This page is the Comparing Quantities chapter of the NCERT Class 8 Mathematics book — Chapter 7, a 14-page chapter on ratios, percentages, discounts, taxes and compound interest.
The official Comparing Quantities Class 8 Chapter 7 PDF is here for instant download. Below it you will find what the chapter contains, how each section works, and worked examples with fresh numbers you will not see in the book.
This page is maintained for the 2026-27 academic session.
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The file below is the official NCERT text hosted on ncert.nic.in — the same Chapter 7 pages your school uses, with sections 7.1 to 7.6, the TRY THESE boxes, Exercises 7.1, 7.2 and 7.3, and the closing WHAT HAVE WE DISCUSSED? list.
Open the Comparing Quantities Class 8 Chapter 7 NCERT PDF to get the chapter exactly as NCERT printed it, with every example, figure and exercise in place for study or printing.
Chapter 7 at a glance: what the PDF contains
Before you start reading, check the table below. It lists what the chapter holds — sections, figures, worked examples and exercise questions — all in one place.
What Comparing Quantities covers, section by section
This walk-through tells you where each idea lives in Chapter 7, so you can jump straight to the part you need. Every section starts from a real-life situation — shopping, bills, bank deposits — and builds the maths from it.
- Section 7.1 — recalling ratios and percentages, using a fruit basket and a school picnic trip.
- Section 7.2 — finding discounts: marked price, sale price, discount percent, plus the estimation trick in 7.2.1.
- Section 7.3 — sales tax, value added tax (VAT) and goods and services tax (GST), all shown on shop bills.
- Section 7.4 — what compound interest means: the bank passbook, fixed deposits, and interest calculated year by year.
- Section 7.5 — deducing the short formula for the amount.
- Section 7.6 — applying the formula to population growth, the growth of bacteria, and depreciation of items like TVs and machinery.
Two devices repeat throughout: TRY THESE boxes give a quick check after each new idea, and the THINK, DISCUSS AND WRITE questions expose common traps. Exercises 7.1, 7.2 and 7.3 collect the practice problems, and the chapter ends with a WHAT HAVE WE DISCUSSED? list.
Ratios and percentages: the two methods in section 7.1
This is the base skill of the whole chapter: comparing two quantities as a ratio, then expressing that comparison as a percentage — a number out of 100.
The chapter opens with a basket of 20 apples and 5 oranges. The ratio of oranges to apples is 5 : 20, which reduces to 1 : 4; the reverse comparison, apples to oranges, is 20 : 5 = 4 : 1, so there are four times as many apples.
To write 5 oranges out of 25 fruits as a percentage, make the denominator 100: \( \frac{5}{25} \times \frac{4}{4} = \frac{20}{100} = 20\% \). NCERT then contrasts two ways to reach the same answer.
- Ashima’s method: multiply the fraction by \( \frac{100}{100} \), which equals 1, so the value does not change — \( \frac{22}{55} = \frac{22}{55} \times \frac{100}{100} = 40\% \).
- John’s unitary method: ask what happens out of 100. Out of 55 km, 22 km are travelled; out of 100 km, that is \( \frac{22}{55} \times 100 = 40 \) km.
The two methods always agree, because both convert the same fraction to a percentage. Since the basket holds only apples and oranges, the two percentages add up: percentage of apples + percentage of oranges = 100, so 20% oranges means 80% apples.
Example 1 then runs the same tools on a school picnic. Sixty per cent girls means 18 girls out of 30 students, so there are 12 boys and the ratio of girls to boys is 18 : 12 = 3 : 2.
The same example finds the cost per head (₹175, with two teachers along) and turns 22 km of a 55 km trip into 40 per cent of the distance, leaving 60 per cent.
Worked example: turning the ratio 7 : 20 into a percentage
- Step 1: Write the ratio as the fraction \( \frac{7}{20} \).
- Step 2: Multiply by 100 to convert to a percentage.
\[ \frac{7}{20} \times 100 = 35 \]
Final answer: 7 : 20 = 35%. The remaining 13 parts out of 20 form the complementary 65%.
Discounts: marked price, sale price and discount percent
A discount is a reduction given on the marked price of an article, usually to attract buyers. The rule that matters most: the discount percent is always calculated on the marked price, never on the sale price.
The definition box in section 7.2 states it as Discount = Marked Price − Sale Price. Example 2 marks an item at ₹840 and sells it for ₹714: the discount is ₹126, and the discount percent is \( \frac{126}{840} \times 100 = 15\% \).
