Percentage

Percent only means this: if the whole is cut into 100 equal parts, how many of those parts do we have?

The word “percent” comes from two Latin pieces meaning per and hundred — for every hundred. When we say “40 percent”, we mean: imagine the whole thing cut into 100 equal parts; 40 of those parts are ours.

The sign for it is %. “40%” and “$\frac{40}{100}$” say exactly the same thing: a percentage is really a fraction whose bottom number (the denominator) is always 100.

Why do we need percent at all?

Because things with different totals cannot be compared directly. 36 out of 45 and 42 out of 60 — who did better? Turn both into “out of 100” and the answer is immediate. Percent is a common yardstick.

Switching between percent, fraction and decimal — and the handful of fractions worth memorising because they will save you minutes.

Half, 0.5 and 50% are not three different things; they are three ways of writing the same part. Convert the given form into whichever one makes the calculation easiest. Only three rules are needed:

Fractions worth memorising

In an exam time is short. If you already know that $\frac{3}{8}$ is 37.5%, you will not spend time dividing 3 by 8 every time. Look at the table below once, carefully; it has patterns.

Less common, but still asked

The most basic job — finding a percentage of a number — in three easy ways.

In percentage questions the word “of” always means multiply. “35% of 480” means: multiply 35% by 480. And 35% means $\frac{35}{100}$.

Two ways to make the calculation light

• Find 1% first. 1% = number ÷ 100 (move the point two places left). Then multiply by the percentage you need.
• Build with pieces of 10% and 5%. For 10% move the point one place left; 5% is half of that; 20% is double; 1% is a tenth of it.

How to find the whole amount when the part and its percentage are known.

So far we were given the whole and asked for a part. Many questions run the other way: “15% of a number is 63; what is the number?” Here the whole is unknown. The method is simple — first find what 1% is, then multiply it by one hundred.

“A is what percent of B” — the method, and how to avoid the most common slip.

Earlier we were given the whole and asked for a part. Now the question goes the other way: both numbers are given and we are asked “what percent is the part of the whole?” The number we compare against is called the base — it is treated as the “hundred”.

How do I spot the base?

One easy rule: the base is the number that comes right after “of”, “than” or “from”.

If A is 25% more than B, why is B not 25% less than A? Change the base and the percentage changes.

Suppose Neha earns ₹ 30,000 and Riya ₹ 24,000. The gap is ₹ 6,000. Yet “by what percent does Neha earn more than Riya” and “by what percent does Riya earn less than Neha” give different answers for the same gap. The reason: the base changes each time.

“New = old × multiplier” — the key to every increase-and-decrease question.

Say a thing costs ₹ 100 and its price rises by 20%. New price = 100 + 20 = ₹ 120 — that is 120% of the old price, or 1.2 times it. If it had fallen by 20% the new price would be 100 − 20 = ₹ 80 — 80% of the old, or 0.8 times.

The benefit is that instead of two steps (find the %, then add or subtract) you do just one multiplication. Here are the multipliers that appear most often.

Finding the percent change from an old and a new value, and finding the original value from the new one.

A percent change is always measured against the original (old) value, never the new one.

How to find the total effect when two or three changes follow one another — two successive discounts, or two years of growth.

Rising 20% and then rising another 10% is not a total rise of 30%. The second 10% is applied to the new, larger value, not the old one. Take the original as 100: 100 → 120 after the first rise → 10% of 120 is 12, so 132. The total rise is 32%, not 30%.

This formula is just the multiplier written short: expanding $(1+\frac{a}{100})(1+\frac{b}{100})-1$ gives $a+b+\frac{ab}{100}$. If you ever forget it, the multiplier is always there to fall back on.

How large a change is needed to win back the original value, or to keep spending steady after a price change.

If a price has fallen by 20%, raising it by 20% is not enough to win it back, because the base is now smaller. The price fell from 100 to 80; to return to 100 it must rise by 20, and 20 is $\frac{20}{80}=25\%$ of the new base 80.

Both formulas come from the same idea: the amount of change ÷ the value after the change.

When a quantity is the product of two things — price × quantity, length × breadth — how their changes combine.

