Percentage

Every basic percentage question is built from three things: the whole (the base), the percentage, and the part. One relation ties them together:

Part = Percentage100 × Whole

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The question hides one of the three and asks for it. So there are only three basic questions in percentage.

1. Find the part. "What is 35% of 800?" Part = 35100 × 800 = 280. Use the fraction chart wherever you can: 35% = 720, so 720 × 800 = 280.

2. Find the percentage. "What percent of 320 is 96?" Percentage = PartWhole × 100 = 96320 × 100 = 30%. The quantity written after of always goes in the denominator.

3. Find the whole. "60% of a number is 54. Find the number." 60% of the whole = 54, so 1% = 5460 = 0.9 and 100% = 90. With fractions it is shorter: 35 of the number = 54, so the number = 54 × 53 = 90.

A very common variation: "25% of a number is 60; what is 70% of it?" You don't need the number at all. 25% ↔ 60, so 70% ↔ 60 × 7025 = 168. When two percentages of the same number are linked, work with the ratio of the percentages.

Then there is the "rest" form. "In a school 55% are boys and there are 360 girls." Girls = 100% − 55% = 45%. So 45% ↔ 360, and the boys, 55%, ↔ 360 × 5545 = 440. Before you calculate anything, check which percentage the given number really belongs to.

Radhika had some chocolates and gave away 70% of them. She is left with 9. Those 9 are the remaining 30%, so 10% = 3, and she had 30 chocolates to begin with.

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Now the questions start joining two or more percentages. The rule does not change: of means multiply.

Percentage of a percentage. "40% of 75% of 5200" means 40100 × 75100 × 5200. Multiply step by step, or multiply the fractions first: 25 × 34 = 310, and 310 × 5200 = 1560. A fraction like 58 inside the chain is handled in exactly the same way.

Difference or sum of two percentages of the same number. "The difference between 35% and 15% of a number is 90." Both percentages are of the same number, so subtract the percentages directly: 35% − 15% = 20% of the number = 90, and the number = 450. In the same way, "15% and 30% of a number add up to 180" means 45% of the number = 180.

A number and its own percentage. "When 40% of a number is added to 90, we get the number itself." Treat the number as 100%. If 40% + 90 = 100%, then 90 has to be the remaining 60%. So 60% ↔ 90 and the number is 150.

A chain of "remaining". "25% are matriculates, 40% of the remaining are graduates, and the remaining 180 are post-graduates." Every "remaining" changes the base. After 25% go, 75% remain. 40% of that 75% go next, so 60% of 75% remain, which is 0.6 × 0.75 = 45% of the total. So 45% ↔ 180 and the total is 400. Whenever you read of the remaining, the base has changed: multiply, don't subtract.

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Take Radhika's pocket money. She spends 50% on books and 50% of the rest on snacks. Many students will say she has spent 100%. She hasn't. She spent 50% + 25% = 75%, and 25% is still with her, because the second 50% was taken of the remaining half.

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Percentage and ratio are two forms of the same concept. "B is 150% of A" means B : A = 150 : 100 = 3 : 2. And A : B = 3 : 5 means A is 35 = 60% of B.

The question you will see most here is: "18% of a = 24% of b. Find a : b."

First reason it out. If 18% of a equals 24% of b, then a must be the bigger number, because a smaller percentage of it gives the same amount. So
18100 a = 24100 b ⇒ 18a = 24b ⇒ ab = 2418 = 43

That gives the rule: if x% of A = y% of B, then A : B = y : x. The percentages cross over. For three quantities, write each one as 1its percentage. If 20% of A = 25% of B = 50% of C, then A : B : C = 120 : 125 : 150, and multiplying by 100 (the LCM) gives 5 : 4 : 2.

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Brackets look scary but are not. For "30% of (P + Q) = 40% of (P − Q)", cancel the percentage sign first: 3(P + Q) = 4(P − Q). Collect the P terms on one side and the Q terms on the other: P = 7Q, so P : Q = 7 : 1.

