- A percentage p% equals p/100.
- Convert fractions and decimals to percentages using a common whole.
- p% of Q=(p/100)×Q.
- Percentage change uses the original quantity as its denominator.
- Profit percentages generally use cost price; discount percentages use marked price.
- GST is calculated using the stated applicable rate.
- Simple interest grows linearly, while compound interest grows multiplicatively.
- Successive percentage changes multiply their factors and cannot simply be added.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
Fractions in Disguise — Class 8 NCERT Mathematics
Direct answer: Fractions in Disguise (Ganita Prakash Part II, Chapter 1; overall Chapter 8) develops percentages from fractions, decimals and ratios. It applies them to comparisons, profit, discount, GST, interest, growth and depreciation.
When a shop announces 25% off, what does that mean for the price you pay? A percentage is simply a fraction out of 100, and the key is knowing which whole the percentage describes.
Source note: The chapter is 32 pages long. The six first-set Figure it Out questions below are supported by accessible chapter text. Later selected textbook problems are paraphrased separately; this is not a complete verified transcription of every NCERT exercise, diagram or in-text activity.
What does per cent mean?
Per cent means per hundred: p%=p/100. A bar divided into 100 equal parts with 35 shaded represents 35%. A percentage is a way of expressing a fraction, not a different kind of quantity.
Convert fractions, decimals and percentages
To convert a fraction a/b to a percentage, calculate (a÷b)×100%, with b≠0. To convert p% to a fraction, write p/100 and simplify. To convert a decimal to a percentage, multiply by 100. For example 3/8=0.375=37.5%; 0.06=6%; 125%=5/4=1.25.
Why percentages use 100
Hundred is convenient in a base-ten number system: 1%=0.01 and 10%=0.1. The reference whole must always be identified. For example 20 students out of 80 means 25%, not 20%, of the class.
Find a percentage of a quantity
p% of Q=(p/100)×Q. Calculate mentally using benchmarks: 50%=half, 25%=quarter, 10%=one tenth, 5%=half of 10%, 1%=one hundredth. For 15% of 240, find 10%=24 and 5%=12, then add to get 36.
Convert ratios to percentages
If red:blue=2:3, the whole has 5 parts, so red is 2/5×100=40% and blue is 3/5×100=60%. Do not mistake 2:3 for 2/3 of the whole: the denominator must be the sum of the parts.
Percentages above 100 and reverse percentages
120% of 50=1.2×50=60. If 40% of a journey is 92 km, the full journey is 92÷0.4=230 km. A quantity that becomes 120% of its old value has increased by 20%, not 120%.
Compare proportions fairly
Different denominators require a common scale. A score of 18/25 is 72%; 35/50 is 70%, so the first is higher despite having fewer raw marks. State the total and unit of comparison.
Percentage increase and decrease
Percentage change=(new−old)/old×100%. From ₹200 to ₹250 is a 25% increase. From ₹250 to ₹200 is a 20% decrease. Reversing the direction changes the denominator, so percentages are not symmetric.
Profit, loss, marked price and discount
Profit=SP−CP; profit%=profit/CP×100. Loss=CP−SP; loss%=loss/CP×100. Discount=MP−SP; discount%=discount/MP×100. Example: buy for ₹400, sell for ₹500: profit=₹100, profit%=25%. A ₹600 shirt at 20% discount costs ₹480.
Gross and net profit
Gross profit is revenue minus the cost of goods sold. Net profit also subtracts other business expenses. If revenue is ₹10,000, goods cost ₹6,000 and other expenses ₹1,500, gross profit=₹4,000 and net profit=₹2,500. Always state the base when calculating a margin.
GST and final price
For an item priced at ₹8250 before 18% GST, tax=0.18×8250=₹1485 and total=₹9735. The shortcut is 8250×1.18. GST rates depend on product classification and applicable law; the 18% here is a given mathematical example, not a universal rate.
Simple interest and compound interest
For principal P, annual rate r as a decimal, and t years: simple-interest amount A=P(1+rt), if the same principal earns interest throughout. Annual compound amount A=P(1+r)^t. On ₹2000 at 10% for two years: simple amount ₹2400; annual compound amount ₹2420. The compounding frequency must be specified.
Growth, decline and depreciation
Repeated growth by r per period gives new value=old value×(1+r)^n; repeated decline gives old value×(1−r)^n, with r as a decimal. For 1000 cells growing 10% twice: 1000×1.1²=1210. For a ₹50,000 item depreciating 10% twice: 50000×0.9²=₹40,500.
