NCERT Solutions for Class 7 Maths Chapter 7: Comparing Quantities — Complete Guide with Step-by-Step Solutions

Why Comparing Quantities Is One of the Most Practical Chapters

If there is one chapter in Class 7 Maths that you will use every single day of your life, it is this one. Ratios help you compare prices. Percentages appear in exam scores, discounts, and battery levels. Profit and loss are the foundation of every business. Simple interest governs bank accounts and loans.

Chapter 7 brings together all these real-world concepts into a systematic framework. It builds on your fraction and decimal skills from Chapter 2 and prepares you for the more advanced Comparing Quantities chapter in Class 8 (which adds compound interest and sales tax).

The NCERT textbook organises this chapter into 4 exercises:
- Exercise 7.1: Ratios and equivalent ratios
- Exercise 7.2: Converting to percentages and finding percentages of quantities
- Exercise 7.3: Profit, loss, and selling price calculations
- Exercise 7.4: Simple interest

This guide covers 8-10 solved problems from each exercise with full working, explains every formula, and gives you a clear exam strategy.

Ratios — The Foundation of Comparison

A ratio compares two quantities of the same kind. The ratio of $a$ to $b$ is written as $a : b$ or $\frac{a}{b}$.

Key rules for ratios:
1. Both quantities must be in the same units before forming a ratio.
2. Ratios have no units (they are pure numbers).
3. Ratios should be expressed in simplest form (divide by the HCF).
4. The order matters: $3 : 5 \neq 5 : 3$.

Equivalent ratios: Just like equivalent fractions, you can multiply or divide both terms by the same number: $2 : 3 = 4 : 6 = 6 : 9$.

Proportion

Two ratios are in proportion if they are equal: $a : b = c : d$ or $\frac{a}{b} = \frac{c}{d}$.

This can also be written as $a : b :: c : d$ (read as "$a$ is to $b$ as $c$ is to $d$").

Cross-multiplication test: $a : b = c : d$ if and only if $a \times d = b \times c$.

Example: Is $3 : 5 = 9 : 15$?
$3 \times 15 = 45$ and $5 \times 9 = 45$. Since $45 = 45$, yes, they are in proportion.

Exercise 7.1 — Ratios and Equivalent Ratios

This exercise tests your ability to form, simplify, and work with ratios.

Solved Example 1: Unit Conversion Before Ratio

Problem: Find the ratio of $3$ km to $300$ m.

Solution:
Convert to the same unit: $3$ km $= 3000$ m.

Answer: $10 : 1$.

Solved Example 2: Map Scale

Problem: A map uses a scale of $1$ cm $: 10$ km. If two cities are $4.5$ cm apart on the map, find the actual distance.

Solution:

Answer: $45$ km.

Solved Example 3: Ratio of Pass to Fail

Problem: Out of $40$ students, $25$ passed. Find the ratio of passed to failed.

Solution:
Passed $= 25$, Failed $= 40 - 25 = 15$.

Answer: $5 : 3$.

Solved Example 4: Dividing in a Given Ratio

Problem: Divide Rs. $600$ in the ratio $2 : 3$.

Solution:
Sum of parts $= 2 + 3 = 5$.

First share $= \frac{2}{5} \times 600 = \text{Rs. } 240$.

Second share $= \frac{3}{5} \times 600 = \text{Rs. } 360$.

Verification: $240 + 360 = 600$. Correct.

Solved Example 5: Finding the Ratio

Problem: A ribbon is cut into two pieces of lengths $45$ cm and $30$ cm. Find the ratio of the longer piece to the shorter piece.

Solution:

Answer: $3 : 2$.

Solved Example 6: Three-Part Ratio

Problem: Divide Rs. $1200$ among $A$, $B$, $C$ in the ratio $2 : 3 : 5$.

Solution:
Sum of parts $= 2 + 3 + 5 = 10$.