When only the discount percent is given, the discount is that percent of the marked price. Example 3 lists a frock at ₹220 with a 20% discount: discount = \( \frac{20}{100} \times 220 \) = ₹44, so the sale price is ₹176.
Rehana’s shortcut (see figure 2 in the walkthrough) skips the discount step: a 20% discount leaves 80% of the marked price, so the sale price is \( \frac{80}{100} \times 220 = \) ₹176. In symbols: Sale Price = \( \frac{100-d}{100} \times \) Marked Price.
Section 7.2.1 adds a mental estimation trick for 15%. For a bill of ₹577.80, round to ₹580, take 10% (₹58), add half of that (₹29), and you have about ₹87 off — roughly ₹495 to pay.
Worked example: a bag marked at ₹950 with a 12% discount
Step 1: Discount = 12% of the marked price.
\[ \text{Discount} = \frac{12}{100} \times 950 = 114 \]
Step 2: Subtract the discount from the marked price.
\[ \text{Sale price} = 950 – 114 = 836 \]
Final answer: discount ₹114, sale price ₹836. Direct check: \( \frac{88}{100} \times 950 = 836 \).
Sales tax, VAT and GST: finding the price before tax
Bills carry tax in two different ways, and each way needs its own method. Sales tax is added on top of the price; VAT and GST are already included in the price you pay.
Sales tax is charged by the government on the sale of an item, collected by the shopkeeper and added to the bill — it is always on the selling price. Example 4 buys roller skates for ₹450 with 5% sales tax: tax = \( \frac{5}{100} \times 450 \) = ₹22.50, so the bill is ₹472.50.
VAT works the other way. If the price without VAT is ₹100, the price with 10% VAT is ₹110 — so to remove the tax from a bill of ₹3,300, divide by \( \frac{110}{100} \): ₹3,300 divided by 1.10 = ₹3,000 (Example 5).
GST behaves like VAT. If ₹100 becomes ₹112 after 12% GST, a bill of ₹784 hides a price of 784 divided by 1.12 = ₹700 (Example 6). One slip to ignore: the printed working of Example 6 shows a wrong fraction, but the logic stated just above it is correct — divide the bill by \( \frac{112}{100} \).
When the rate is r%, the single rule is: price before tax = bill divided by \( \frac{100+r}{100} \). GST itself was introduced from 1 July 2017 and is levied on the supply of goods or services or both.
The bill format in section 7.3 shows where sales tax sits: columns for S.No., Item, Quantity, Rate and Amount, then a line for “Bill amount + ST (5%)” before the Total. The tax is a separate, visible line.
Worked example 1: a book priced at ₹1,800 with 5% sales tax
Step 1: Sales tax = 5% of the price.
\[ \text{Tax} = \frac{5}{100} \times 1800 = 90 \]
Step 2: Add the tax to the price.
\[ \text{Bill} = 1800 + 90 = 1890 \]
Final answer: the bill is ₹1,890.
Worked example 2: a bill of ₹2,712 including 13% GST
Step 1: Divide the bill by \( \frac{113}{100} \), since ₹100 becomes ₹113 after 13% GST.
\[ \text{Price before GST} = 2712 \div 1.13 = 2400 \]
Step 2: Check by multiplying back.
\[ 2400 \times 1.13 = 2712 \]
Final answer: the article cost ₹2,400 before GST.
Compound interest: how the formula is built
Compound interest is simple interest recalculated each year on the previous year’s amount. Because every year’s interest joins the principal, the amount grows a little faster each year.
Example 7 recaps simple interest first: ₹10,000 at 15% per annum for 2 years earns ₹1,500 each year, ₹3,000 in total, so ₹13,000 must be repaid. The TRY THESE right after it — ₹15,000 at 5% for 2 years — is still simple interest; compound interest begins only after that box.
Heena’s loan shows the year-by-year method on ₹20,000 at 8% compounded annually. Four steps: find one year’s simple interest, add it to make the next year’s principal, repeat, then total the yearly interests.
- First year: \( \frac{8}{100} \times 20000 = \) ₹1,600; amount = ₹21,600, which becomes the new principal.
- Second year: \( \frac{8}{100} \times 21600 = \) ₹1,728; amount = ₹23,328.
- Compound interest = 1600 + 1728 = ₹3,328.
Reeta compares the same sum under simple interest: \( \frac{8}{100} \times 20000 \times 2 = \) ₹3,200. Compound interest is ₹128 more, because the second year’s interest is charged on ₹21,600, not on ₹20,000.
