Fractions in Disguise | CBSE Class 8 Maths Notes

This note covers percentages as fractions, decimal conversions, mental calculations, ratios, finding an unknown whole, comparisons, percentage change, profit and loss, discounts, taxes, interest, compounding, depreciation and successive percentage changes.

What does a percentage mean?

A percentage expresses a quantity per hundred. The symbol % is read as “per cent”, meaning “out of hundred”. Thus, 25% means 25 out of every 100, whether the quantities being compared are people, rupees or marks.

Definition: A percentage is a fraction with denominator 100. The denominator is the number below the fraction bar; the numerator is the number above it. The whole is the complete quantity being considered.

For example, 20% means 20/100, which equals 1/5. Here the slash denotes division. Likewise, 33% means 33/100. These are different ways of expressing a proportion, or the size of one quantity compared with another.

Result: A percentage is a fraction per hundred

Let z be the number written before the percentage sign. Then z% = z/100. To find that percentage of a quantity, multiply the quantity by this fraction. For instance, 50% of a quantity means half of it.

The whole corresponds to 100%. A percentage therefore needs a reference quantity: half of one quantity need not have the same size as half of another. Identifying what counts as the whole is the first step in interpreting a percentage.

The common denominator makes comparisons easier. The choice of 100 also fits conveniently with decimals in our base-ten number system, which groups numbers using powers of ten. For example, 31% is 31/100, or 0.31.

How do fractions, decimals and percentages connect?

Equivalent fractions have the same value even when their numerators and denominators differ. To express a fraction as a percentage, find its equivalent with denominator 100. Alternatively, multiply the fraction by 100 to obtain the number before the percentage sign.

Worked example 1. Red paint forms 3/4 of Surya’s deep orange mixture. Express the red share as a percentage.

Answer: Multiply numerator and denominator by 25: 3/4 = 75/100 = 75%. Thus, red paint forms 75% of the mixture.

Multiplying both parts of a fraction by the same non-zero number leaves its value unchanged. In this example, the denominator changes from 4 to 100, while the numerator changes from 3 to 75. The quantity of paint being described has not changed.

The horizontal bar is divided into four equal parts. Three parts are red and one is yellow. Fraction labels above align with hundredths and percentages below, ending at 1 and 100%.

Reference: NCERT Class 8, Chapter 1, page 2, unnumbered diagram

Worked example 2. Surya plans to save 2/5 of his prize money. What percentage will he save?

Answer: 2/5 = 20/50 = 40/100 = 40%. Saving two fifths therefore means saving 40% of the prize money.

How is a percentage converted back?

Write the percentage over 100, then simplify if useful. Thus, 24% = 24/100 = 6/25. Other equivalent forms include 12/50 and 48/200. A percentage does not have just one possible fraction representation.

A decimal writes fractional quantities using place values such as tenths and hundredths. Since 50/100 = 1/2 = 0.5, finding 50% of a value can be done by multiplying by either 1/2 or 0.5.

How can percentages of quantities be calculated mentally?

To calculate a percentage of a quantity, turn the percentage into a fraction or decimal and multiply. The word “of” indicates multiplication in this calculation. Keep the unit of the original quantity in the answer, such as grams for a mass.

Worked example 3. Madhu eats 120 grams of biscuits containing 25% sugar. Madhav eats 95 grams of biscuits containing 35% sugar. Who consumes more sugar?

Answer: Madhu consumes 25/100 × 120 = 30 grams of sugar. Madhav consumes 35/100 × 95 = 33.25 grams. Madhav therefore consumes more sugar in this comparison.

The percentages describe sugar proportions. The biscuit masses tell us the wholes to which those proportions apply. Comparing the percentages alone would not establish who consumed more sugar unless the biscuit quantities were also known.

Result: Percentages of the same whole can be added

Let y represent the quantity being considered. Then (20/100 × y) + (5/100 × y) = 25/100 × y. Thus, 20% of y plus 5% of y equals 25% of y.