$A = \frac{2}{10} \times 1200 = \text{Rs. } 240$.

$B = \frac{3}{10} \times 1200 = \text{Rs. } 360$.

$C = \frac{5}{10} \times 1200 = \text{Rs. } 600$.

Verification: $240 + 360 + 600 = 1200$. Correct.

Solved Example 7: Comparing Ratios

Problem: Which is a better deal: $3$ kg for Rs. $45$ or $5$ kg for Rs. $80$?

Solution:
Price per kg in Deal 1: $\frac{45}{3} = \text{Rs. } 15$ per kg.
Price per kg in Deal 2: $\frac{80}{5} = \text{Rs. } 16$ per kg.

Deal 1 is cheaper per kg, so Deal 1 is better.

Solved Example 8: Ratio with Time

Problem: Find the ratio of $45$ minutes to $1.5$ hours.

Solution:
Convert to the same unit: $1.5$ hours $= 90$ minutes.

Answer: $1 : 2$.

Exercise 7.2 — Percentages

Percentage means "per hundred." A percentage is a fraction with denominator $100$.

Solved Example 1: Fraction to Percentage

Problem: Convert $\frac{3}{8}$ to a percentage.

Solution:

Answer: $37.5\%$.

Solved Example 2: Decimal to Percentage

Problem: Convert $0.45$ to a percentage.

Solution:

Answer: $45\%$.

Solved Example 3: Percentage of a Number

Problem: Find $15\%$ of $250$.

Solution:

Answer: $37.5$.

Solved Example 4: Percentage Increase

Problem: A price increased from Rs. $200$ to Rs. $250$. Find the percentage increase.

Solution:
Increase $= 250 - 200 = \text{Rs. } 50$.

Answer: $25\%$ increase.

Solved Example 5: Percentage Decrease

Problem: The number of students decreased from $80$ to $68$. Find the percentage decrease.

Solution:
Decrease $= 80 - 68 = 12$.

Answer: $15\%$ decrease.

Solved Example 6: Finding the Whole from Percentage

Problem: $30\%$ of a number is $45$. Find the number.

Solution:

Answer: The number is $150$.

Solved Example 7: Percentage of Marks

Problem: Rahul scored $420$ out of $500$. Find his percentage.

Solution:

Answer: $84\%$.

Solved Example 8: Boys and Girls

Problem: A school has $800$ students. If $60\%$ are girls, how many boys are there?

Solution:
Girls $= 60\%$ of $800 = \frac{60}{100} \times 800 = 480$.

Boys $= 800 - 480 = 320$.

Alternatively: Boys $= 40\%$ of $800 = \frac{40}{100} \times 800 = 320$.

Answer: $320$ boys.

Solved Example 9: Percentage to Fraction

Problem: Convert $12.5\%$ to a fraction.

Solution:

Answer: $\frac{1}{8}$.

Solved Example 10: Successive Percentage Changes

Problem: A price of Rs. $100$ increases by $10\%$ and then decreases by $10\%$. Is the final price Rs. $100$?

Solution:
After $10\%$ increase: $100 + 10 = \text{Rs. } 110$.
After $10\%$ decrease: $110 - 11 = \text{Rs. } 99$.

The final price is Rs. $99$, NOT Rs. $100$.

Key insight: A percentage increase followed by the same percentage decrease does NOT bring you back to the original value. There is always a net decrease of $\left(\frac{x}{10}\right)^2 \%$ for an $x\%$ change.

Exercise 7.3 — Profit, Loss, and Discount

Profit and loss are always calculated on the Cost Price (CP).

Key formulae:
- Profit $=$ SP $-$ CP (when SP $>$ CP)
- Loss $=$ CP $-$ SP (when CP $>$ SP)
- Profit $\% = \frac{\text{Profit}}{\text{CP}} \times 100$
- Loss $\% = \frac{\text{Loss}}{\text{CP}} \times 100$
- SP $= \text{CP} \times \left(1 + \frac{\text{Profit}\%}{100}\right)$ or $\text{CP} \times \left(1 - \frac{\text{Loss}\%}{100}\right)$

Solved Example 1: Finding Profit Percentage

Problem: A shopkeeper buys a toy for Rs. $250$ and sells it for Rs. $300$. Find the profit percentage.

