Percentage, Ratio and Proportion — CLAT Quantitative Techniques

If you master one area of CLAT Quant, make it this one. Percentage, ratio and proportion underlie the majority of caselets — sales changes, vote shares, mixtures, splits of money. The maths is Class-10, but a caselet dresses it in words. This chapter drills the core operations and the two ideas students most often trip on: reverse percentages and successive change.

1. Percentage — the three conversions

A percentage is just a fraction out of 100.

$x\%=\frac{x}{100},\text{fraction}\to \text{percent}:\times 100$

• Percent of a whole: $\text{part}=\frac{\text{rate}}{100}\times \text{whole}$. E.g. $15\%$ of $240=36$.
• Fraction to percent: $\frac{3}{5}\times 100=60\%$.
• Keep the common ones memorised: $\frac{1}{4}=25\%,\frac{1}{3}\approx 33.3\%,\frac{1}{8}=12.5\%$.

2. Percentage change

$\text{percentage change}=\frac{\text{change}}{\text{original}}\times 100$

• Sales rise from ₹80,000 to ₹1,00,000: change $=20,000$, so $\frac{20,000}{80,000}\times 100=25\%$ increase.
• Always divide by the original (the starting value), not the new one.

A rise from 400 to 500 is a 25% increase; a fall from 500 to 400 is a 20% decrease. The base differs, so the percentages differ.

3. Reverse percentages — finding the original

When the result after a change is given, work backwards by division.

• "A number increased by 25% becomes 250. Find the original."
• $\text{original}\times 1.25=250\Rightarrow \text{original}=\frac{250}{1.25}=200.$

Rule: to undo a $+p\%$ change, divide by $\left(1+\frac{p}{100}\right)$; to undo a $-p\%$ change, divide by $\left(1-\frac{p}{100}\right)$. Do not subtract the same percent back.

4. Successive percentage change

Two changes in a row multiply, they do not add.

$\text{net factor}=\left(1+\frac{a}{100}\right)\left(1+\frac{b}{100}\right)$

• A price rises 10% then 20%: net factor $=1.10\times 1.20=1.32$, i.e. a 32% increase (not 30%).
• A rise of 20% then a fall of 10%: $1.20\times 0.90=1.08$, a 8% net increase.

5. Ratio — sharing a total

A ratio $a:b$ splits a whole into $a+b$ parts.

$\text{share}=\frac{\text{your parts}}{\text{total parts}}\times \text{total}$

• ₹5,000 in the ratio $3:2$ → larger share $=\frac{3}{5}\times 5000=3000$; smaller $=2000$.
• Combining ratios: to join $a:b=2:3$ and $b:c=4:5$, make $b$ common — $2:3=8:12$ and $4:5=12:15$, so $a:b:c=8:12:15$.

6. Proportion — direct and inverse

• Direct: as one rises, the other rises in step — $\frac{x_1}{y_1}=\frac{x_2}{y_2}$. (More items, more cost.)
• Inverse: as one rises, the other falls — $x_1{y}_1=x_2{y}_2$. (More workers, fewer days.)

6 workers finish a job in 10 days; how long for 4 workers? Work $=6\times 10=60$ worker-days; time $=\frac{60}{4}=15$ days. Inverse proportion.

7. Exam protocol

1. For "percent of", multiply; for "what percent", divide by the original.
2. Reverse a change by dividing by the factor — never subtract the same percent.
3. Multiply successive percentage changes; do not add them.
4. Split a total by your parts over total parts.
5. Join ratios by making the common term equal.
6. Decide direct vs inverse before setting up a proportion.