The phrase “of the same whole” matters. Both calculations above use y as their reference quantity. This result supports mental methods: find 10%, double it for 20%, halve 10% for 5%, and combine the appropriate portions.

Recognising familiar fractions also helps. Finding 25% of 40 means finding a quarter of 40. Finding 50% means halving the quantity. These methods use the meaning of the percentage instead of treating the percentage sign as a separate operation.

Worked example 4. Zubin’s test has a maximum of 75 marks. An A grade requires 80% or above. Find the minimum score needed.

Answer: 80/100 × 75 = 4/5 × 75 = 60. Zubin needs at least 60 marks. The decimal calculation, 0.8 × 75, gives the same result.

How is a ratio converted into percentages?

A ratio compares quantities by their relative sizes. The colon in 2:7 means “2 to 7”. When a mixture contains ingredients in this ratio, the complete mixture contains 2 + 7 = 9 equal parts.

For a part’s percentage of the mixture, compare that part with the total number of parts. A ratio of millet to water is a comparison between two ingredients. A fraction of the mixture compares one ingredient with their combined total.

Worked example 5. A millet kanji mixture uses millet and water in the ratio 2:7. Find the millet percentage and the millet needed for 500 millilitres of mixture, using 22.22% for its share.

Answer: Millet forms 2/9 of the mixture, expressed approximately as 22.22%. Water forms 77.78%. Using that millet percentage, 500 × 22.22/100 = 111.1 millilitres of millet are needed.

Why estimate before calculating?

An estimate is an approximate value used to judge the likely size of an answer. Half of 9 is 4.5 and a quarter is 2.25, so 2/9 is below 25%. Since 20% of 9 is 1.8, the answer lies between 20% and 25%.

This check helps distinguish the correct fraction, 2/9, from the ingredient-to-ingredient comparison, 2/7. The denominator for a percentage of the mixture must represent the whole mixture. Often, exact values are unnecessary, and a quick estimate can be useful.

The figures 22.22%, 77.78% and 111.1 millilitres use decimal approximations. Keep track of that approximation when interpreting the calculated amount rather than treating a shortened decimal as an exact fraction.

How can a known percentage reveal the whole?

Sometimes a percentage and its corresponding quantity are known, while the whole is unknown. In that case, work backwards from the given part. Divide the known quantity by the percentage written as a decimal, or scale the percentage up to 100%.

Worked example 6. A cyclist has travelled 92 kilometres from Delhi towards Agra, completing 40% of the journey. Find the distance still to travel.

Answer: If 40% is 92 kilometres, 20% is 46 kilometres. The remaining 60% is 92 + 46 = 138 kilometres. Alternatively, the total is 92 ÷ 0.4 = 230 kilometres, leaving 230 − 92 = 138 kilometres.

A bar runs from Delhi to Agra. Its first portion is labelled 92 km and ends at 40%. The complete bar ends at 100%, with a question mark over the remaining portion. Here km means kilometres.

Reference: NCERT Class 8, Chapter 1, page 10, unnumbered diagram

What does a percentage above 100 mean?

A value above 100% exceeds the reference quantity. Kishanlal’s daily sales target is ₹5000, where ₹ denotes rupees. Sales of ₹6000 are 6000/5000 × 100 = 120% of the target, meaning 20% more than the target.

Similarly, a farmer’s harvest rises from 260 kilograms to 650 kilograms. The new harvest is 650/260 × 100 = 250% of the earlier harvest, or 2.5 times as much. “Percentage of the original” and “percentage increase” answer different questions.

How do percentages compare scores and describe change?

Percentages allow proportions with different totals to be compared on a common scale. A raw score by itself does not show the share of the available marks obtained. Both the score and the maximum possible score are needed.

Worked example 7. Eesha scores 42 out of 50 in English and 70 out of 80 in Science. In which test is her percentage higher?