The chapter’s ₹100 chart makes the difference visible. At 10% for 3 years, simple interest finishes at ₹130, compound interest at ₹133.10 — you earn interest on the interest already accumulated.
Section 7.5 turns the year-by-year method into a formula. After one year, \( A_1 = P\left(1+\frac{R}{100}\right) \); treat that as the next year’s principal, and \( A_2 = P\left(1+\frac{R}{100}\right)^2 \). Repeating for n years gives \[ A = P\left(1+\frac{R}{100}\right)^n \]
and compound interest is simply \( CI = A – P \). Example 8 checks it on ₹12,600 at 10% for 2 years: amount ₹15,246, CI ₹2,646.
The passbook in figure 3 shows the real-world version: each year’s interest entry is bigger than the last because interest is added to the account before the next year’s calculation.
Worked example: a ₹24,000 loan at 6% per annum compounded for 2 years
Step 1: Year 1 interest on the original principal.
\[ \frac{6}{100} \times 24000 = 1440 \]
Step 2: Amount at the end of year 1 — this becomes the principal for year 2.
\[ 24000 + 1440 = 25440 \]
Step 3: Year 2 interest on ₹25,440.
\[ \frac{6}{100} \times 25440 = 1526.40 \]
Step 4: Amount at the end of year 2.
\[ 25440 + 1526.40 = 26966.40 \]
Step 5: Compound interest = amount − principal.
\[ 26966.40 – 24000 = 2966.40 \]
Final answer: compound interest ₹2,966.40, amount payable ₹26,966.40. The formula agrees: \( 24000 \times (1.06)^2 = 26966.40 \).
Compound interest in real life: population, bacteria and depreciation
Any quantity that grows by a fixed percent each period grows in compounded form, because each new period starts from a new base. Population and bacteria grow this way; values that fall — like machines and TVs — shrink the same way, and that fall is called depreciation.
Example 9 takes a city of 20,000 growing at 5% per annum from 1997 to the end of 2000, that is, for 3 years. The book’s logic: every new year has a new population, so the growth is compounded.
- Year by year: 20,000 → 21,000 → 22,050 → 23,152.5, which the book rounds to 23,153.
- Formula: \( 20000 \times \left(1+\frac{5}{100}\right)^3 = 20000 \times \frac{21}{20} \times \frac{21}{20} \times \frac{21}{20} = 23152.5 \).
Bacteria follow the same rule with a faster clock: the rate is per hour, so n counts hours. Exercise 7.3 Q2 starts with 5,06,000 bacteria growing at 2.5% per hour for 2 hours.
Depreciation is the falling case. Example 10 defines it as reduction of value due to use and age; a TV bought for ₹21,000 depreciating at 5% is worth \( 21000 \times \left(1-\frac{5}{100}\right) = 21000 \times \frac{19}{20} = \) ₹19,950 after one year. The bracket is \( 1 – \frac{R}{100} \), not \( 1 + \frac{R}{100} \).
Worked example: a machine worth ₹40,000 depreciating at 10% per annum
Step 1: Value after 1 year.
\[ 40000 \times \left(1-\frac{10}{100}\right) = 40000 \times 0.9 = 36000 \]
Step 2: Value after 2 years.
\[ 36000 \times \left(1-\frac{10}{100}\right) = 40000 \times (0.9)^2 = 32400 \]
Final answer: ₹36,000 after one year and ₹32,400 after two years. Each value is smaller than the one before, which is the natural check for depreciation.
Reading the figures in this chapter
The figures in this chapter are few, and each one carries a point worth reading. Here is what each shows and what to take from it.
This box is the definition that section 7.2 runs on. It fixes the marked price as the starting point, which is why a discount percent is always a percent of the marked price. Test every discount answer against this: the sale price must always come out smaller than the marked price.
This is Rehana’s shortcut from Example 3: a 20% discount leaves 80% of the marked price, so the sale price is \( \frac{80}{100} \times 220 = \) ₹176 in one multiplication. When the discount percent is given, the discount step is optional — go straight to the sale price.
The passbook is section 7.4’s real-world observation: interest is added every year, and each year’s interest is a little more than the last. That is why interest is compounded — it is calculated on the amount of the previous year.
Each link of the chain is the result of the one before it: 20,000 is the principal for year one, 21,000 for year two, 22,050 for year three. Because every new year has a new population, growth is compounded — the reason the amount formula applies to population.
