Percentages Class 8th ICSE Maths / Selina Chapter 7

Introduction

In our previous classes, we have learned about fractions and decimals. We know how to compare two fractions by making their denominators the same. However, in our daily lives, we often need a much simpler and quicker way to compare different quantities. This is where the concept of Percent becomes very important. Percent is a powerful mathematical tool used to compare results, even when the total marks or quantities are different.

Let us understand this with the help of an example. Suppose Student A scores 25 marks out of 50, and Student B scores 75 marks out of 100. At first glance, it is hard to say who performed better because their total marks are different. But if we convert these scores into a standard form “out of 100,” we find that Student A scored 50% and Student B scored 75%. Now, the comparison is very easy! We can clearly see that Student B performed better.

We use percentages everywhere in real life. When you see a “25% Discount” sign at a shop, or when a bank says the interest rate for a gold loan is 11%, they are using percentages. Even the news reports changes in petrol prices or your father’s salary increase in the form of percentages. Because it is so useful for comparing data and tracking changes, understanding this chapter is essential for every student.

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What is Percent?

The word Percent comes from two words: “Per” (which means “for every”) and “Cent” (which means “100”). So, the literal meaning of percent is “for every 100” or “out of 100”.

In simple words, if we say a student got 52% marks, it means they scored 52 marks for every 100 marks available. If we say 30% of students are present in a class of 100, it means exactly 30 students are sitting in the classroom.

Key Conditions for Percent:

1. The base or total must always be considered as 100.
2. The symbol ‘%’ is used to represent percent.
3. When a fraction has 100 as the denominator, the numerator is called the Rate Percent.

Examples of Percent:

• 40% students means 40 out of 100 are absent.
• 82% marks means 82 out of 100.
• 10% discount means ₹10 off for every ₹100.
• 100% marks means scoring the full marks (e.g., 50 out of 50).

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Examples (Identification Type)

Let us look at some examples to identify valid percentage representations:

1. 82/100: This is a valid percentage (82%) because the denominator is 100.
2. 30/100: This is 30% because it is 30 out of 100.
3. 12/20: This is NOT yet a percentage because the denominator is 20, not 100.
4. 42/100: This is 42% because it is out of 100.
5. 3/5: This is a fraction, not a percentage, until we multiply it by 100.
6. 0.5: This is a decimal. To make it a percentage, we must multiply by 100.
7. 60%: This is a valid percentage form.
8. 200/100: This is 200%, a valid percentage.
9. 50 out of 50: This is 100% because it equals the whole amount.

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General Form / Standard Form

In mathematics, we represent percentages in a standard way using fractions.

The Rate Percent Form: Any percentage x% can be written as a fraction: Percentage=100Numerator (Rate Percent)​

For example, if the rate percent is 82, the fraction is 82/100.

Conversion Rules:

1. To change a fraction to a percentage: Multiply the fraction by 100.

• Example: 3/5×100=60%.

1. To change a percentage to a fraction: Remove the ‘%’ sign and divide by 100.

• Example: 80%=80/100=4/5.

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Which are NOT Percentages?

It is important to know what is not a percentage so you don’t get confused:

1. Pure Numbers (e.g., 50): Without the ‘%’ sign or a denominator of 100, this is just a whole number.
2. Fractions with non-100 denominators (e.g., 12/20): While this represents a part, it is not “percent” until converted.
3. Ratios (e.g., 2:3): This is a comparison but not a percentage until calculated out of 100.
4. Decimals (e.g., 0.45): This must be multiplied by 100 to become 45%.
5. Algebraic Expressions (e.g., x+5): This is an expression, not a fixed percentage value.

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Parts and Important Terms

When solving percentage problems, especially word problems, we use terms from Linear Equations:

1. Variable: We use a letter like ‘x’ to represent an unknown value, such as a man’s total income.
2. Constant: These are fixed numbers given in the problem, like “₹160” or “800”.
3. Coefficient: The number multiplied by the variable, like 1/5 in the term x/5.
4. LHS (Left Hand Side): The part of the equation before the equal sign.
5. RHS (Right Hand Side): The part of the equation after the equal sign.
6. Solution: The final value of the variable (e.g., x=200) that makes the equation true.

