NCERT Solutions Class 8 Maths Comparing Quantities Ex 8.1

Understanding Ratios and Percentages: The Core Concepts

Before jumping straight into the numericals, let's understand the math behind comparing quantities. A ratio is a way of comparing two quantities of the same kind by division. For example, if we compare the speed of a cycle (15 km/h) to the speed of a scooter (30 km/h), the ratio is 15 to 30, which simplifies to 1:2. The most critical rule for ratios is that both quantities must be in the identical unit before comparing them. If you try to compare meters to kilometers directly, your math will go wrong! Percentages are simply fractions with a denominator of 100. The word 'percent' comes from 'per centum', meaning 'for every hundred'. Converting ratios to percentages helps us visualize proportions on a standard scale of 100, making comparisons incredibly intuitive.

Step-by-Step Process to Find Ratios with Different Units

1. Step 1: Identify the Units — Look closely at the two given quantities. Note down their respective units (e.g., meters and kilometers, or paise and rupees).
2. Step 2: Convert to a Common Unit — Always convert the larger unit to the smaller unit to avoid working with messy decimals. For example, since ₹1 = 100 paise, ₹5 becomes 500 paise.
3. Step 3: Express as a Fraction — Write the first quantity in the numerator and the second in the denominator. For example: 50 paise / 500 paise.
4. Step 4: Simplify to Lowest Terms — Divide both numbers by their Greatest Common Divisor (GCD) to get the final reduced ratio. 50/500 simplifies to 1/10, written as 1:10.

Exam Tips & Common Pitfalls to Avoid

When preparing for comparing quantities ex 8 1 class 8 ncert, pay extra attention to these typical exam traps:

1. Forgetting Unit Conversions: This is the #1 mistake. If a question asks for the ratio of 5 m to 10 km, writing 5:10 (or 1:2) is completely incorrect. You must convert 10 km to 10,000 m first!
2. Misreading 'Not' in Word Problems: If a question says '72% of students are good at math, how many are not good?', students often calculate 72% of the total and stop. Remember to subtract that count from the total, or directly calculate (100 - 72)% = 28% of the total.
3. Ratio Order Matters: The ratio of A to B is written as A:B (or A/B). It is not the same as B:A. Always write the term mentioned first as the numerator.

Practice Questions with Solutions

• Q: Find the ratio of the speed of a cycle 15 km per hour to the speed of scooter 30 km per hour. A: Step 1: Check if the units are the same. Both speeds are in km per hour (km/h), so no unit conversion is needed. Step 2: Write the speed of the cycle as the numerator and the speed of the scooter as the denominator. Fraction = 15 / 30 Step 3: Simplify the fraction by dividing both terms by their common factor, which is 15. 15 ÷ 15 = 1 30 ÷ 15 = 2 So, the fraction is 1/2. Final answer: The ratio of the speed of the cycle to the speed of the scooter is 1:2.
• Q: Convert the ratio 3:4 into a percentage. A: Step 1: Write the given ratio as a fraction. The ratio 3:4 is written as 3/4. Step 2: To convert any fraction to a percentage, multiply it by 100 and add the percent symbol (%). Percentage = (3/4) × 100% Step 3: Solve the math. Divide 100 by 4, which is 25. Then multiply 3 by 25. 3 × 25 = 75% Final answer: The ratio 3:4 converted to percentage is 75%.
• Q: Out of 25 students, 72% are interested in mathematics. How many students are not interested in mathematics? A: Step 1: Find the percentage of students who are not interested in mathematics. Total percentage = 100% Percentage of students not interested = 100% - 72% = 28% Step 2: Calculate 28% of the total number of students (25). Number of students = 28% of 25 Number of students = (28 / 100) × 25 Step 3: Simplify the terms. Since 25 goes into 100 four times: Number of students = 28 / 4 = 7 Final answer: 7 students are not interested in mathematics.
• Q: A football team won 10 matches out of the total number of matches they played. If their win percentage was 40, how many matches did they play in all? A: Step 1: Let the total number of matches played by the team be 'x'. Step 2: According to the problem, 40% of the total matches played equals 10 matches. So, 40% of x = 10 Step 3: Express this relation as an algebraic equation. (40 / 100) × x = 10 Step 4: Solve for x by isolating the variable. x = (10 × 100) / 40 x = 1000 / 40 x = 25 Final answer: The football team played 25 matches in all.

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