Say 30% of Radhika's savings equals 20% of her sister's savings. Then Radhika : sister = 20 : 30 = 2 : 3. The person with the bigger percentage has the smaller amount.

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When something changes from one value to another, the question asks "by what percent did it change?" The answer is always measured on the old (original) value, because the change happened to the old value.

Percentage change = ChangeOriginal value × 100

A price goes from ₹600 to ₹750. Change = 150, original = 600, so the increase = 150600 × 100 = 25%. Now the price comes back from ₹750 to ₹600. The change is again 150, but the original is now 750, so the decrease = 150750 × 100 = 20%. The change is the same, the base is different, and so the answer is different.

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Finding the new value. Increase by r% ⇒ new = old × 100 + r100. Decrease by r% ⇒ new = old × 100 − r100. ₹800 increased by 15% = 800 × 1.15 = ₹920.

Finding the original value. This is the step where marks are lost. If a salary after a 20% increase is ₹42,000, the original is not 42,000 − 20% of 42,000. The 20% was of the old salary. So old × 1.2 = 42,000 ⇒ old = 42,0001.2 = ₹35,000. To go back, divide by the multiplying factor.

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Percentage error. When a value is measured or calculated wrongly, the error is compared with the true (correct) value:
Percentage error = |Wrong value − Correct value|Correct value × 100
Bring both values to the same unit before you calculate: hours and minutes into minutes, kg and g into g, metres and cm into cm.

Radhika's height was recorded as 150 cm, but she is actually 160 cm tall. Error = 10 cm and the true value = 160 cm, so the percentage error = 10160 × 100 = 6.25%.

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"A is 20% more than B." Who is 100 here? The word than is attached to B, so B is the base. Take B = 100, and then A = 120.

Now the question turns around: "By what percent is B less than A?" The difference is still 20, but we are now comparing with A, so A is the base:
B is less than A by 20120 × 100 = 1623%.

Keep this one firmly in mind. 20% more one way does not mean 20% less the other way. The difference stays the same; the base changes.

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This gives two results worth learning:

• If A is r% more than B, then B is less than A by r100 + r × 100%.
• If A is r% less than B, then B is more than A by r100 − r × 100%.

With fractions you can do it mentally. 25% more = 14 more, so B : A = 4 : 5 and B is less by 15 = 20%. 20% less = 15 less, so B : A = 5 : 4 and B is more by 14 = 25%. 40% less = 25 less, so B : A = 5 : 3 and B is more by 23 = 6623%.

Chains. "A is 30% more than B, and B is 20% more than C." Start from the last one. Take C = 100; then B = 120 and A = 120 × 1.3 = 156. So A is 56% more than C. You can also multiply the factors directly: 1.3 × 1.2 = 1.56.

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Finding the base from a value. "A is 35% less than B and A = 520." A is 65% of B, so B = 5200.65 = 800.

A word on "times more". In SSC questions, "A is 3 times more than B" is taken as A = B + 3B = 4B, and the options are made on that reading. If the question says "3 times B" or "3 times of B", then A = 3B.

Radhika earns 50% more than her brother. Brother = 100, Radhika = 150. So the brother earns 50150 = 3313% less than Radhika, not 50% less.

When one change comes after another, the second change acts on the new value, not on the original one. That is why two changes can never simply be added.

Take ₹100 and increase it by 10%: it becomes 110. Now decrease it by 10%. 10% of 110 is 11, so it becomes 99. The net result is a 1% decrease, not zero. The decrease was taken on a bigger base, so it removed more than the increase had added.

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The cleanest method is the multiplying factor from the foundation:
Final = Original × 100 ± a100 × 100 ± b100 × …
+10% then −10% = 1.1 × 0.9 = 0.99 ⇒ 1% decrease.
+50% then −20% = 32 × 45 = 65 = 1.2 ⇒ 20% increase.

For two changes there is also a one-line formula. If the changes are a% and b% (with a minus sign for a decrease):
Net change = a + b + ab100 % (the ab rule of successive percentage)
+50%, −20%: 50 − 20 + 50 × (−20)100 = 30 − 10 = +20%.

