Finding the Whole From a Part Runs Every Step Backwards

**$240$ — and the way to get there is to run the ordinary calculation backwards.

Finding a percentage of a number multiplies** by $\frac{35}{100}$. So finding the number from the percentage must divide by $\frac{35}{100}$, which is the same as multiplying by $\frac{100}{35}$:

$$84\times \frac{100}{35}=240$$

Checking forwards: $35\%$ of $240=\frac{35}{100}\times 240=84$. Correct.

This is the one percentage question students most often attempt by taking a percentage of the part, which gives a smaller number when the answer must be bigger. This page covers the first part of the ICSE Class 8 Mathematics chapter on percentage: the conversions, the two directions of percentage of, and finding the whole.

These are the two directions of the same relationship, and telling them apart is most of the chapter.

**Direction 1 — a percentage of a quantity.** Multiply:

$$x\%\text{ of }Q=\frac{x}{100}\times Q$$

Worked example 1. Find $15\%$ of $\text{₹}840$.

$$\frac{15}{100}\times 840=\frac{12600}{100}=\text{₹}126$$

Worked example 2. Find $12.5\%$ of $360\text{kg}$.

Since $12.5\%=\frac{1}{8}$:

$$\frac{360}{8}=45\text{kg}$$

Recognising the fraction turned a multiplication into a single division — which is why the common conversions are worth memorising.

**Direction 2 — one quantity as a percentage of another.** Divide, then multiply by $100$:

$$\frac{\text{the part}}{\text{the whole}}\times 100$$

Worked example 3. Express $45\text{kg}$ as a percentage of $360\text{kg}$.

$$\frac{45}{360}\times 100=\frac{1}{8}\times 100=12.5\%$$

Notice this is exactly worked example 2 reversed, which is a useful way to check either one.

Worked example 4 — marks. A student scores $18$ marks out of $25$. What percentage is that?

$$\frac{18}{25}\times 100=72\%$$

**The critical point about of. The quantity after the word of is the whole, and it goes on the bottom**. So *$45$ as a percentage of $360$* puts $360$ underneath and gives $12.5\%$; putting them the other way round gives $800\%$, which is plainly wrong for a part smaller than the whole.

A size check that never fails. If the part is smaller than the whole, the answer must be **under $100\%$. If it is larger, the answer must be over $100\%$. Running that check takes a moment and catches an upside-down fraction every time.

Both quantities must be in the same unit.** Express $250\text{g}$ as a percentage of $2\text{kg}$: convert first, since $2\text{kg}=2000\text{g}$, and then

$$\frac{250}{2000}\times 100=12.5\%$$

Dividing $250$ by $2$ without converting would have given $12500\%$ — an answer whose absurd size is itself the warning.

**Divide the part by the percentage and multiply by $100$:**

$$\text{whole}=\frac{\text{part}}{x}\times 100$$

This is the reverse of finding a percentage of a quantity, so it divides where the forward calculation multiplied.

Worked example 1. $35\%$ of a number is $84$. Find the number.

$$\frac{84}{35}\times 100=2.4\times 100=240$$

Checking: $35\%$ of $240=84$. Correct.

Worked example 2. $15\%$ of a man's salary is $\text{₹}4500$. Find his salary.

$$\frac{4500}{15}\times 100=300\times 100=\text{₹}30000$$

Checking: $15\%$ of $30000=4500$. Correct.

**Worked example 3 — the percentage is the part that is left.** After spending $80\%$ of her money, a girl has $\text{₹}1500$ left. How much did she have?

The $\text{₹}1500$ is not $80\%$ — it is what remains, which is

$$100\%-80\%=20\%$$

So

$$\text{total}=\frac{1500}{20}\times 100=\text{₹}7500$$

Checking: $80\%$ of $7500=6000$ spent, leaving $7500-6000=1500$. Correct.

This is the commonest form of the question and the commonest place to go wrong. The percentage given and the amount given must refer to the same part, and here they did not until $80\%$ was turned into $20\%$.

Worked example 4 — a two-step version. In a school, $45\%$ of the students are girls and there are $605$ boys. How many students are there?

Boys are $100\%-45\%=55\%$, so

$$\text{total}=\frac{605}{55}\times 100=11\times 100=1100$$

Checking: $45\%$ of $1100=495$ girls, and $495+605=1100$. Correct.

The habit that makes all of these safe. Write down what percentage the given amount represents before doing any arithmetic. If the question says spent, remaining, boys or failed, decide first whether that is the percentage quoted or its complement — and only then divide.

And always check forwards. Take your answer, find the stated percentage of it, and confirm you get the given part back. It is one line, and it distinguishes a correct answer from a plausible one.