EQN-01-03

Percentages

Percent of a number, percent change, and reverse-percent problems — one of the most frequently tested ideas on the GAT-E.

~5 min read•Difficulty 2/5

The one definition you need

"Percent" means "per hundred". So $25% = 25/100 = 0.25.$ Every percent problem becomes easy once you turn the percent into a fraction or decimal.

Percent of a number

p % of N = (p /100) \times N

20% of 80 = 0.20 \times 80 = 16

Percent change

percent change = (new - old) / old \times 100

A price rises from 40 to 50: change $= (50 - 40) /40 \times 100 = 25%$ increase.

Successive changes do NOT add

A 10% increase followed by a 10% decrease is not zero.

100 \to + 10% \to 110 \to - 10% \to 99 (a net 1% decrease)

Always apply each change to the running value, not the original.

Reverse percent (a classic trap)

"After a 20% discount the price is 240. What was the original?" The 240 is 80% of the original, so original $= 240/0.80 = 300.$ Do not add 20% of 240.

Worked examples

Example 1 (difficulty 1/5)

A shirt costing 60 is discounted by 15%. What is the new price?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

After a 25% increase, a salary becomes 5000. What was it before?

◎ Predict the answer first:

Tips

Convert the percent to a decimal first — it removes the "/100" bookkeeping.
For reverse-percent, ask "the given number is what percent of the original?" then divide.
10% of any number is just the number with the decimal point moved one place left — build other percents from it (5% is half of 10%).

Common mistakes

Adding successive percentages (10% up then 10% down

=

no change)

Why: Percentages look additive.

Fix: Apply each change to the new running value; the bases differ.

In reverse percent, taking the rate off the final value

Why: The discount feels like it should be subtracted from the price you see.

Fix: The final value is

(100 -

rate)% of the original — divide, don’t subtract.

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