A Percentage Is Just a Fraction Wearing a Different Hat

Because percent means per hundred, so $25\%$ is simply the fraction $\frac{25}{100}$, which reduces to $\frac{1}{4}$.

The word is a unit, not a new kind of number — which is why percentages, fractions and decimals convert into one another freely. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: conversions, finding a percentage of a quantity, finding the whole, and turning a ratio into percentages.

Multiply the quantity by the percentage written as a fraction or decimal.

Worked examples.

$$15\%\text{ of }400=\frac{15}{100}\times 400=60$$

$$8\%\text{ of }₹2500=\frac{8}{100}\times 2500=₹200$$

$$40\%\text{ of }350=140,12.5\%\text{ of }800=100$$

Using an easy fraction is often quicker than the decimal. Since $25\%=\frac{1}{4}$:

$$25\%\text{ of }640=\frac{640}{4}=160$$

and since $20\%=\frac{1}{5}$, $20\%$ of $450$ is $90$.

Expressing one quantity as a percentage of another — divide, then multiply by 100:

$$45\text{ as a percentage of }180=\frac{45}{180}\times 100=25\%$$

A marks example: a student scoring $63$ out of $75$ has

$$\frac{63}{75}\times 100=84\%$$

Across units, convert first. What percentage is $250$ g of $2$ kg?

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

The order of the two quantities decides the answer, and reversing it is the standard error. The quantity after the word "of" goes on the bottom — so 45 of 180 is $25\%$, while 180 of 45 would be $400\%$, a completely different statement.

Divide the known part by the percentage as a fraction — or equivalently multiply by $\frac{100}{\text{percent}}$.

Worked example. If $20\%$ of a number is $80$, the number is

$$80\times \frac{100}{20}=400$$

Checking: $20\%$ of $400$ is indeed $80$.

Worked example. A student scores $84\%$ and gets $462$ marks. The total is

$$462\times \frac{100}{84}=550\text{ marks}$$

Worked example. $35\%$ of a sum of money is ₹1400, so the whole sum is

$$1400\times \frac{100}{35}=₹4000$$

Quantity remaining. A man spends $65\%$ of his salary and is left with ₹1400. The ₹1400 is the **remaining $35\%$**, so his salary is

$$1400\times \frac{100}{35}=₹4000$$

Two spendings. A woman spends $30\%$ on rent and $25\%$ on food, and ₹2700 is left. The part left is $100-30-25=45\%$, so her income is

$$2700\times \frac{100}{45}=₹6000$$

The step that decides these questions is identifying which percentage the given amount represents. In the salary problem the ₹1400 is not $65\%$ but the **$35\%$ left over — and using 65 would give about ₹2154, an income smaller than the amount supposedly spent, which a moment's sense check rejects. The whole must always exceed the part**.

Treat each term as a share of the total parts, then convert each share to a percentage.

Worked example. A quantity is split in the ratio $2:3$. The total parts are $5$, so:

$$\frac{2}{5}\times 100=40\%,\frac{3}{5}\times 100=60\%$$

Check: $40+60=100\%$.

Worked example with three parts. A ratio of $3:5:2$ has $10$ total parts:

$$\frac{3}{10}\times 100=30\%,\frac{5}{10}\times 100=50\%,\frac{2}{10}\times 100=20\%$$

Check: $30+50+20=100\%$.

Worked example with awkward numbers. A ratio of $1:2:5$ has $8$ parts:

$$\frac{1}{8}\times 100=12.5\%,\frac{2}{8}\times 100=25\%,\frac{5}{8}\times 100=62.5\%$$

Check: $12.5+25+62.5=100\%$.

Applied to a real quantity. If ₹2400 is divided in the ratio $3:5$, the percentages are $37.5\%$ and $62.5\%$, giving $₹900$ and $₹1500$ — which add back to ₹2400.

Running it backwards, percentages convert to a ratio: $40\%$ and $60\%$ give $40:60=2:3$.

The total of 100 percent is the check to run every time, and it is what catches the commonest mistake. A ratio of $2:3$ does not give $\frac{2}{3}=66.7\%$ and $\frac{3}{2}=150\%$ — those sum to well over 100. The denominator must be the total parts, which is 5, not the other term of the ratio.