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How to Solve Percentage Problems

To solve complex percentage problems, we follow a step-by-step method similar to solving equations:

Step 1: Assume the Unknown Usually, we start by saying “Let the original number/income be x”.

Step 2: Find the Percentage Value Calculate the specific percentage of x. For example, 20% of x is x×20/100=x/5.

Step 3: Form the Equation If a number is decreased, we subtract: x−x/5=Result.

Step 4: Balance the Equation

• On transposing: If a number is dividing on the LHS, it will multiply on the RHS.
• On dividing both sides: If a number is multiplying the variable, we divide the RHS by that number to find x.

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Solved Examples

Example 1) A bag has 8 white balls and 12 black balls. Find the percentage of black balls. Solution — Total balls = 8+12=20. Fraction of black balls = 12/20. To find the percentage, multiply by 100: (12/20)×100=60. Ans. 60%.

Example 2) Convert 3/5 into a percentage. Solution — Multiply the fraction by 100: 3/5×100=60. Ans. 60%.

Example 3) Convert 45% into a fraction in its simplest form. Solution — Remove the ‘%’ sign and divide by 100: 45/100. Divide both by 5: 9/20. Ans. 9/20.

Example 4) A number increases by 20% and the result then decreases by 20%. Find the net change percent. Solution — Let the original number be 100. Increase of 20% = 100+20=120. Now, decrease the result (120) by 20%: 20% of 120=24. New number = 120−24=96. Change = 100−96=4. Since it is out of 100, the change is 4%. Ans. 4% decrease.

Example 5) After paying 20% of his income, a man is left with ₹160. Find his income. Solution — Let the income be x. Payment = 20% of x=x/5. Remaining = x−x/5=160. (5x−x)/5=160. 4x/5=160. On transposing 5 to the RHS: 4x=160×5=800. On dividing by 4: x=800/4=200. Ans. ₹200.

Example 6) A number is decreased by 20%. If the resulting number is 800, find the original number. Solution — Let the number be x. x−(20/100)x=800. x−x/5=800. 4x/5=800. 4x=4000. x=1000. Ans. 1000.

Example 7) Total of A, B, and C is 400. A is 40% and B is 45%. Find the quantity of C. Solution — Quantity of A = 40% of 400=160. Quantity of B = 45% of 400=180. Total of A + B = 160+180=340. Quantity of C = 400−340=60. Ans. 60.

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Practice Questions

1. Convert 4/25 into a percentage.
2. Express 75% as a fraction in the lowest terms.
3. A student scores 18 marks out of 25. What is their percentage?
4. If a salary of ₹2000 increases by 10%, find the new salary.
5. Find x if 15% of x is 45.
6. A bag contains 5 red and 15 blue marbles. Find the percentage of red marbles.
7. A shop offers a 30% discount on a shirt worth ₹500. Find the discount amount.
8. A number is increased by 10% and then decreased by 10%. Find the total change.

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Note Section

• Note 1: Always ensure the “Original Quantity” is in the denominator when calculating increase or decrease percentage.
• Note 2: To convert any fraction to a percentage, simply multiply by 100.
• Note 3: When solving for an unknown income or number, it is often easiest to assume the variable is ‘x’ or the original number is ‘100’.
• Note 4: Percentage means “out of 100,” so 100% represents the whole or complete amount.

Conclusion

In this chapter, we learned that Percentage is simply a way of expressing numbers as a fraction of 100. Whether we are comparing marks, calculating bank interest, or finding discounts at a shop, percentages make math easy and practical. By mastering the conversion between fractions and percentages and learning to solve equations for unknown values, you can handle any problem in Exercise 7(a) with confidence!