Recovering the whole from a known percentage

Introduction

In percentage problems, you are often given a part of a quantity and asked to find the whole. This is the reverse of finding a percentage of a number. The key idea is simple: if a known part represents a certain percent, then the whole represents 100%.

Core idea

If p% of a number is known and equals part, then

whole = part × 100 / p

This formula works because percentage means “per 100.” When you know the part and the percent, you can scale back to the full amount.

How to think about it

1. Identify the known percentage.
2. Write the part value.
3. Divide by the percentage as a fraction of 100, or multiply by 100 and divide by the percent.
4. Check whether the answer is reasonable.

Worked example 1

A shopkeeper says 18 candies are 30% of the total candies in a jar. Find the total number of candies.

Since 30% of the whole is 18,

whole = 18 × 100 / 30

whole = 60

So, the jar has 60 candies.

Worked example 2

45 students are 75% of the total students in a class. Find the total number of students.

whole = 45 × 100 / 75

whole = 60

So, the total class strength is 60 students.

Common mistakes

• Treating the given part as the whole.
• Using multiplication instead of reversing the percentage relation.
• Forgetting that 100% represents the entire quantity.
• Mixing up the part percentage and the total percentage.

Practice

1. 24 books are 40% of the total books on a shelf. Find the total number of books.
2. 36 seats are 60% of the seats in a hall. Find the total number of seats.

Recap

When a percentage value of the whole is known, recover the whole by using whole = part × 100 / percent. Always identify the part, the percent, and then scale up to 100%.

MCQ Practice

Question 1

27 is 30% of which number?

A. 70

B. 80

C. 90

D. 100

Answer: C. 90

Solution: If 27 is 30% of the whole, then whole = 27 × 100 / 30 = 90.

Question 2

A number has 56 as 70% of it. What is the number?

A. 70

B. 75

C. 80

D. 85

Answer: C. 80

Solution: whole = 56 × 100 / 70 = 80.

Analogy

You are holding the red beads, and you are told they are 30 out of every 100 beads on the string. 18 red beads at 30% means the full string is 18 x 100 / 30 = 60.

Quick reference

• whole = part x 100 / percent.
• 27 is 30% of 90.
• 56 is 70% of 80.