Answer: English gives 42/50 × 100 = 84%. Science gives 70/80 × 100 = 87.5%. Her Science percentage is higher, so her performance relative to the available marks is better in Science.

Which amount is the base for change?

The base is the reference quantity treated as 100%. For percentage increase or decrease, use the original amount. First find the amount of change, then divide by the original amount and multiply by 100.

When the price of one kilogram of tomatoes rises from ₹30 to ₹42, the increase is ₹12. The percentage increase is 12/30 × 100 = 40%. Dividing by ₹42 instead would use a different reference amount.

Average footfall means the average number of visitors. If a theatre’s average footfall falls from 160 to 100, the decrease is 60. Its percentage decrease is 60/160 × 100 = 37.5%, measured against the earlier footfall.

Result: Increase and final percentage differ by the original whole

A quantity that becomes 165% of its original value has increased by 65%. The final quantity contains the original 100% plus the additional 65%. Conversely, an 85% decrease leaves 15% of the original quantity.

For a stated percentage rise, adding the change to the original quantity and multiplying by the corresponding final decimal are equivalent methods. Choosing the correct base gives both methods the same meaning.

How are profit, loss and their percentages found?

The cost price (CP) is the price paid to obtain an item. The marked price (MP) is the price quoted to the customer. The selling price (SP) is the amount the customer actually pays after any discount, meaning a reduction from the quoted price.

A profit occurs when the selling price exceeds the cost price. A loss occurs when the selling price is lower. For the usual comparison with the amount invested in buying goods, calculate the profit or loss percentage using the cost price as the base.

Worked example 8. Kishanlal buys a sweater for ₹300, marks it at ₹480 and sells it for ₹430. Find his profit and its percentage of cost.

Answer: Profit is ₹430 − ₹300 = ₹130. Profit percentage is 130/300 × 100, approximately 43.3%. The marked price is not the amount received and is not the base for this profit calculation.

Raghu buys rice at ₹35 per kilogram and sells 10 kilograms for ₹300. His cost is ₹350 and his loss is ₹50. The percentage loss is 50/350 × 100, expressed as 14.28% in the worked calculation.

Worked example 9. Shyamala buys a vase for ₹2650. It is slightly damaged, so she sells it at an 18% loss. Find the selling price.

Answer: The loss is 0.18 × 2650 = ₹477. The selling price is ₹2650 − ₹477 = ₹2173. Equivalently, 82% of the cost remains: 0.82 × 2650 = ₹2173.

How do gross profit and net profit differ?

Revenue is the money received from sales. Gross profit is sales revenue minus the cost of the goods sold. Net profit is the amount remaining after other expenses, such as transport, salaries and electricity, are deducted from gross profit.

Kishanlal’s sales of ₹80,000 and goods cost of ₹48,000 give gross profit of ₹32,000. Other expenses of ₹8000 leave net profit of ₹24,000. Unless otherwise specified here, profit means gross profit.

The base can change with the question. Manisha’s monthly net profit of ₹45,000 on revenue of ₹1,50,000 is 30% of revenue. This answers a different question from profit as a percentage of the purchase cost.

How do discounts and taxes change a price?

A discount percentage expresses a reduction as a share of the quoted price. A 30% discount removes 30% of that price. To find the amount payable, calculate the discount and subtract it, or multiply by the fraction of the price that remains.

The maximum retail price (MRP) may be the marked price of an item. Distinguish this quoted amount from its cost to the shopkeeper and the discounted price paid by the customer. These amounts play different roles in a calculation.

Worked example 10. A shirt originally priced at ₹300 has a 25% discount. How much does Anwar pay?

Answer: The discount is 25/100 × 300 = ₹75. Anwar pays ₹300 − ₹75 = ₹225. This is 75% of the original shirt price.

How is a specified tax added?