Solution:
Profit $= 300 - 250 = \text{Rs. } 50$.

Answer: $20\%$ profit.

Solved Example 2: Finding Loss Percentage

Problem: A shirt bought for Rs. $400$ is sold for Rs. $380$. Find the loss percentage.

Solution:
Loss $= 400 - 380 = \text{Rs. } 20$.

Answer: $5\%$ loss.

Solved Example 3: Finding SP from CP and Loss%

Problem: CP of an article is Rs. $1500$ and loss is $10\%$. Find the SP.

Solution:
Loss $= \frac{10}{100} \times 1500 = \text{Rs. } 150$.

Answer: Rs. $1350$.

Solved Example 4: Finding CP from SP and Profit%

Problem: SP $=$ Rs. $540$ and profit $= 8\%$. Find the CP.

Solution:
SP $= \text{CP} + 8\%$ of CP $= 1.08 \times \text{CP}$.

Answer: Rs. $500$.

Solved Example 5: Finding SP for a Desired Profit

Problem: A merchant bought goods for Rs. $2000$. At what price should he sell to make $15\%$ profit?

Solution:

Answer: Rs. $2300$.

Solved Example 6: Selling at a Gain vs. Loss

Problem: An article bought for Rs. $600$ is sold at a gain of $25\%$. Find the SP. If instead it was sold at a loss of $25\%$, what would the SP be?

Solution:
With gain: SP $= 600 \times 1.25 = \text{Rs. } 750$.

With loss: SP $= 600 \times 0.75 = \text{Rs. } 450$.

Note: The same CP and same percentage give different SPs depending on profit or loss.

Solved Example 7: Discount Problem

Problem: Marked price of a bag is Rs. $800$. A discount of $15\%$ is offered. Find the selling price.

Solution:
Discount $= \frac{15}{100} \times 800 = \text{Rs. } 120$.

Answer: Rs. $680$.

Solved Example 8: Finding CP When Only Profit and SP Are Known

Problem: A man sold a bicycle for Rs. $1540$ making a profit of Rs. $140$. Find the CP and profit percentage.

Solution:

Answer: CP $=$ Rs. $1400$, Profit $= 10\%$.

Exercise 7.4 — Simple Interest

Simple Interest (SI) is the interest calculated only on the original principal amount.

where $P =$ Principal, $R =$ Rate of interest per annum (%), $T =$ Time in years.

Amount $= P + \text{SI}$ (the total money after interest).

Solved Example 1: Basic SI Calculation

Problem: Find the simple interest on Rs. $5000$ at $8\%$ per annum for $3$ years.

Solution:

Answer: Rs. $1200$.

Solved Example 2: Finding the Amount

Problem: Find the amount to be paid after $2$ years on Rs. $10000$ at $12\%$ per annum.

Solution:

Answer: Rs. $12400$.

Solved Example 3: Finding the Rate

Problem: At what rate will Rs. $4000$ earn Rs. $640$ as SI in $2$ years?

Solution:

Answer: $8\%$ per annum.

Solved Example 4: Finding the Time

Problem: In how much time will Rs. $2000$ amount to Rs. $2600$ at $5\%$ per annum?

Solution:
SI $= 2600 - 2000 = \text{Rs. } 600$.

Answer: $6$ years.

Solved Example 5: Finding the Principal

Problem: The SI on a certain sum for $3$ years at $10\%$ is Rs. $900$. Find the principal.

Solution:

Answer: Rs. $3000$.

Solved Example 6: SI with Months

Problem: Find the SI on Rs. $6000$ at $10\%$ per annum for $6$ months.