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Two results are worth keeping ready:

• Increase by x% and then decrease by x% (or the other way round), and the net result is always a decrease of x²100%.
• To come back to the original after an increase of r%, the decrease needed is r100 + r × 100%. After a decrease of r%, the increase needed is r100 − r × 100%. This is the same change of base you saw in the previous type.

Area and product. The area of a rectangle is length × breadth, so a change in length and a change in breadth behave exactly like two successive changes. Length +50% with the breadth unchanged ⇒ area +50%. If the area has to stay the same, the breadth must fall by 50150 = 3313%, because 32 × 23 = 1. In a square or a circle both dimensions change by the same r%, so the area changes by 2r + r²100%.

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Radhika's pocket money went up by 10% in January and by another 10% in July. The total rise is 10 + 10 + 10 × 10100 = 21%, not 20%.

Solids: volume and surface area. The same product idea works in three dimensions. The volume of a cuboid is length × breadth × height, so three changes multiply, one factor for each dimension. If every edge of a cube grows by 20%, the volume becomes 1.2 × 1.2 × 1.2 = 1.728 times, an increase of 72.8%, while each face (an area) becomes 1.2 × 1.2 = 1.44 times, an increase of 44%.

Some dimensions count more than once. In a cylinder V = πr²h, so the radius is used twice and the height once. In a sphere the volume goes with r³ and the surface area with r². In a cone the volume goes with r² and h. So in every such question write the factor for each dimension, raise it to the power it has in the formula, multiply, and read the answer as (final factor − 1) × 100.

Radhika's cylindrical tin has its radius increased by 10% and its height decreased by 10%. Volume factor = 1.1 × 1.1 × 0.9 = 1.089, so the volume rises by 8.9%. The 10% and 10% did not cancel, because the radius is squared.

Water level. When the same water is poured into a wider vessel, the volume of water does not change, so (area of the base) × (height of the water) stays constant. This is the "product stays the same" rule from the area example above. If the radius of the vessel becomes 1.25 times, the base area becomes 1.5625 times, so the water level falls to 11.5625 = 0.64 of its old value, a fall of 36%.

Every household question of this type rests on one line:
Expenditure = Price × Consumption

So expenditure is a product of two things, and a product behaves exactly like the area of a rectangle in the previous type. If the price goes up and you want to spend the same money, consumption has to come down. The question is, by how much?

A 20% rise in price means the price becomes 65 times. For the expenditure to stay the same, consumption must become 56 times, so it falls by 16 = 1623%, not by 20%. The base has changed once again.

• Price increases by r% ⇒ to keep the expenditure the same, cut consumption by r100 + r × 100%.
• Price decreases by r% ⇒ consumption can go up by r100 − r × 100%.

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If both price and consumption change, the expenditure changes by the successive formula a + b + ab100.

"A fall in price lets him buy x kg more." Read this one slowly. The money spent is fixed, say ₹M. When the price falls by r%, the same goods cost r% less, so the money saved is r% of M. The saved money buys the extra x kg at the new price. So
New price = r% of Mx, and Original price = New price1 − r100

Radhika buys sugar for ₹100. The price falls by 20%, so on her old quantity she now saves ₹20. With that ₹20 she buys 1 kg more, so the new price is ₹20 per kg, and the old price was 200.8 = ₹25 per kg.

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Fuel questions are plain products: fuel used in a month = distancemileage, and the extra expenditure = fuel used × rise in price per litre.

One equation runs this whole type:
Income = Expenditure + Saving

To begin, always take income = 100. If a man spends 75%, then expenditure = 75 and saving = 25. Every change can now be applied to these three numbers separately.

When income and expenditure change by different percentages, write the new income and the new expenditure, and get the new saving by subtraction. Income 100, expenditure 80, saving 20. Income rises 10% ⇒ 110. Expenditure rises 15% ⇒ 92. New saving = 110 − 92 = 18. The saving fell from 20 to 18, that is by 220 = 10%.

