A Price Raised 20 Percent Does Not Come Back Down by 20

No. A ₹500 item rising 20 percent becomes ₹600, and falling 20 percent from ₹600 loses ₹120 — leaving ₹480, not ₹500.

The two percentages were taken of different amounts, and that is the idea at the heart of this chapter. This page covers everything in the ICSE Class 7 Mathematics chapter on percentage: conversions, finding a percentage of a quantity, working back to the whole, and percentage increase and decrease.

Percent means per hundred, so $x\%$ is simply $\frac{x}{100}$.

Percentage to fraction — divide by 100 and reduce:

$$25\%=\frac{25}{100}=\frac{1}{4},12.5\%=\frac{12.5}{100}=\frac{1}{8}$$

Percentage to decimal — divide by 100, moving the point two places left:

$$25\%=0.25,7\%=0.07$$

Fraction to percentage — multiply by 100:

$$\frac{3}{8}=\frac{3}{8}\times 100=37.5\%$$

Decimal to percentage — multiply by 100:

$$0.45=45\%,1.2=120\%$$

Ratio to percentage — write it as a fraction, then multiply by 100:

$$2:5=\frac{2}{5}\times 100=40\%$$

A bank's interest rate of 7 percent and a shirt's 40 percent discount are the same idea in daily use.

A percentage can exceed 100, and that is not an error. $1.2=120\%$ simply means more than the whole — a price that has risen above its original value, for instance.

A percentage of a quantity — multiply:

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

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

One quantity as a percentage of another — divide the first by the second, 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

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

And 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. "45 as a percentage of 180" puts 180 on the bottom, giving 25 percent, whereas "180 as a percentage of 45" gives $\frac{180}{45}\times 100=400\%$ — a completely different statement. The quantity after the word of is always the denominator.

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

Worked example. If 20 percent of a number is 80, the number is

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

Worked example. A student scores 84 percent and gets 462 marks. The total is

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

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

$$1400\times \frac{100}{35}=₹4000$$

Two spendings. A woman spends 30 percent on rent and 25 percent on food, and ₹2700 is left. The part left is $100-30-25=45$ percent, 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 percent but the 35 percent left over — and using 65 would give ₹2154, which is not even bigger than what was spent. A quick sense check catches it: the whole must always exceed the part.

Both are worked out as a percentage of the original value:

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

Increase. A price rises from ₹250 to ₹300. The change is ₹50, so

$$\frac{50}{250}\times 100=20\%\text{ increase}$$

Decrease. A quantity falls from 80 to 60. The change is 20, so

$$\frac{20}{80}\times 100=25\%\text{ decrease}$$

Applying a change. A ₹1500 item increased by 12 percent becomes

$$1500\times \frac{112}{100}=₹1680$$

and decreased by 12 percent becomes $1500\times \frac{88}{100}=₹1320$.

Finding the original. After a 20 percent increase a price is ₹600. The original was

$$600\times \frac{100}{120}=₹500$$

Note that ₹600 is 120 percent of the original, not 100 percent — subtracting 20 percent of 600 would wrongly give ₹480.

That is precisely the trap in this page's opening question. The 20 percent rise was taken on ₹500 and the 20 percent fall on ₹600, so the two changes were ₹100 and ₹120 — unequal, which is why the price ends below where it began.