NCERT Solutions for Class 7 Maths Chapter 8 Exercise 8.1 — Comparing Quantities

Welcome to the exciting world of Comparing Quantities! In our daily lives, we often compare heights, weights, prices, and speeds. But how do we do it mathematically? CBSE Class 7 Maths Chapter 8 introduces us to the powerful concept of Ratios, and Exercise 8.1 is the perfect starting point to master it. This exercise focuses on comparing two quantities by division and finding equivalent ratios. Whether you are finding the ratio of speed between a cycle and a car, or comparing your pocket money, these foundational skills are crucial. In this guide, our YoLearn AI Tutor will walk you through the essential rules of ratios, step-by-step conversion techniques, and fully worked-out solutions. By the end of this page, you will not only solve comparing quantities ex 8 1 class 7 ncert problems with ease but also build a solid foundation for percentages and proportion. Let's start sketching out these concepts and solve them together!

Understanding Ratios & Unit Rules

To compare two quantities, we look at how many times one quantity is of another. This comparison by division is known as a ratio. We denote ratios using the colon symbol ( : ). For example, if there are 20 boys and 40 girls in a class, the ratio of boys to girls is 20:40, which simplifies to 1:2.

The most critical rule to remember in ratios is that we can only compare two quantities when they are expressed in the same unit. If one quantity is in kilograms and the other is in grams, we cannot divide them directly. We must first convert them into a common unit. Ratios are pure numbers, which means ratios do not have any units of measurement like 'kg' or 'cm' at the end. They simply tell us the relative scale of the two quantities being compared.

Step-by-Step Guide: How to Find a Ratio

  1. Identify the Units — Look at both quantities. For example, if you are comparing 5 km to 500 m, identify that one is in kilometers (km) and the other is in meters (m).
  2. Convert to a Common Unit — Convert the larger unit to the smaller unit to avoid decimals. Since 1 km = 1000 m, convert 5 km to 5 * 1000 = 5000 m.
  3. Write as a Fraction — Express the quantities as a fraction: Numerator (First Quantity) / Denominator (Second Quantity). Here, it becomes 5000 / 500.
  4. Simplify to Lowest Terms — Divide both the numerator and denominator by their Highest Common Factor (HCF). 5000 / 500 simplifies to 10 / 1. Hence, the ratio is 10:1.

Equivalent Ratios & Lab Problems

  • Example 1: Are 1:2 and 2:3 equivalent ratios? To check, write them as fractions: 1/2 and 2/3. Convert them to like fractions by taking the LCM of denominators (2 and 3), which is 6. 1/2 = (1 3) / (2 3) = 3/6 2/3 = (2 2) / (3 2) = 4/6 Since 3/6 is not equal to 4/6, the ratios 1:2 and 2:3 are NOT equivalent.
  • Example 2: In a computer lab, there are 3 computers for every 6 students. How many computers will be needed for 24 students? Step 1: Ratio of computers to students = 3 / 6 = 1 / 2 (meaning 1 computer for every 2 students). Step 2: Let the number of computers needed for 24 students be 'x'. Step 3: Set up equivalent ratios: 1 / 2 = x / 24. Step 4: Cross-multiply: 2 x = 24 1 => x = 24 / 2 = 12. Thus, 12 computers are needed for 24 students.

Exam Alert: Avoid These Common Mistakes!

  1. Order Matters: The ratio of A to B is written as A:B (or A/B). Writing it as B:A is incorrect and will cost you marks. For example, the ratio of 5 to 10 is 1:2, not 2:1!
  2. Forgetting to Convert Units: Never divide numbers directly if their units are different. Always convert them first. Writing the ratio of ₹5 to 50 paise as 5:50 (which is 1:10) is a very common error. The correct calculation requires converting ₹5 to 500 paise, giving a ratio of 500:50 = 10:1.
  3. Do Not Write Units in Final Ratio: Remember, ratios are dimensionless comparisons. Never write your final answer as '10:1 kg' or '2:3 cm'. Write it simply as '10:1' or '2:3'.

Practice Questions with Solutions

  • Q: Find the ratio of ₹ 5 to 50 paise. A: Step 1: Identify units. One quantity is in rupees (₹) and the other is in paise. Step 2: Convert rupees to paise. Since ₹ 1 = 100 paise, we get ₹ 5 = 5 * 100 = 500 paise. Step 3: Express as a fraction: 500 paise / 50 paise. Step 4: Simplify the fraction. 500 / 50 = 10 / 1. Final answer: The ratio of ₹ 5 to 50 paise is 10:1.
  • Q: Find the ratio of 15 kg to 210 g. A: Step 1: Identify units. One is in kilograms (kg) and the other is in grams (g). Step 2: Convert kg to grams. Since 1 kg = 1000 g, we get 15 kg = 15 * 1000 = 15000 g. Step 3: Express as a fraction: 15000 g / 210 g = 15000 / 210. Step 4: Simplify by dividing both by 30 (their common factor). 15000 / 30 = 500, and 210 / 30 = 7. Final answer: The ratio of 15 kg to 210 g is 500:7.
  • Q: Find the ratio of 9 m to 27 cm. A: Step 1: Identify units. One is in meters (m) and the other is in centimeters (cm). Step 2: Convert meters to centimeters. Since 1 m = 100 cm, we get 9 m = 9 * 100 = 900 cm. Step 3: Express as a fraction: 900 cm / 27 cm = 900 / 27. Step 4: Simplify by dividing the numerator and denominator by 9. 900 / 9 = 100, and 27 / 9 = 3. Final answer: The ratio of 9 m to 27 cm is 100:3.
  • Q: Find the ratio of 30 days to 36 hours. A: Step 1: Convert days to hours. Since 1 day = 24 hours, we get 30 days = 30 * 24 = 720 hours. Step 2: Express as a fraction of hours: 720 hours / 36 hours. Step 3: Simplify by dividing both by 36. 720 / 36 = 20, and 36 / 36 = 1. Final answer: The ratio of 30 days to 36 hours is 20:1.

Frequently Asked Questions

Why do we need to convert quantities to the same unit before finding their ratio?

Ratios compare the relative sizes of two quantities. If the units are different (e.g., comparing meters to kilometers directly), the numerical division will not represent the true relationship between the sizes.

Does a ratio have any units?

No, ratios do not have any units. Since a ratio is obtained by dividing two quantities of the same unit, the units cancel each other out, leaving a pure numerical value.

What are equivalent ratios?

Equivalent ratios are ratios that express the same relationship between numbers. You can find them by multiplying or dividing both terms of a ratio by the same non-zero number.