Comparing Quantities — for Class 7-8

What Is Comparing Quantities — and Why Should You Care?

When you go to a shop and the shopkeeper says “10% off”, that’s comparing quantities. When you see “Sachin scored 50 runs out of 80 balls” and your friend asks “was that good?”, you’re comparing quantities. This chapter is about making sense of ratios, percentages, profit-loss, and simple interest — things that show up in real life and in your board exam every single year.

Class 7 introduces ratios and percentages. Class 8 builds on that with compound situations, discount, tax, and interest calculations. Both years, this chapter carries solid weightage in CBSE term exams.

The good news: once you understand that all of comparing quantities is just fractions in disguise, the chapter clicks into place. A percentage is a fraction with denominator 100. A ratio is a fraction. Profit percentage is a fraction of cost price. Same idea, different language.

Key Terms and Definitions

Ratio — a comparison of two quantities of the same kind by division. If a class has 20 boys and 30 girls, the ratio of boys to girls is $20:30 = 2:3$ . Always simplify to lowest terms.

Proportion — two ratios are equal. If $a:b = c:d$ , then $a, b, c, d$ are in proportion and $ad = bc$ (this is the cross-multiplication rule).

Percentage — a ratio with denominator 100. " $35\%$ " means $\frac{35}{100}$ . The word “per cent” literally means “per hundred” — cent comes from Latin centum.

Cost Price (CP) — what you paid to buy something.

Selling Price (SP) — what you received when you sold it.

Profit — $SP > CP$ , so Profit $= SP - CP$ .

Loss — $CP > SP$ , so Loss $= CP - SP$ .

Marked Price (MP) — the price written on the tag (also called list price). After discount, you pay the selling price.

Discount — reduction on marked price. Discount $= MP - SP$ .

Simple Interest (SI) — interest calculated only on the original principal each year.

Principal (P) — the original amount deposited or borrowed.

Core Concepts and Methods

Ratio and Proportion

A ratio $a:b$ means for every $a$ units of the first quantity, there are $b$ units of the second.

To find a quantity given ratio and total:

If two quantities are in ratio $3:5$ and their sum is $640$ , then:

Total parts $= 3 + 5 = 8$
First quantity $= \frac{3}{8} \times 640 = 240$
Second quantity $= \frac{5}{8} \times 640 = 400$

This partition method is the fastest way to handle ratio problems in Class 7.

Percentage Conversions

\text{Fraction} \to \text{Percentage} : \frac{a}{b} \times 100

\text{Percentage} \to \text{Fraction} : \frac{x\%}{100} = \frac{x}{100}

\text{Percentage of a number} : x\% \text{ of } N = \frac{x}{100} \times N

Memorise these fraction-percentage pairs — they save 30 seconds per question:

Fraction	Percentage
$\frac{1}{2}$	$50\%$
$\frac{1}{4}$	$25\%$
$\frac{3}{4}$	$75\%$
$\frac{1}{5}$	$20\%$
$\frac{1}{3}$	$33.33\%$
$\frac{2}{3}$	$66.67\%$

Profit and Loss

\text{Profit\%} = \frac{\text{Profit}}{\text{CP}} \times 100

\text{Loss\%} = \frac{\text{Loss}}{\text{CP}} \times 100

SP = CP \times \frac{(100 + \text{Profit\%})}{100}

SP = CP \times \frac{(100 - \text{Loss\%})}{100}

CP = \frac{SP \times 100}{100 + \text{Profit\%}}

CP = \frac{SP \times 100}{100 - \text{Loss\%}}

The profit/loss percentage is always calculated on CP, not SP. Students mix this up constantly — we’ll revisit it in the common mistakes section.

Discount and Tax (Class 8)

\text{Discount} = MP - SP

\text{Discount\%} = \frac{\text{Discount}}{MP} \times 100

SP = MP \times \frac{(100 - \text{Discount\%})}{100}

\text{Price after tax} = SP \times \frac{(100 + \text{Tax\%})}{100}

When both discount and tax are given, apply discount first to get SP, then apply tax on that SP. The two are not interchangeable — sequence matters.

Simple Interest (Class 7 and 8)

SI = \frac{P \times R \times T}{100}

A = P + SI

where $P$ = Principal, $R$ = Rate per annum, $T$ = Time in years

If time is given in months, convert: $T = \frac{\text{months}}{12}$ .

Solved Examples

Example 1 — Easy (CBSE Class 7)

A bag contains 20 red and 30 blue marbles. What percentage are red?

Total marbles $= 20 + 30 = 50$

\text{Red\%} = \frac{20}{50} \times 100 = 40\%

Clean, direct. No formula needed beyond “part divided by whole, times 100.”

Example 2 — Easy (CBSE Class 7)

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

\frac{3}{8} \times 100 = \frac{300}{8} = 37.5\%

Example 3 — Medium (CBSE Class 8)

A shopkeeper buys a shirt for ₹400 and sells it for ₹480. Find the profit percentage.