Key terms defined in one line each
These are the words the chapter uses most. The table gives a plain definition and the section where each term first appears.
| Term | Plain definition | Where it appears |
|---|---|---|
| Ratio | Comparing two quantities by division, as in 5 : 20. | Section 7.1 |
| Percentage | A fraction with denominator 100, written with a % sign. | Section 7.1 |
| Marked price | The price printed on the article before any discount. | Section 7.2 |
| Sale price | The price actually paid after the discount. | Section 7.2 |
| Discount | Reduction given on the marked price; Discount = Marked Price − Sale Price. | Section 7.2 |
| Discount percent | The discount expressed as a percent of the marked price. | Section 7.2 |
| Overhead expenses | Extra costs after buying an article; they go into the cost price: CP = Buying price + Overhead expenses. | Closing summary |
| Sales tax | Tax charged by the government on a sale, added to the bill amount. | Section 7.3 |
| VAT | Value added tax, already included in the price you pay. | Section 7.3 |
| GST | Goods and Services Tax, levied on the supply of goods or services or both. | Section 7.3 |
| Principal | The sum of money on which interest is calculated. | Section 7.4 |
| Amount | Principal + interest; the total after interest is added. | Section 7.4 |
| Interest | The extra money paid on money deposited or borrowed. | Section 7.4 |
| Simple interest | Interest calculated only on the original principal, year after year. | Section 7.4 |
| Compound interest | Interest calculated on the previous year’s amount, so the principal changes each year. | Section 7.4 |
| Depreciation | Reduction of value due to use and age of the item. | Section 7.6 |
Mistakes students make in this chapter
Each row names the wrong move, the correct rule, and a way to check yourself. Read it before you start the exercises.
| Mistake | Correct rule | How to check your answer |
|---|---|---|
| Calculating discount percent on the sale price | Discount % is on the marked price: \( \frac{\text{discount}}{\text{MP}} \times 100 \). | Your discount percent, applied to the marked price, must reproduce the given discount. |
| Assuming percent change is symmetric | ₹2,000 is \( 16\frac{2}{3}\% \) less than ₹2,400, while ₹2,400 is 20% more than ₹2,000 — the base changes. | Divide the difference by the value you are comparing against, not the other one. |
| Confusing a 100% increase with taking half | Two times a number is a 100% increase; half the number is a 50% decrease. | 100% increase means the new value is double the old; 50% decrease means the new value is half the old. |
| Subtracting the tax rate from the bill to find the pre-tax price | Price before tax = bill divided by \( \frac{100+r}{100} \). | Multiply your answer back by \( \frac{100+r}{100} \); it must return the bill. |
| Treating the principal as fixed under compound interest | Each year’s amount becomes the next year’s principal. | Simple interest on the same sum must come out smaller than CI. |
| Mixing units inside a ratio | Use one unit first: 50 paise : ₹5 = 50 : 500 = 1 : 10; 5 m : 10 km = 5 : 10,000 = 1 : 2,000. | Rewrite both quantities in the same unit before reducing the ratio. |
| Using \( 1 + \frac{R}{100} \) when the value falls | Depreciation uses \( 1 – \frac{R}{100} \) inside the bracket. | A depreciated value must be less than the starting value. |
One more warning from the chapter’s closing summary: overhead expenses — extra costs made after buying an article — are added into the cost price (CP = Buying price + Overhead expenses). The body of the chapter states this rule without working a full example of it, which is exactly why students forget it exists.
How to use this chapter for exam practice
The three exercises each practise one skill, so you can aim your revision precisely.
- Exercise 7.1 — ratios and percentages. Q1 tests matching units inside a ratio; Q2 converts ratios to percentages; Q3–Q6 apply “x% of a quantity” forwards and backwards.
- Exercise 7.2 — discounts and taxes together. Q1 joins a discount percent to a multi-item bill; Q2 is sales tax; Q3 works backwards from sale price to marked price; Q4 is VAT; Q5 is GST.
- Exercise 7.3 — the compound interest formula in new settings. Q1 is population, forwards and backwards; Q2 is bacteria with an hourly rate; Q3 is depreciation.
Notice the recurring three-out-of-four shape: the question gives three of {marked price, sale price, discount percent} — or {principal, rate, time, amount} — and asks for the fourth. Writing the same relation each time, Discount = MP − SP or \( A = P\left(1+\frac{R}{100}\right)^n \), is the whole method.
The chapter ends with its own WHAT HAVE WE DISCUSSED? list — the best revision skeleton, one result at a time. For any numerical, a complete answer states the relation, substitutes the given values, and ends with the unit and a clear conclusion, such as “sale price is ₹836”.