Successive discounts and percentage traps
Two successive 10% discounts produce 0.9×0.9=0.81 of the marked price, an overall 19% discount, not 20%. A 20% increase followed by 20% decrease leaves 1.2×0.8=0.96 of the original, a 4% decrease. The changing base matters.
Formula reference table
| Concept | Formula | Example |
|---|---|---|
| Fraction to percent | (a/b)×100% | 3/4=75% |
| Percent of quantity | (p/100)×Q | 20% of 150=30 |
| Percentage change | (new−old)/old×100% | 80→100: +25% |
| Profit percentage | (SP−CP)/CP×100% | 400→500: 25% |
| Discount percentage | (MP−SP)/MP×100% | 500→400: 20% |
| Simple interest | A=P(1+rt) | ₹1000, 10%, 2y: ₹1200 |
| Compound interest | A=P(1+r)^t | ₹1000, 10%, 2y: ₹1210 |
Common mistakes and quick revision
Always label the whole or base. Distinguish '120% of' from '120% increase'. Profit percentage is normally calculated on cost price, discount percentage on marked price. Repeated percentage changes multiply factors, not percentage points. Convert units before using percentages. Estimate whether an answer is plausible before calculating.
Textbook activities and visual learning
Use a 100-square grid to shade 25%, 60% or 125% (one full grid plus 25 squares). Draw bar models for part-to-whole problems. For a race-position diagram, estimate the fraction of the whole track completed, then compare to the given percentage options. For the Peaceful Knights puzzle, refer to the original chessboard; a text-only answer cannot reliably reproduce the constraints.
NCERT Figure it Out — first exercise, page 3 (six questions)
Question 1. Convert 3/5, 7/14, 9/20, 72/150, 1/3 and 5/11 to percentages.
Step-by-step solution: Multiply each fraction by 100%. (i) 3/5=60%; (ii) 7/14=1/2=50%; (iii) 9/20=45%; (iv) 72/150=48%; (v) 1/3=33⅓%; (vi) 5/11≈45.45%.
Final answer: 60%; 50%; 45%; 48%; 33⅓%; approximately 45.45%.
Question 2. Nandini has 25 marbles, 15 white. What percentage are white?
Step-by-step solution: White fraction=15/25=3/5. Multiply by 100 to obtain 60%.
Final answer: 60% (option iv).
Question 3. In a school, 15 of 80 students walk to school. What percentage walk?
Step-by-step solution: 15÷80×100=1500÷80=18.75.
Final answer: 18.75%.
Question 4. Match four runners' approximate progress to percentages in the textbook race diagram.
Step-by-step solution: Read the position of each marker as a fraction of the start-to-finish length. The diagram-based matching given by the worked reference is A≈38%, B≈55%, C≈72%, D≈93%. The original figure must be checked for exact marker positions.
Final answer: A≈38%; B≈55%; C≈72%; D≈93% (diagram-based, pending visual confirmation).
Question 5. Compare (i) 50% and 5%, (ii) 5/10 and 50%, (iii) 3/11 and 61%, (iv) 30% and 1/3.
Step-by-step solution: (i) 50>5. (ii) 5/10=0.5=50%. (iii) 3/11≈27.27%, less than 61%. (iv) 1/3≈33.33%, greater than 30%.
Final answer: (i) >; (ii) =; (iii) <; (iv) <.
Question 6. Why use a denominator of 100 for percentages?
Step-by-step solution: Our decimal place-value system uses powers of 10, so 100 provides a convenient standard for comparisons: 31%=31/100=0.31. Other denominators work mathematically, but 100 is a widely shared convention.
Final answer: 100 gives a convenient common scale linked to decimal notation.
Selected textbook-based applications and original exam practice (not an exhaustive NCERT exercise list)
Question 1. Find 25%, 16%, 62%, 140%, 1% and 7% of the stated quantities: 160, 250, 360, 40, one hour and 10 kg.
Step-by-step solution: (i) 0.25×160=40. (ii) 0.16×250=40. (iii) 0.62×360=223.2. (iv) 1.4×40=56. (v) 0.01×60 min=0.6 min=36 s. (vi) 0.07×10 kg=0.7 kg.
Final answer: 40; 40; 223.2; 56; 36 seconds; 0.7 kg.
Question 2. A geometry box costs ₹75 and sells for ₹110. Find profit percentage on cost.
Step-by-step solution: Profit=110−75=35. Profit%=35/75×100=46⅔%.
Final answer: 46⅔% profit.
Question 3. A chair costs ₹475 to make. Find selling price for 50% profit on cost.
Step-by-step solution: Profit=0.5×475=237.50. Selling price=475+237.50.
Final answer: ₹712.50.