Profit $= 480 - 400 = \text{₹}80$

\text{Profit\%} = \frac{80}{400} \times 100 = 20\%

CP is always the base for profit/loss percentage. This is a favourite CBSE question type.

Example 4 — Medium (CBSE Class 8)

A television is marked at ₹15,000. A discount of $12\%$ is offered. Find the selling price.

$SP = 15000 \times \frac{(100 - 12)}{100} = 15000 \times \frac{88}{100} = \text{₹}13,200$

Alternatively: Discount $= 12\%$ of $15000 = \text{₹}1800$ , so $SP = 15000 - 1800 = \text{₹}13,200$ .

Example 5 — Medium (CBSE Class 8)

Ramesh deposits ₹6,000 at $8\%$ per annum simple interest for 3 years. Find the amount he receives.

$SI = \frac{6000 \times 8 \times 3}{100} = \frac{144000}{100} = \text{₹}1440$

$A = 6000 + 1440 = \text{₹}7440$

Example 6 — Hard (CBSE Class 8 / ICSE)

After allowing a discount of $15\%$ on the marked price, a shopkeeper still earns a profit of $19\%$ on cost price. If the marked price is ₹3,570, find the cost price.

Step 1: Find SP from MP and discount.

$SP = 3570 \times \frac{85}{100} = \text{₹}3034.50$

Step 2: SP is $119\%$ of CP (since profit is $19\%$ ).

$CP = \frac{SP \times 100}{119} = \frac{3034.50 \times 100}{119} = \text{₹}2550$

This “discount on MP, profit on CP” combo question appears almost every year in CBSE Class 8 annual exams and ICSE. The key is working backward from SP to CP using the profit formula.

Example 7 — Hard (CBSE Class 8)

Two numbers are in ratio $3:4$ . If $6$ is subtracted from each, the ratio becomes $5:8$ . Find the numbers.

Let the numbers be $3x$ and $4x$ .

After subtracting 6: $\frac{3x - 6}{4x - 6} = \frac{5}{8}$

Cross multiply: $8(3x - 6) = 5(4x - 6)$

$24x - 48 = 20x - 30$

$4x = 18$ , so $x = 4.5$

Numbers: $3 \times 4.5 = 13.5$ and $4 \times 4.5 = 18$ .

Check: $(13.5 - 6):(18 - 6) = 7.5:12 = 5:8$ ✓

Exam-Specific Tips

CBSE Class 7

SA/Term exams typically have 1 MCQ on percentage conversion, 1 short answer on profit/loss, and 1 long answer on simple interest.
The “find what percent A is of B” phrasing trips many students — it always means $\frac{A}{B} \times 100$ .
Marks are awarded for formula + substitution + answer. Show all three steps clearly.

CBSE Class 8

Discount and tax questions are almost always worth 3–4 marks.
Expect at least one question where you’re given SP and profit% and asked to find CP — backwards application of the formula.
In long answers, write the formula before substituting. It gets you a method mark even if the arithmetic goes wrong.

In CBSE Class 8, the chapter “Comparing Quantities” accounts for roughly $8-10$ marks in the annual exam across MCQs, short answers, and long answers. It’s among the highest-scoring chapters in the Number System strand because the formulas are fixed and the question patterns repeat predictably.

ICSE Class 8

ICSE pushes harder on the combined discount + profit/loss type (like Example 6 above). They also test successive discounts — two discounts applied one after another, not added. $20\% + 10\%$ successive discount is NOT $30\%$ — it’s $1 - 0.8 \times 0.9 = 28\%$ .

Common Mistakes to Avoid

Mistake 1: Profit% calculated on SP, not CP. Always: $\text{Profit\%} = \frac{\text{Profit}}{CP} \times 100$ . The denominator is CP, every single time.

Mistake 2: Adding discount% and tax% directly. If an item has $10\%$ discount and $5\%$ GST, you cannot say the net is $5\%$ off. Apply discount first to get SP, then apply tax on SP.

Mistake 3: Forgetting to simplify ratios. Ratio $24:36$ should be simplified to $2:3$ . Examiners deduct marks for unsimplified final answers.

Mistake 4: Using months directly in the SI formula. Time must be in years. 8 months $= \frac{8}{12} = \frac{2}{3}$ years. Substitute $\frac{2}{3}$ , not $8$ .

Mistake 5: Reversing ratio order. “Ratio of boys to girls” and “ratio of girls to boys” are different. $20:30 \neq 30:20$ . Read the question carefully about which quantity comes first.

Practice Questions

Q1. In a class of 40 students, 16 are girls. What percentage of students are boys?

Boys $= 40 - 16 = 24$

Boys% $= \frac{24}{40} \times 100 = 60\%$

Q2. Express $\frac{5}{6}$ as a percentage. (Round to 2 decimal places.)

