ÷ (2/3) = 54 × 3/2 = 81. Change “÷ a fraction” to “× its reciprocal.” - Answer:Capacity = 150 people
Explanation: If the whole is 1, then 80% (= 4/5) entered, so the remaining 20% (= 1/5) is 30 people. Capacity = 30 ÷ (1/5) = 30 × 5 = 150 people.