A tax is an amount paid to the government. Goods and Services Tax (GST) is one tax expressed using percentages. Where a question says tax is added to a price, calculate the tax on that price and add it.

Worked example 11. A mobile phone costs ₹8250 before an added GST of 18%. Find the price including this specified tax.

Answer: The tax is 8250 × 0.18 = ₹1485. The final price is ₹8250 + ₹1485 = ₹9735. The equivalent calculation is 8250 × 1.18 = ₹9735.

The decimal 0.18 represents the added part; 1.18 represents the original price plus that part. Similarly, a discount calculation needs the amount remaining after the reduction. Confusing the changed part with the final whole produces an incorrect selling price.

How does interest work without compounding?

Interest is extra money paid on a deposit or charged on borrowed money. The principal is the amount on which interest is calculated. An interest rate expresses that interest as a percentage of the principal for a specified period.

A fixed deposit (FD) keeps a specified sum deposited for a chosen period at a predetermined interest rate. The maturity is the end of that agreed period. The abbreviation p.a. means “per annum”, or “for every year”.

Worked example 12. A deposit of ₹6000 earns 10% p.a. Calculate the interest for one year and the amount including that interest.

Answer: Interest is 0.10 × 6000 = ₹600. Including the principal, the amount is ₹6000 + ₹600 = ₹6600, or 110% of the original deposit.

What happens when each year’s interest is paid out?

Without compounding, meaning adding earned interest back to the deposit, the interest is paid out and the principal stays unchanged. With ₹6000 at 10% p.a., ₹600 is paid out each year. After three years, the principal is returned as well.

Total interest received is ₹1800, and total money received is ₹1800 + ₹6000 = ₹7800. The ₹7800 includes the interest payments received across the three years. It is not all held in the deposit when interest is paid out annually.

Compared with the original deposit, ₹7800 is 130%. The gain is 30%, because the original ₹6000 accounts for 100%. This distinction between total amount and interest gained remains important when comparing deposit options.

How does compounding change the amount received?

With compounding, earned interest is added back to the deposit for the next period. The next calculation therefore uses an increased principal. At a positive interest rate, later interest payments grow because they are calculated on a larger amount.

Worked example 13. Deposit ₹6000 at 10% p.a., adding each year’s interest back to the deposit. Find the amount after three years.

Answer: After Year 1, ₹600 interest gives ₹6600. Year 2 interest is ₹660, giving ₹7260. Year 3 interest is ₹726, giving ₹7986. Equivalently, 6000 × 1.1 × 1.1 × 1.1 = ₹7986.

The final amount is 133.1% of the initial ₹6000, so the percentage gain is 33.1%. Without compounding, the corresponding gain over three years is 30%. The difference arises because retained interest contributes to later calculations.

How are the two patterns written algebraically?

Let p represent the initial principal, r the interest rate per term written as a decimal, and t the number of terms. A term is one interest-calculation period. For annual calculations at 10%, r is 0.10 and t counts years.

Without compounding, interest per term is p × r. Across t terms it is p × r × t. Adding the original principal gives p(1 + rt), where adjacent letters mean multiplication.

With compounding, multiply by (1 + r) once per term. The amount is p × (1 + r)ᵗ. The superscript t is an exponent: here it tells how many times the factor (1 + r) is multiplied together.

Note: Enter a percentage rate as its decimal in these formulas. At 10% per year, use 0.10, not 10. Keep the period for the rate consistent with the periods counted by t.

How do percentage decreases model depreciation and decline?

Depreciation is a reduction in an item’s value due to use and age. Its extent could depend on factors such as years since purchase, use, damage and replacement of parts. A stated percentage decrease gives a way to model a particular reduction.

Worked example 14. A television costs ₹21,000. Its value depreciates by 5% after one year. Find its value then.

Answer: The reduction is 0.05 × 21,000 = ₹1050. Its value becomes ₹21,000 − ₹1050 = ₹19,950. Equivalently, 95% remains, so 0.95 × 21,000 = ₹19,950.