$\frac{5}{6} \times 100 = \frac{500}{6} = 83.33\%$

Q3. A cycle costs ₹1,200 and is sold for ₹1,050. Find the loss percentage.

Loss $= 1200 - 1050 = \text{₹}150$

Loss% $= \frac{150}{1200} \times 100 = 12.5\%$

Q4. Two numbers are in ratio $5:7$ . Their sum is $360$ . Find the numbers.

Total parts $= 5 + 7 = 12$

First number $= \frac{5}{12} \times 360 = 150$

Second number $= \frac{7}{12} \times 360 = 210$

Check: $150 + 210 = 360$ ✓ and $150:210 = 5:7$ ✓

Q5. Find the SI on ₹5,000 at $9\%$ per annum for 2 years 6 months.

Time $= 2\frac{6}{12} = 2.5$ years

$SI = \frac{5000 \times 9 \times 2.5}{100} = \frac{112500}{100} = \text{₹}1125$

Amount $= 5000 + 1125 = \text{₹}6125$

Q6. A shopkeeper marks goods at ₹2,500 and allows a $16\%$ discount. Find the SP.

$SP = 2500 \times \frac{84}{100} = \text{₹}2100$

Or: Discount $= 16\%$ of $2500 = \text{₹}400$ , so $SP = 2500 - 400 = \text{₹}2100$

Q7. At what rate of interest will ₹4,000 amount to ₹4,600 in 3 years under simple interest?

$SI = 4600 - 4000 = \text{₹}600$

$R = \frac{SI \times 100}{P \times T} = \frac{600 \times 100}{4000 \times 3} = \frac{60000}{12000} = 5\%$ per annum

Q8. A seller bought a watch for ₹800 and spent ₹120 on repairs. He sold it for ₹1,000. Find profit or loss percentage.

Total CP $= 800 + 120 = \text{₹}920$ (repair cost adds to CP)

SP $= \text{₹}1000$

Profit $= 1000 - 920 = \text{₹}80$

Profit% $= \frac{80}{920} \times 100 = 8.70\%$ (approx)

Q9. (ICSE level) Two successive discounts of $20\%$ and $15\%$ are given on an article marked at ₹5,000. Find the final SP.

After first discount: $SP_1 = 5000 \times \frac{80}{100} = \text{₹}4000$

After second discount: $SP_2 = 4000 \times \frac{85}{100} = \text{₹}3400$

Do not calculate $20 + 15 = 35\%$ off — successive discounts are never simply added.

Equivalent single discount $= 100 - 68 = 32\%$ , confirming $\text{₹}5000 \times 0.68 = \text{₹}3400$ .

Q10. An amount doubles itself in 10 years under simple interest. Find the rate of interest per annum.

Let principal $= P$ . Amount $= 2P$ , so $SI = P$ .

$R = \frac{SI \times 100}{P \times T} = \frac{P \times 100}{P \times 10} = 10\%$ per annum

This is a classic question. If it doubles in $n$ years, $R = \frac{100}{n}\%$ .

FAQs

What is the difference between ratio and proportion?

A ratio is a comparison of two quantities: $3:4$ . A proportion says two ratios are equal: $3:4 = 6:8$ . Proportion is an equation; ratio is just a relationship.

How do we calculate percentage increase or decrease?

\text{Percentage change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100

If positive, it’s an increase; if negative, it’s a decrease. Always divide by the original value.

Why is profit percentage always on cost price?

Because profit percentage tells you how much extra you earned relative to what you spent. What you spent is CP — so CP is the reference. SP is what you received, not your reference point.

What is the formula to find CP when SP and profit% are given?

CP = \frac{SP \times 100}{100 + \text{Profit\%}}

Similarly, for loss: $CP = \frac{SP \times 100}{100 - \text{Loss\%}}$

Can discount be calculated on cost price?

No. Discount is always on marked price (MP), not cost price. Profit is on CP. These two have different bases — confusing them is one of the most common Class 8 exam errors.

What is the difference between simple interest and compound interest?

Simple interest is calculated on the principal alone, every year. Compound interest is calculated on the principal plus accumulated interest — so interest grows each year. Class 8 covers SI fully; compound interest comes up more formally in Class 9 onwards.

How do we convert percentage back to a fraction?

Divide by 100 and simplify. $45\% = \frac{45}{100} = \frac{9}{20}$ .

If two items are bought at the same price and one is sold at $x\%$ profit and the other at $x\%$ loss, is the overall result profit, loss, or neither?

Always a loss. This is a classic tricky question. If each CP is ₹100 and $x = 10\%$ : one sells at ₹110, the other at ₹90. Total SP $= \text{₹}200$ , total CP $= \text{₹}200$ , so… wait, that’s no loss? Only if the SPs are the same does the famous formula apply. When CPs are the same, there’s no net profit or loss. But when the SPs are the same, the result is always a loss of $\left(\frac{x}{10}\right)^2\%$ .