Solution:
$6$ months $= \frac{6}{12} = 0.5$ years.

Answer: Rs. $300$.

Solved Example 7: Doubling the Principal

Problem: In how many years will Rs. $5000$ double itself at $10\%$ SI per annum?

Solution:
For the amount to double, SI must equal the principal: SI $= \text{Rs. } 5000$.

Answer: $10$ years.

Shortcut: Time to double at $R\%$ SI $= \frac{100}{R}$ years.

Solved Example 8: Comparing Two Investments

Problem: Which gives more interest: Rs. $5000$ at $12\%$ for $2$ years, or Rs. $6000$ at $10\%$ for $2$ years?

Solution:
Investment 1: SI $= \frac{5000 \times 12 \times 2}{100} = \text{Rs. } 1200$.

Investment 2: SI $= \frac{6000 \times 10 \times 2}{100} = \text{Rs. } 1200$.

Both give the same interest of Rs. $1200$.

Answer: Both investments give equal interest.

Common Mistakes Students Make in Comparing Quantities

1. Not Converting to Same Units Before Ratio:
2. * Mistake: Ratio of $3$ km to $300$ m $= 3 : 300 = 1 : 100$.
3. * Fix: Convert first: $3$ km $= 3000$ m. Ratio $= 3000 : 300 = 10 : 1$.
4. Profit/Loss Percentage on SP Instead of CP:
5. * Mistake: Profit $\% = \frac{\text{Profit}}{\text{SP}} \times 100$.
6. * Fix: Profit/loss percentage is ALWAYS calculated on CP, not SP.
7. Confusing SI Formula Parameters:
8. * Mistake: Using months directly as $T$ without converting to years.
9. * Fix: $T$ must be in years. $6$ months $= 0.5$ years. $18$ months $= 1.5$ years.
10. Wrong Percentage Calculation:
11. * Mistake: $\frac{3}{8} \times 100 = \frac{3}{800}$.
12. * Fix: $\frac{3}{8} \times 100 = \frac{300}{8} = 37.5\%$.
13. Thinking Equal Increase and Decrease Cancel Out:
14. * Mistake: $+10\%$ then $-10\%$ gives original price.
15. * Fix: No! $100 \to 110 \to 99$. There is a net loss of $1\%$.
16. Forgetting That Discount Is on Marked Price:
17. * Mistake: Calculating discount on CP.
18. * Fix: Discount is always on the Marked Price (MRP), not on CP.

Exam Strategy for Chapter 7

Practice Problems for Self-Assessment

Problem 1: Convert $\frac{5}{8}$ to a percentage.

Problem 2: Divide Rs. $750$ in the ratio $3 : 2$.

Problem 3: CP $=$ Rs. $800$, SP $=$ Rs. $920$. Find profit percentage.

Problem 4: Find SI on Rs. $8000$ at $6\%$ for $2.5$ years.

Problem 5: At what rate will Rs. $3000$ give SI of Rs. $450$ in $3$ years?

Problem 6: A shirt marked at Rs. $1200$ is sold at $20\%$ discount. Find SP.

Answers to Practice Problems

Answer 1: $\frac{5}{8} \times 100 = 62.5\%$.

Answer 2: Sum $= 5$. First $= \frac{3}{5} \times 750 = 450$. Second $= \frac{2}{5} \times 750 = 300$.

Answer 3: Profit $= 920 - 800 = 120$. Profit $\% = \frac{120}{800} \times 100 = 15\%$.

Answer 4: SI $= \frac{8000 \times 6 \times 2.5}{100} = \frac{120000}{100} = \text{Rs. } 1200$.

Answer 5: $450 = \frac{3000 \times R \times 3}{100}$. $R = \frac{450 \times 100}{9000} = 5\%$.

Answer 6: Discount $= \frac{20}{100} \times 1200 = 240$. SP $= 1200 - 240 = \text{Rs. } 960$.