Question 4. A company earns ₹2.5 crore revenue and has a 25% profit margin measured against revenue. Find its expenses.
Step-by-step solution: Profit=25%×2.5=0.625 crore. Expenditure=2.5−0.625=1.875 crore. Note that margin on revenue differs from markup on cost.
Final answer: ₹1.875 crore.
Question 5. A ₹300 shirt is discounted by 25%. What is the amount payable?
Step-by-step solution: Discount=300×0.25=75. Payable=300−75.
Final answer: ₹225.
Question 6. Petrol price rises from ₹60 to ₹100. Find percentage increase.
Step-by-step solution: Increase=100−60=40. Percentage increase=40/60×100=66⅔%.
Final answer: 66⅔%.
Question 7. A car costs ₹4,40,000 after a 15% discount. Find the marked price.
Step-by-step solution: Sale price is 85% of marked price. Marked price=440000/0.85≈517647.06.
Final answer: Approximately ₹5,17,647.06.
Question 8. A phone costs ₹8250 before 18% GST. Find its final price.
Step-by-step solution: GST=8250×0.18=1485. Total=8250+1485=9735.
Final answer: ₹9735.
Question 9. Population is 50 lakh and later becomes 250% of the original. Find the new population.
Step-by-step solution: 250%=2.5. New population=50×2.5=125 lakh.
Final answer: 125 lakh or 1.25 crore.
Question 10. Compare simple and annual compound interest on ₹2000 at 10% for two years.
Step-by-step solution: Simple amount=2000(1+0.1×2)=2400. Compound amount=2000(1.1)²=2420. Difference=2420−2400=20.
Final answer: ₹2400 simple; ₹2420 compound; difference ₹20.
Question 11. A product is discounted 10% twice. Find the overall discount.
Step-by-step solution: After two reductions, fraction remaining=0.9×0.9=0.81. Overall discount=1−0.81=0.19.
Final answer: 19%.
Question 12. A population changes +5%, −2%, −3% over three months. Find the net change.
Step-by-step solution: Multiplier=1.05×0.98×0.97=0.99813. Net change=(0.99813−1)×100%=−0.187%.
Final answer: A 0.187% decrease.
Question 13. A rectangle's length increases by 20% and width decreases by 20%. How does area change?
Step-by-step solution: New area factor=1.2×0.8=0.96, so the area becomes 96% of its old value.
Final answer: 4% decrease.
Question 14. MCQ: Which equals 125%? (A) 1/4 (B) 5/4 (C) 4/5 (D) 1/5
Step-by-step solution: 125/100=5/4=1.25.
Final answer: Option B.
Question 15. HOTS: A price rises by 25%. What percentage decrease restores the original price?
Step-by-step solution: New price=1.25P. Required reduction=(1.25P−P)/(1.25P)×100=0.25/1.25×100=20%.
Final answer: 20%.
Question 16. Case study: 80 students take a test, 60 pass. If 10 more pass after rechecking, what is the new pass percentage?
Step-by-step solution: Initially 60/80=75%. New passes=70. New percentage=70/80×100=87.5%.
Final answer: 87.5%.
Source, chapter scope and exercise coverage
Official NCERT textbook: Ganita Prakash Part II Chapter 1 — Fractions in Disguise. The chapter includes several further Figure it Out sets on pages 12–14, 19–20, 22–24 and 28–30, as well as bar-model illustrations and the Peaceful Knights puzzle. These are not yet reproduced in full here. Figure-dependent results must be checked against the printed figures. Supplementary practice is distinguished from textbook questions.
Frequently asked questions
What is Fractions in Disguise about?
Percentages, their relationship to fractions and decimals, and applications including discounts, GST and interest.
What does per cent mean?
Per hundred, so 25%=25/100=1/4.
How do you convert a fraction to a percentage?
Divide numerator by denominator and multiply by 100%.
What is 3/8 as a percentage?
37.5%.
How do you find 15% of 240?
0.15×240=36.
What is percentage change?
(new−old)/old×100%, where old is the original nonzero quantity.
What is profit percentage?
(selling price−cost price)/cost price×100%.
What is discount percentage?
(marked price−selling price)/marked price×100%.
What is the price of ₹8250 plus 18% GST?
₹9735.
What is the formula for annual compound interest?
A=P(1+r)^t, with r as a decimal annual rate and t in years.
Are two 10% discounts equal to a 20% discount?
No. Successive 10% discounts give an overall 19% reduction.
Does this page contain every NCERT textbook exercise?
Not yet. The first Figure it Out set and selected further applications are covered; remaining diagram-based and exercise sets require a full textbook audit.