Many quantities are the product of two things. When one of them changes, its effect combines with the other’s. The rule is the same as always: multiply the multipliers.

The minimum percentage needed to pass, and “failed by / passed by so many marks” questions.

The big idea in exam questions is that the pass mark is one fixed number, even when it is given as a percentage. A student who “fails by some marks” is that many marks short of it; one who “passes by some marks” is that many marks above it.

When a population, output or price rises or falls by a fixed percent each year, how to find its value after several years.

A population that grows 10% every year does not grow 30% in 3 years. Each year’s growth is applied to that year’s starting number, which has already been raised by the earlier growth. So the multiplier is applied once more every year — it is the same “successive change” idea, just identical each time.

The ladder of votes — from total voters down to valid votes, and then to the winner and the winning margin.

Election questions come in layers, and each layer’s percent applies to the layer just above it. Do not start multiplying until you have decided which percent applies to what.

• Invalid votes are not counted; candidates share only the valid votes.
• With only two candidates, if one gets $w\%$ the other gets $(100-w)\%$.
• Winning margin = winner’s votes − loser’s votes = $(2w-100)\%$ of the valid votes.

Splitting a salary into expenses, percent “of the remainder”, and what happens to savings when income and spending change.

Salary questions use two kinds of sentence: “25% of the salary” and “40% of the remainder”. In the first the base is the whole salary; in the second the base is what is left after the first expense. Spotting that difference is the whole question.

When some people have both of two qualities — say passed in both subjects — how to count the total percent without counting anyone twice.

Suppose 70% of students passed Maths and 65% passed English. Adding gives 135% — impossible! The reason: those who passed both were counted twice. So we must subtract them once.

And: at least one + neither = 100%. These two lines untangle every question of this kind.

The percent of a substance in a solution, and how it changes when water is added or boiled off.

“A 25% salt solution” means every 100 parts of the solution hold 25 parts of salt. The key observation for such questions is this: adding or boiling off water does not change the amount of salt — only the total quantity of solution changes. So find the amount of salt, set it aside, and then take its percent of the new total.

When different percents of two numbers are equal, the ratio of the numbers drops out directly.

It is given that x% of A equals y% of B. That means $\frac{x}{100}\times A=\frac{y}{100}\times B$. The 100 cancels on both sides and what remains is $x\times A=y\times B$. Now set A against B:

When a rate (like an interest rate) is itself a percent and it changes, the difference between “how many points” and “how many percent”.

Suppose a bank’s interest rate goes from 8% to 10%. This can be described in two ways:

The art of assuming a convenient number (like 100) when only percents are given, and questions that blend several percents.

Many questions give no real number, only percents. In that case assume a convenient total — usually 100, though 200 or 400 also works if the percents are like 25 or 20. If the answer is asked as a percent, the assumption does not change the answer, because a percent stays the same whatever the total.

Time is short in an exam. The table below takes you straight from the wording to the method.

Before solving, ask yourself one question: “What is given, what is asked, and which number is the base?” Then pick the matching row below.

Marks in percentage questions are rarely lost to hard arithmetic; they are lost to a wrong base or a wrong addition. Learn to spot these seven traps.

In an exam with negative marking one wrong answer can cost more than two right ones can earn — so build the habit of checking.

1. Look at the size. After a rise the number must be bigger, after a fall smaller. A part cannot be bigger than its whole.
2. Estimate roughly. To find 18% of 1,250, remember 20% of 1,250 is 250, so the answer must be a little less. 225 is close, so it looks right.
3. Put the answer back into the question. If you found an original amount, apply the given percent to it and see whether you get the number in the question.
4. Check the limits. A part cannot exceed 100% and a fall cannot exceed 100% (unless the question speaks of a rise).

One-step questions.

Each of these needs a single idea. Aim to do each in 30–40 seconds.

Changing the form

A percentage of a number

What percent?

Rise, fall and the whole

Two-step, exam-level questions.

These blend two ideas. For each question first write down what the base is.

Changes and the original value

Chains and combined percents

Groups and mixtures

Questions at the hard end of competitive exams.

These questions take thinking time. Do not hurry: first assume 100, then go step by step.

Multi-step changes

Thinking backwards

Ratios, mixtures and limits