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Look at what happened there. The change in saving is not "10% − 15%". Saving is what is left over, so its percentage change can be much bigger or much smaller than the changes in income and expenditure. Never subtract the percentages; subtract the amounts.

The weighted form. Since income = expenditure + saving, the actual change in income = the change in expenditure + the change in saving. With income 100, expenditure 60 and saving 40:
A 20% rise in income = 20 units = (60 × e%) + (40 × s%)
If the saving rises by 35%, that is 14 units, so the expenditure must have risen by 6 units, which is 660 = 10%. When a question gives the change in income and in saving and asks for the change in expenditure, this form saves a lot of work.

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How the income is divided. "25% on rent, 30% on food, 15% on other things, and he saves ₹6,600." Saving = 100 − 70 = 30%, so 30% ↔ 6,600 and income = ₹22,000. This is Type 1 again, told as a household story.

Radhika gets ₹1,000 as pocket money and spends ₹800. If the pocket money rises by 10% to ₹1,100 and she still spends ₹800, her saving goes from ₹200 to ₹300. A 10% rise in income has given a 50% rise in saving.

Population and depreciation are successive percentage change spread over years. Each year's change acts on the value at the start of that year, so the multiplying factors just multiply.

The same rate every year. If a population P grows at r% per year, then after n years
Population = P × (1 + r100)ⁿ
A machine worth V that loses value (depreciates) at r% per year is worth
Value = V × (1 − r100)ⁿ
This is the compound interest formula, and you will meet it again in that chapter.

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Different rates in different years. Multiply the factors one by one: +10%, −20%, +25% ⇒ P × 1.1 × 0.8 × 1.25.

Going back in time. If the present population is given and the population n years ago is asked, divide by the factor: past = Present(1 + r100)ⁿ. The ratio form keeps the numbers clean. 8% growth means 25 : 27 each year, so over two years it is 625 : 729.

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Birth rate and death rate. If births add 10% and deaths remove 4% every year, the net growth is 10 − 4 = 6% per year, and the usual formula applies after that.

Coming back to the original. After a 20% fall, a rise of 2080 × 100 = 25% is needed to come back. It is the same change of base as before.

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Radhika's village has 1,000 people and the number grows by 10% each year. After one year there are 1,100, and after two years there are 1,210, not 1,200, because the second 10% is taken on 1,100.

In exam questions the maximum marks are the base. Marks obtained, pass marks, and the marks by which someone fails or passes are all parts of the maximum marks.

• Marks obtained = (percentage scored) × maximum marks
• Pass marks = (pass percentage) × maximum marks
• "Fails by 8 marks" ⇒ pass marks = marks obtained + 8
• "Passes by 10 marks" (or gets 10 more than the pass marks) ⇒ pass marks = marks obtained − 10

Two-situation questions. "A student gets 30% and fails by 15 marks. With 45% he would pass by 30 marks." The pass mark is the same in both situations. Going from 30% to 45%, the student moves from 15 below the pass mark to 30 above it, a jump of 15 + 30 = 45 marks. So 15% of the maximum marks = 45, and the maximum = 300.

If one situation is a fail and the other a pass, add the marks; if both are on the same side, subtract them. The gap between the two percentages always equals that sum or difference.

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Comparing performance. To see which subject went better, compare percentages, not raw marks. 13.2 out of 16 is 82.5%, while 16 out of 20 is only 80%.

"Both subjects" questions. If a% pass in English, b% in Maths and c% in both, then the students who passed in at least one subject = a + b − c. The "both" group was counted twice, so it is removed once. In the same way, failed in at least one = (failed in the first) + (failed in the second) − (failed in both), and passed in both = 100 − (failed in at least one).

In Radhika's class, 70% passed in Hindi, 60% passed in English and every student passed in at least one. Then 70 + 60 − both = 100, so 30% passed in both.

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Election questions look long, but they are a chain of "of the remaining", just like Type 2. Draw the chain once and the rest follows:

Total voters (on the list) → votes polled (cast) → invalid votes removed → valid votes → shared by the candidates

• If 30% did not vote, the votes polled = 70% of the total voters.
• If 10% of the polled votes were invalid, the valid votes = 90% of the polled votes.
• The candidates' percentages are usually of the valid votes. Read the line carefully, because sometimes they are of all the votes polled.