Why does repeated decline use a changing base?

If a percentage decrease applies in each successive period, the next decrease is calculated on the quantity remaining. Repeatedly subtracting the same amount would represent a different pattern. A repeated percentage decline works through repeated multiplication by the remaining fraction.

A village’s population was observed to fall by about 10% every decade, where a decade is ten years. Starting from 1250, the estimated population after three decades is calculated using 1250 × 0.9 × 0.9 × 0.9 = 911.25.

The expected population is around 910. This is an estimate based on an approximate rate, not a claim that a fractional person can be counted. The value 0.9 represents the 90% remaining after each decade’s approximate 10% decrease.

Growth and decline calculations both require attention to what each rate applies to. A forecast also retains the uncertainty of its starting assumptions. An observed rate described as “about” a given percentage does not justify an exact prediction.

Why can successive percentages be misleading?

Successive percentage changes are changes applied one after another. Each may use the result of the previous change as its base. Adding or subtracting the quoted percentage numbers without checking their bases can therefore give the wrong final amount.

Worked example 15. A ₹200 cake receives successive discounts of 30% and 20% at Cakely. Cakify offers a single 50% discount on a ₹200 cake. Which gives the lower price?

Answer: Cakely’s first discount is ₹60, leaving ₹140. The second is 20% of ₹140, or ₹28, leaving ₹112. Cakify’s single 50% discount gives ₹100, which is cheaper.

Although 30% + 20% equals 50% when added as numbers, successive discounts use different prices. The second discount above is calculated on ₹140, not ₹200. This is why the two advertised offers give different payable amounts.

Can equal percentage increases and decreases cancel?

Surbhi adds a 50% profit margin to her purchase cost, meaning she sets a price 50% above that cost. She then offers a 50% discount on the increased price. Let x denote the purchase cost of the cookware.

The increased price is 1.5x. Halving it leaves 0.75x, or three quarters of the cost. The result is a 25% loss. If she receives ₹12,000 after discount, then 0.75x = 12,000, giving a cost of ₹16,000 and a loss of ₹4000.

Why distinguish percentage gain from money gained?

A contest offers ₹300 back after a ₹100 deposit, or ₹1500 back after a ₹1000 deposit. The first option gives a larger percentage gain, while the second gives a larger amount of money gained. The deposits, and therefore the bases, differ.

For any comparison, identify what the question asks: a fraction of the starting amount, the final amount, or an absolute gain. Percentages compare proportions. They do not by themselves establish which option gives the larger quantity in rupees.

Glossary

• Percentage — A fraction expressed per hundred, showing a quantity relative to a stated whole.
• Equivalent fractions — Fractions with different numerators and denominators that nevertheless represent the same numerical value.
• Base — The reference quantity treated as one hundred per cent in a percentage comparison.
• Ratio — A comparison of quantities by their relative sizes, such as millet to water.
• Cost price — The price a seller pays to buy the item being considered in a transaction.
• Marked price — The price quoted to a customer before any negotiated reduction or discount is applied.
• Selling price — The amount received by the seller when the customer purchases an item.
• Discount — A reduction from a quoted price, often expressed as a percentage of that price.
• Gross profit — Sales revenue minus the purchase cost of the goods that have been sold.
• Net profit — The profit remaining after other business expenses are deducted from the gross profit.
• Principal — The amount on which interest is calculated for the period being considered.
• Interest — Extra money received on deposits or paid for the use of borrowed money.
• Per annum — For every year, specifying the yearly period associated with a stated interest rate.
• Compounding — Adding earned interest back so that it contributes to the principal for the next period.
• Depreciation — A reduction in the value of an item due to its use and age.

