Introduction
Understanding percentages is crucial for students preparing for competitive exams. Percentages allow us to compare quantities, analyze data, and make informed decisions in various scenarios. One essential skill is finding the whole from a given percentage. This skill not only aids in exam preparation but also enhances everyday mathematical competence. In this article, we’ll explore this concept in a clear, step-by-step manner, with practical examples, exercises, and visual aids like tables to enhance understanding.
What Does It Mean to Find the Whole from a Given Percentage?
When we talk about finding the whole from a percentage, we’re essentially trying to determine the original quantity before the percentage was applied. For instance, if we know that 30% of a quantity is 60, our goal is to find out what the total quantity (the whole) is.
The Formula
To find the whole from a given percentage, we can use the formula:
Whole = (Part / Percentage) × 100
Where:
- Part is the known quantity (the amount you have).
- Percentage is the percentage represented as a whole number (not a decimal).
This formula is derived from the basic concept that a percentage is a fraction of 100.
Step-by-Step Guide to Finding the Whole
Step 1: Identify the Part and Percentage
The first step is to identify the part (the known quantity) and the percentage. Let’s take an example for better clarity.
Example:
Suppose you have 45, which represents 15% of a certain quantity. Here:
- Part = 45
- Percentage = 15%
Step 2: Substitute Values into the Formula
Now, substitute the values you identified into the formula:
Whole = (45 / 15) × 100
Step 3: Perform the Calculation
- Divide the part by the percentage:
(45 / 15) = 3 - Multiply the result by 100:
3 × 100 = 300
So, the whole quantity is 300.
Practical Examples
Let’s look at a few more examples to reinforce this concept.
Example 1: Finding the Whole from a Given Percentage
Problem: If 25% of a quantity is 50, what is the whole quantity?
- Part = 50
- Percentage = 25%
Using the formula:
Whole = (50 / 25) × 100
Calculating:
(50 / 25) = 2
and 2 × 100 = 200
Answer: The whole quantity is 200.
Example 2: Finding the Whole When the Percentage is 50%
Problem: If 50% of a quantity is 150, what is the whole quantity?
- Part = 150
- Percentage = 50%
Using the formula:
Whole = (150 / 50) × 100
Calculating:
(150 / 50) = 3
and 3 × 100 = 300
Answer: The whole quantity is 300.
Example 3: More Complex Scenario
Problem: A student scores 75 marks, which is 60% of the total marks. What are the total marks?
- Part = 75
- Percentage = 60%
Using the formula:
Whole = (75 / 60) × 100
Calculating:
(75 / 60) = 1.25
and 1.25 × 100 = 125
Answer: The total marks are 125.
Example 4: Understanding Different Scenarios
Problem: A shop sells 120 items, which accounts for 30% of its total inventory. How many items does the shop have in total?
- Part = 120
- Percentage = 30%
Using the formula:
Whole = (120 / 30) × 100
Calculating:
(120 / 30) = 4
and 4 × 100 = 400
Answer: The total inventory is 400 items.
Visual Representation: Understanding Percentages with Tables
To further illustrate how to find the whole from a given percentage, here’s a table that summarizes the relationship between the part, percentage, and whole.
| Part | Percentage (%) | Whole |
|---|---|---|
| 20 | 10% | 200 |
| 30 | 15% | 200 |
| 45 | 15% | 300 |
| 50 | 25% | 200 |
| 75 | 60% | 125 |
| 120 | 30% | 400 |
| 100 | 25% | 400 |
The above table helps students visualize how different parts correspond to their respective whole quantities based on various percentages.
Another Visual Representation: Percentage Breakdown
To make the concept clearer, let’s also visualize the concept using a breakdown of percentages:
| Whole Quantity | Percentage | Part |
|---|---|---|
| 200 | 10% | 20 |
| 200 | 15% | 30 |
| 300 | 20% | 60 |
| 400 | 30% | 120 |
| 500 | 25% | 125 |
| 800 | 50% | 400 |
Practice Exercises
To reinforce your understanding, try solving these exercises:
- If 25% of a number is 75, what is the original number?
- 40% of a school’s total students are boys, which is 320 students. What is the total number of students in the school?
- A shop sold 60% of its stock, which amounted to 240 items. What was the total stock before the sale?
- 75% of a tank’s capacity is filled, and this amounts to 450 liters. What is the full capacity of the tank?
- A person saves 20% of their income, which equals Rs. 5,000. What is the total income?
- In an election, 65% of the votes went to the winning candidate. If the candidate received 13,000 votes, what was the total number of votes cast?
- A factory produces 120 defective products, which is 15% of its total production. How many products did the factory produce in total?
- 85% of a country’s population is literate, which corresponds to 340 million people. What is the total population of the country?
- 35% of a class scored above 90 marks, which is 14 students. What is the total number of students in the class?
- A company’s profits dropped by 30%, amounting to a loss of Rs. 3,000. What were the original profits before the drop?
- If 60% of a number is 90, what is the original number?
- A vehicle consumes 50% of its fuel, which is 25 liters. What is the full tank capacity?
- 12% of a town’s population is elderly, which equals 4,800 people. What is the total population of the town?
- A student completed 80% of their homework, which is 40 pages. How many pages of homework were assigned in total?
- 90% of a company’s workforce are permanent employees, totaling 1,800 employees. What is the total workforce of the company?
Answers:
- 300
- 800
- 400
- 600 liters
- Rs. 25,000
- 20,000 votes
- 800 products
- 400 million people
- 40 students
- Rs. 10,000
- 150
- 50 liters
- 40,000 people
- 50 pages
- 2,000 employees
Conclusion
Finding the whole from a given percentage is a fundamental skill that enhances mathematical understanding and practical problem-solving abilities. Mastering this concept prepares students for various challenges in competitive exams and real-life scenarios. Regular practice with examples and exercises will help solidify this knowledge. Remember to break down the problem into manageable steps, apply the formula accurately, and visualize the relationships through tables and graphs for better comprehension. As you continue your preparation for banking exams, ensuring a strong grasp of percentages will undoubtedly benefit you in both the exam room and your everyday life.