Quick Revision: All Formulas at a Glance

Connecting to Other Chapters

Chapter 2 (Fractions and Decimals): Percentage calculations use fraction and decimal operations extensively.

Class 8 (Comparing Quantities): Extends to compound interest, sales tax/VAT, and successive discounts.

Class 8 (Direct and Inverse Proportions): Ratio concepts from this chapter lead directly to proportion and variation.

Mastering ratios, percentages, and simple interest now prepares you for all commercial mathematics in higher classes.

Common Mistakes Students Make in Comparing Quantities

This chapter is a marks goldmine if you know the formulas, and a marks wasteland if you do not read the question carefully. Here are ten mistakes that ruin otherwise perfect answer sheets.

Mistake 1: Calculating profit percentage on selling price
❌ Wrong: CP $=$ Rs. $400$, SP $=$ Rs. $500$. Profit $\% = \tfrac{100}{500} \times 100 = 20\%$.
✅ Correct: Profit $\% = \tfrac{100}{400} \times 100 = 25\%$.
Profit and loss percentages are always calculated on cost price, never on selling price.

Mistake 2: Forgetting to convert units before forming a ratio
❌ Wrong: Ratio of $50$ cm to $2$ m is $50 : 2 = 25 : 1$.
✅ Correct: Convert $2$ m to $200$ cm. Ratio $= 50 : 200 = 1 : 4$.
Both quantities in a ratio must be in the same unit.

Mistake 3: Using time in months when computing simple interest
❌ Wrong: For $P = 2000$, $R = 10\%$, $T = 6$ months, writing SI $= \tfrac{2000 \times 10 \times 6}{100} = 1200$.
✅ Correct: Convert $6$ months to $0.5$ years. SI $= \tfrac{2000 \times 10 \times 0.5}{100} = 100$.
The $T$ in the SI formula is always in years.

Mistake 4: Writing percentage without a percent sign
❌ Wrong: $45$ when the answer should be a percentage.
✅ Correct: $45\%$.
CBSE markers deduct half a mark for the missing symbol.

Mistake 5: Adding percentages instead of compounding
❌ Wrong: After a $10\%$ rise followed by a $10\%$ fall, the final value is the same as the original.
✅ Correct: Starting at $100$, after $10\%$ rise you have $110$. After $10\%$ fall you have $99$. So the value drops by $1\%$ overall.
Percentages do not add when applied to different bases.

Mistake 6: Forgetting that discount is on marked price
❌ Wrong: Discount $\%$ is calculated on the selling price.
✅ Correct: Discount $\% = \tfrac{\text{Discount}}{\text{Marked Price}} \times 100$.

Mistake 7: Confusing increase and decrease problems
❌ Wrong: Price increased from $200$ to $250$. Percentage change $= \tfrac{50}{250} \times 100 = 20\%$.
✅ Correct: $\tfrac{50}{200} \times 100 = 25\%$.
Percentage change is on the original value, not the new value.

Mistake 8: Wrong ratio when dividing a quantity
❌ Wrong: Dividing Rs. $600$ in the ratio $2 : 3$ as $300$ and $300$.
✅ Correct: Total parts $= 5$. First share $= \tfrac{2}{5} \times 600 = 240$. Second share $= \tfrac{3}{5} \times 600 = 360$.
Always sum the ratio parts first, then split.

Mistake 9: Forgetting the amount formula
❌ Wrong: In an SI problem asking for the final amount, only writing the SI.
✅ Correct: Amount $=$ Principal $+$ SI.
Read the question carefully to see whether it asks for SI or total amount.

Mistake 10: Using percentages of the wrong total
❌ Wrong: $40\%$ of $60\%$ of $500 = 40\%$ of $500 = 200$.
✅ Correct: $60\%$ of $500 = 300$. Then $40\%$ of $300 = 120$.
When you see nested percentages, apply them one at a time.

