Percentage changes: choose the right base before using a shortcut

A percentage compares a part with a base of 100. To find 15% of 240, calculate $240 × 15 ÷ 100 = 36$. Here 240 is the base.

When a value changes, the percentage change usually compares the difference with the original value. A price rises from 400 to 460. The increase is 60, so the percentage rise is $60 ÷ 400 × 100 = 15$.

If the price falls from 460 back to 400, the decrease is still 60, but its base is now 460. The percentage fall is $60 ÷ 460 × 100$, approximately 13.04%. Equal differences do not mean equal percentage changes when their bases differ.

Before calculating, write “percentage of ___” in the margin. If the question says profit as a percentage of cost, the base is cost. If it asks the discount on marked price, the base is marked price. Learn those definitions in profit, loss and discount.

A 20% increase retains the original 100% and adds 20%, so its multiplier is 1.20. A 20% decrease retains 80%, so its multiplier is 0.80.

Starting with 500, a 20% rise followed by a 20% fall gives $500 × 1.20 × 0.80 = 480$. This is a 20 decrease from the original 500, or 4% overall.

For signed changes $a$ and $b$, the combined percentage is $a + b + ab ÷ 100$. Use positive values for increases and negative values for decreases. With 20 and −20, the result is $20 - 20 - 4 = -4$.

The multiplier method is often easier to audit. For a 10% increase followed by a 10% decrease, $1.10 × 0.90 = 0.99$, a 1% decrease. This assumes that each change applies to the value produced by the previous change. If two discounts apply independently to the original price, read the wording before multiplying.

After a 25% increase, a fee is 750. The original fee is not 750 minus 25% of 750: that would use the wrong base.

Let the original fee be $x$. Then $1.25x = 750$, so $x = 750 ÷ 1.25 = 600$. Check forward: 25% of 600 is 150, and 600 + 150 is 750.

After a 20% discount, a book costs 640. Its marked price is $640 ÷ 0.80 = 800$. Check: 20% of 800 is 160, leaving 640.

A useful boundary check is that a positive increase makes the final value larger than the original; a discount makes it smaller. If your reverse answer violates this, examine the multiplier or division.

A test accuracy rate rises from 40% to 50%. The difference is 10 percentage points. The relative increase in the rate is $10 ÷ 40 × 100 = 25\%$.

These answer different questions. Percentage points subtract two rates. Percentage change divides the difference by the original rate. Do not say “10% increase” when the question asks for the relative change from 40% to 50%.

For a population of 200 students, 40% is 80 and 50% is 100. The increase of 20 students is 25% of the original 80, which confirms the relative calculation. This example assumes the population stays at 200.

Try these with the answers covered:

1. A quantity rises from 800 to 920. Find its percentage increase.
2. A price of 1,000 receives successive discounts of 10% and 20%. Find the final price and overall discount.
3. A value is 900 after a 25% decrease. Find the original value.
4. An accuracy rate rises from 60% to 72%. Find the percentage-point difference and relative increase.

Worked answers: (1) $120 ÷ 800 × 100 = 15\%$. (2) $1000 × 0.90 × 0.80 = 720$, so the discount is 28%. (3) $900 ÷ 0.75 = 1200$. (4) 12 percentage points; $12 ÷ 60 × 100 = 20\%$ relative increase.

If you got a wrong answer, record the base you used in your error notebook. Then retry with different numbers. Continue with the percentage topic and put a closed-book review in the revision planner.