Common errors and misconceptions

• Misconception: A percentage cannot exceed 100. Correct: 120% of a sales target means the sales exceed the target by 20% of that target.
• Misconception: A larger percentage necessarily means a larger amount. Correct: The wholes may differ; calculate the quantities before comparing absolute amounts.
• Misconception: Millet in a 2:7 millet-to-water mixture forms 2/7 of the whole. Correct: The mixture has nine parts, so millet forms 2/9.
• Misconception: Percentage increase uses the final quantity as its base. Correct: Divide the increase by the original quantity when calculating change from that original.
• Misconception: A 10% rate means using 10 in p(1 + rt). Correct: In this formula r is the decimal rate, so use 0.10.
• Misconception: Total amount and interest are the same. Correct: Total amount includes the principal; interest is the additional money earned.
• Misconception: Successive discounts of 30% and 20% equal one 50% discount. Correct: The second reduction applies to the already reduced price.
• Misconception: A 50% rise followed by a 50% fall restores the original value. Correct: The final value is 0.75 times the original, a 25% decrease.

Exam-style questions with model answers

Q1. Red paint forms 3/4 of Surya’s paint mixture. Convert this fraction to a percentage, showing the equivalent fraction with denominator 100. [2 marks]

1. Multiply the numerator and denominator by 25: 3/4 = 75/100.
2. A fraction of 75 out of 100 means 75%, so red paint forms 75% of the mixture.

Q2. Madhu eats 120 grams of biscuits containing 25% sugar. Madhav eats 95 grams containing 35% sugar. Calculate each sugar quantity and compare them. [3 marks]

1. Madhu’s sugar quantity is 25/100 × 120 = 30 grams. The percentage applies to the 120 grams of biscuits he eats.
2. Madhav’s sugar quantity is 35/100 × 95 = 33.25 grams, using his own biscuit quantity as the whole.
3. Madhav consumes more sugar because 33.25 grams exceeds 30 grams. The comparison uses calculated sugar masses, rather than percentages alone.

Q3. A cyclist has covered 92 kilometres, which is 40% of the journey from Delhi to Agra. Find the total distance and the distance remaining, then verify the result. [4 marks]

1. The full journey represents 100%, and the completed part represents 40%. Therefore, 60% of the journey remains.
2. Convert 40% to 0.4. Dividing the known distance by this fraction gives the total distance: 92 ÷ 0.4 = 230 kilometres.
3. Subtract the completed distance from the total: 230 − 92 = 138 kilometres remain.
4. Check by calculating 60% of 230: 0.6 × 230 = 138 kilometres, agreeing with the subtraction method.

Q4. Kishanlal buys a sweater for ₹300, quotes ₹480 and sells it for ₹430. Identify the three prices, find the profit and calculate profit as a percentage of cost. [3 marks]

1. The cost price is ₹300, the marked price is ₹480 and the selling price is ₹430. The actual sale receipt is the last of these.
2. Profit equals selling price minus cost price: ₹430 − ₹300 = ₹130.
3. Using cost as the base, profit percentage is 130/300 × 100, approximately 43.3%. The marked price is not the base for the requested comparison.

Q5. A deposit of ₹6000 earns 10% per annum for three years. Compare total receipts when interest is paid out each year with the final amount when interest is added back annually. Explain why they differ. [5 marks]

1. Without compounding, each year’s interest is 0.10 × ₹6000 = ₹600 because the principal used for every calculation remains ₹6000.
2. Three interest payments total ₹1800. Including the returned principal, total receipts are ₹1800 + ₹6000 = ₹7800.
3. With annual compounding, the first year adds ₹600, giving ₹6600. This increased amount becomes the principal for the second year.
4. The second year adds ₹660, giving ₹7260. The third year adds ₹726, giving a final amount of ₹7986.
5. Compounding gives the larger amount because previously earned interest remains invested and earns further interest. Annual payouts leave the deposit’s principal unchanged.