One honest note: textbook contents and the examinable syllabus are not always identical — check the current official syllabus before deciding what to prioritise.
For condensed last-minute practice, see the Class 8 Mathematics revision notes, the revision notes index, or the Class 8 study notes hub.
What to remember from Chapter 7
The chapter closes with its own summary. Here are the same takeaways in plainer words, one result per line.
- Discount is a reduction given on the marked price: Discount = Marked Price − Sale Price.
- When the discount percent d is given, Discount = d% of Marked Price, and Sale Price = \( \frac{100-d}{100} \times \) Marked Price.
- Overhead expenses are extra costs after buying; they are included in the cost price: CP = Buying price + Overhead expenses.
- Sales tax is charged by the government on the sale of an item and is added to the bill amount: Sales tax = Tax% of Bill Amount.
- GST is levied on the supply of goods or services or both.
- Compound interest is interest calculated on the previous year’s amount (A = P + I), so the principal changes every year.
- Amount formula: \( A = P\left(1+\frac{R}{100}\right)^n \), and Compound Interest = A − P.
- For depreciation, the bracket changes sign: \( 1 – \frac{R}{100} \).
Related NCERT Class 8 Mathematics chapters
Comparing Quantities sits between two chapters of the same book. The NCERT Class 8 Mathematics book page lists every chapter; the Class 8 NCERT books hub covers all subjects of the class. The chapter before is Cubes and Cube Roots (Chapter 6), and the one after is Algebraic Expressions and Identities (Chapter 8).
For revision-style support on those chapters, see the Cubes and Cube Roots revision notes and the Algebraic Expressions and Identities revision notes.
The authoritative source for every NCERT textbook and edition is ncert.nic.in, where the official PDFs live.
Sources and data verification
The contents and formulas on this page describe the NCERT Class 8 Mathematics textbook, Chapter 7 (Comparing Quantities), in the official edition published on ncert.nic.in.
This page covers Chapter 7 of that book only; other chapters and other Class 8 subjects have their own pages in this directory.
This listing is maintained for the 2026-27 academic session.
NCERT settles textbooks, editions and PDFs; CBSE settles the curriculum, syllabus and examinations. This page does not state exam weightage for the chapter, because the syllabus — not the textbook — decides what is examinable.
Common questions about Comparing Quantities
What is the compound interest formula in Class 8 Comparing Quantities?
The amount formula is \( A = P\left(1+\frac{R}{100}\right)^n \), where P is the principal, R the rate per cent per annum and n the number of years. Compound interest is then CI = A − P. Section 7.5 builds this formula by repeating the year-by-year calculation.
How do I find the marked price when the sale price and the discount percent are given?
Use Sale Price = Marked Price × \( \frac{100-d}{100} \) and rearrange: Marked Price = Sale Price divided by \( \frac{100-d}{100} \). If the sale price is ₹1,600 after a 20% discount, the marked price is \( 1600 \div \frac{80}{100} = \) ₹2,000 — the same logic as Exercise 7.2 Q3.
Why is compound interest higher than simple interest on the same amount?
Under compound interest, each year’s interest is calculated on the previous year’s amount, so you earn interest on interest. The chapter’s chart shows ₹100 at 10% for 3 years ending at ₹130 under simple interest but ₹133.10 under compound interest; in Heena’s example the difference is ₹128.
What is the difference between sales tax and GST in this chapter?
Sales tax is added on top of the price and appears as a separate line on the bill, while VAT and GST are already included in the price you pay. To remove an included tax, divide the bill by \( \frac{100+r}{100} \). GST was introduced from 1 July 2017 and is levied on the supply of goods or services or both.
How do I convert a ratio like 3 : 4 to a percentage?
Write it as a fraction and multiply by 100: \( \frac{3}{4} \times 100 = 75\% \). A ratio a : b converts to the percentage \( \frac{a}{b} \times 100\% \).
| What the chapter holds | Count | Where it is used |
|---|---|---|
| Printed pages | 14 | |
| Sections in the chapter | 7 | |
| Figures with NCERT captions | 4 | |
| Tables | 2 | |
| Worked examples | 3 | solved step by step in our NCERT Solutions |
| Exercise questions | 14 | answered in our NCERT Solutions |
| In-text questions | 6 | |
| Official NCERT PDF | Download the chapter PDF | the chapter exactly as NCERT publishes it |
Reference: NCERT Class 8 Mathematics textbook, chapter 7, official edition on ncert.nic.in.