Two candidates. If the winner gets w% of the valid votes, the loser gets (100 − w)%. The margin of victory = w − (100 − w) = (2w − 100)% of the valid votes. A winner with 65% beats the loser (35%) by 30% of the valid votes.

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Linking the margin to the total. Multiply the factors in one line: margin = (margin %) × (valid fraction) × (polled fraction) × total voters.

Radhika's school has 500 students, and 80% of them vote in the school election. 10% of those votes are invalid, so the valid votes = 500 × 0.8 × 0.9 = 360. Radhika gets 55% of them, which is 198 votes, and she wins by 10% of 360 = 36 votes.

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Percentage composition. "Element A is 12% by weight in a mixture" means every 100 kg of the mixture contains 12 kg of A. So the amount of A = 12% of the mixture, and the mixture needed for 54 kg of A = 540.12 = 450 kg. It is a Type 1 question, with the mixture as the base.

Weighted percentage. When two groups have different percentages, the overall percentage is not the simple average of the two unless the groups are of equal size. Each percentage has to be weighted by the size of its group:
Overall % = (Size₁ × %₁) + (Size₂ × %₂)Size₁ + Size₂

70% of a class are boys and 30% of them pass; 30% are girls and 60% of them pass. Overall pass % = 0.7 × 30 + 0.3 × 60 = 21 + 18 = 39%. The answer sits closer to 30 than to 60 because the boys form the bigger group.

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Only when the two groups are equal (equal maximum marks, or 50% men and 50% women) is the overall percentage the simple average.

Alligation: working backwards. If the overall percentage is known and the ratio of the groups is asked, use alligation. Men change by −4%, women by +8%, and the overall change is +4%. Then
Men : Women = (8 − 4) : (4 − (−4)) = 4 : 8 = 1 : 2
The group whose percentage is farther from the average is the smaller group. You will use the same method again in the Mixture and Alligation chapter.

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Radhika scored 90% in a 50-mark test and 70% in a 150-mark test. Overall = 45 + 105200 = 75%, not 80%, because the second test carries more marks.

A tax is a percentage of an amount, and the whole difficulty is to know which amount that percentage is taken on. Three situations come up again and again.

1. GST or VAT added on a price. The price before tax is the base, 100%. GST at g% adds g% of that base, so the bill is base × 100 + g100. A phone priced at ₹20,000 before 18% GST costs 20,000 × 1.18 = ₹23,600, and ₹3,600 of it is GST.

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2. Getting the price or the tax out of a bill. When the bill already contains the GST, the bill is (100 + g)% of the base. So

base = bill × 100100 + g and GST = bill × g100 + g

A bill of ₹5,900 with 18% GST: base = 5,900 × 100118 = ₹5,000 and GST = ₹900. Notice that the GST is 18118 of the bill, not 18% of the bill.

3. Slab (bracket) tax. Income tax and electricity bills charge different rates on different portions of the amount. Only the part that falls inside a slab is charged at that slab's rate, and the earlier parts keep their own rates. Suppose there is no tax up to ₹3,00,000, 5% on the next ₹3,00,000 and 10% on whatever is above ₹6,00,000. On an income of ₹9,00,000 the tax is nil on the first 3 lakh, 5% of 3 lakh = ₹15,000 on the next 3 lakh, and 10% of 3 lakh = ₹30,000 on the last 3 lakh. Total ₹45,000. The tax paid on the whole income is 5%, not 10%: this is called the effective rate.

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When the rate changes. If GST on an article goes from 12% to 18%, the bill goes from 112 to 118 for every 100 of base. The rise is 6112 × 100 ≈ 5.36%, not 6%, because the change is measured against the old bill.

Discount and GST together. A 20% discount and a 10% GST are both multiplying factors on the same base, so base × 0.8 × 1.1 = base × 1.1 × 0.8. The order does not change the final bill. What matters is only which amount each percentage is taken on.