Q6. Cakely offers successive discounts of 30% and 20% on a ₹200 cake. Cakify offers a single 50% discount on a ₹200 cake. Calculate both prices and explain which offer is cheaper. [5 marks]

1. At Cakely, the first discount is 30/100 × ₹200 = ₹60. This reduction is calculated on the original cake price.
2. After the first reduction, the cake price is ₹200 − ₹60 = ₹140. This becomes the base for the next discount.
3. The second discount is 20/100 × ₹140 = ₹28, so Cakely’s final price is ₹140 − ₹28 = ₹112.
4. At Cakify, a single 50% discount halves the original ₹200 price, giving a final price of ₹100.
5. Cakify is cheaper because ₹100 is less than ₹112. The successive discounts do not equal a single 50% discount because their bases differ.

Q7. A television is bought for ₹21,000 and depreciates by 5% after one year. Calculate the reduction and the value remaining, and show a second method. [3 marks]

1. The reduction in value is 5/100 × ₹21,000 = ₹1050. The original purchase value is the base for this percentage.
2. Subtract that reduction from the original value: ₹21,000 − ₹1050 = ₹19,950 after one year.
3. Alternatively, 95% of the original value remains. Multiplying ₹21,000 by 0.95 also gives ₹19,950, confirming the first calculation.

Q8. Surbhi sets cookware prices 50% above purchase cost and then discounts those prices by 50%. She receives ₹12,000 after discount. Find the purchase cost, loss and loss percentage, explaining the two percentage bases. [5 marks]

1. Let x represent the purchase cost in rupees. Adding 50% of this cost gives an initial selling price of 1.5x.
2. The discount is half of that increased price, so the final receipt is 0.5 × 1.5x = 0.75x.
3. The given final receipt is ₹12,000. Hence 0.75x = 12,000 and x = 12,000 ÷ 0.75 = ₹16,000.
4. The loss is ₹16,000 − ₹12,000 = ₹4000. Relative to cost, the loss percentage is 4000/16,000 × 100 = 25%.
5. The increase uses purchase cost as its base, whereas the discount uses the increased price. Equal percentage numbers therefore do not cancel in this situation.

Key takeaways

• A percentage is a fraction per hundred, and its meaning depends on the reference quantity treated as the whole.
• Convert a percentage to a fraction or decimal before multiplying it by the quantity whose part is required.
• For a mixture given as a ratio, add the parts before calculating each ingredient’s fraction of the whole mixture.
• A percentage above one hundred means the quantity exceeds its base; it does not make the comparison invalid.
• Calculate percentage increase or decrease using the original amount, and distinguish the change from the final percentage.
• Cost price, marked price and selling price have different roles; profit relative to purchase cost uses cost as the base.
• Compounding adds interest back to the deposit, increasing the principal used for the next interest calculation.
• Successive percentage changes use successive bases, so equal percentage increases and decreases need not cancel each other.

Test yourself

What fraction in simplest form represents 24%?

It is 24/100 = 6/25, found by dividing numerator and denominator by four.

What percentage is represented by 2/5?

It represents 40%, because 2/5 has the equivalent fraction 40/100.

A test is out of 75 and an A grade requires at least 80%. What score is needed?

At least 60 marks are needed, since 0.8 × 75 = 60.

Sales are ₹6000 against a ₹5000 target. What percentage of the target is achieved?

The result is 120%, because 6000/5000 × 100 = 120. This is 20% above the target.

A price rises from ₹30 to ₹42. What is the percentage increase?

The increase is ₹12, so the percentage increase is 12/30 × 100 = 40%.

What does 10% p.a. mean for a deposit?

It means an interest rate of ten per cent for every year, applied to the relevant principal.

A ₹6000 deposit earns 10% annually with compounding. What amount starts the second year?

The second year starts with ₹6600, comprising the ₹6000 deposit and ₹600 first-year interest.

Why does a 50% price rise followed by a 50% fall give a loss relative to the original?

The fall applies to the increased price. Multiplying by 1.5 and then 0.5 leaves 0.75 of the original value.