Previous Year Questions (PYQs) with Solutions

These questions represent the CBSE Class 7 paper style for Comparing Quantities.

Q1. Convert $\tfrac{3}{4}$ to a percentage. `(CBSE 2021, 1 mark)`
Solution: $\tfrac{3}{4} \times 100 = 75\%$.
Answer: $75\%$

Q2. Divide Rs. $900$ in the ratio $2 : 3 : 4$. `(CBSE 2022, 3 marks)`
Solution: Total parts $= 9$.
First $= \tfrac{2}{9} \times 900 = 200$.
Second $= \tfrac{3}{9} \times 900 = 300$.
Third $= \tfrac{4}{9} \times 900 = 400$.
Answer: Rs. $200$, Rs. $300$, Rs. $400$.

Q3. A man buys a watch for Rs. $800$ and sells it for Rs. $920$. Find his profit percentage. `(CBSE 2021, 2 marks)`
Solution: Profit $= 920 - 800 = 120$.
Profit $\% = \tfrac{120}{800} \times 100 = 15\%$.
Answer: $15\%$

Q4. Find the simple interest on Rs. $5000$ at $8\%$ per annum for $3$ years. `(CBSE 2022, 2 marks)`
Solution: SI $= \tfrac{5000 \times 8 \times 3}{100} = 1200$.
Answer: Rs. $1200$

Q5. A shopkeeper marks the price of a shirt at Rs. $1500$ and offers a discount of $20\%$. Find the selling price. `(CBSE 2023, 3 marks)`
Solution: Discount $= \tfrac{20}{100} \times 1500 = 300$.
SP $= 1500 - 300 = 1200$.
Answer: Rs. $1200$

Q6. At what rate of simple interest will Rs. $6000$ amount to Rs. $7800$ in $3$ years? `(CBSE 2023, 3 marks)`
Solution: SI $= 7800 - 6000 = 1800$.
$R = \tfrac{\text{SI} \times 100}{P \times T} = \tfrac{1800 \times 100}{6000 \times 3} = 10\%$.
Answer: $10\%$

Q7. A shopkeeper sells an article at a loss of $10\%$. Had he sold it for Rs. $60$ more, he would have earned a profit of $5\%$. Find the cost price. `(CBSE 2024, 4 marks)`
Solution: Let CP $= x$.
SP at $10\%$ loss $= 0.9x$.
SP at $5\%$ profit $= 1.05x$.
$1.05x - 0.9x = 60$
$0.15x = 60$
$x = 400$
Answer: CP $=$ Rs. $400$

Q8 (HOTS). The population of a town is $25000$. It decreases by $4\%$ in the first year and increases by $5\%$ in the second year. Find the population at the end of the second year. `(CBSE 2024, 4 marks)`
Solution:
End of year 1 $= 25000 \times 0.96 = 24000$.
End of year 2 $= 24000 \times 1.05 = 25200$.
Answer: $25200$

Practice This Chapter on SparkEd

Comparing Quantities rewards speed as much as accuracy. When you sit for the annual exam, you should be able to convert a fraction to a percentage, find a profit margin, and compute simple interest without hesitation. That level of fluency only comes from practising dozens of problems with immediate feedback.

The SparkEd Comparing Quantities module has more than 60 practice questions split across three levels. Level 1 is where you drill the basic conversions and single step profit or loss calculations. Level 2 layers in discount problems, ratio division, and SI problems where you have to find an unknown rate or time. Level 3 is the CBSE exam zone with multi step word problems, successive percentage changes, and tricky HOTS style questions.

The instant AI feedback catches the most common errors in this chapter, such as computing profit on SP or forgetting to convert time to years, and tells you exactly what the correct step would have been. The gamified coins and pets keep you coming back, and the adaptive engine gives you more of whichever question type you need more practice on.

Practice Comparing Quantities on SparkEd →

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Boost Your Preparation with SparkEd

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