1.1.18 · D5Arithmetic & Number Systems
Question bank — Percentages — finding %, % of a quantity, % increase - decrease
Before we start, one reminder that every item leans on: a percent is a fraction whose denominator is fixed at 100 (see Fractions and Decimals), and every percentage is measured against a base — the quantity that counts as the whole 100%. Half the traps below are really "which base?" in disguise.
True or false — justify
A 20% increase followed by a 20% decrease returns you to the start.
False. The rise multiplies by , but the fall of 20% acts on that larger value, so overall — a net 4% loss (see Figure 1, where the final bar sits below the dashed start line).
If A is 50% of B, then B is 50% of A.
False. If then of . Swapping the base flips the relationship completely.
"Increasing by 100%" and "doubling" are the same thing.
True. means multiply by , which is exactly doubling.
A quantity can decrease by 150%.
False. A decrease caps at 100% (that removes everything, leaving zero). To go below zero you'd need a signed quantity like temperature or profit — not a plain count.
A quantity can increase by 150%.
True. Increases have no upper limit; multiplies by , e.g. 40 becomes 100.
30% of 50 equals 50% of 30.
True. Both are ; "% of" is just multiplication, which is commutative.
If prices rise 10% every year, after 3 years they've risen 30%.
False. They multiply by , a 33.1% rise — the third year's 10% is taken on an already-grown amount (Figure 2 shows the extra sliver; this is the compounding in Simple and Compound Interest).
Two 25% discounts stacked give a 50% discount.
False. , so it's a 43.75% discount — the second 25% comes off the already-reduced price.
Saying "the number of cases doubled" is a bigger jump than "cases rose 90%".
True. Doubling is , which exceeds .
A 100% decrease and a 100% increase cancel out.
False. A 100% decrease gives zero, and no percentage increase can lift zero back up ( anything ). They never cancel.
Spot the error
"Shirt was ₹800, now ₹1000, that's a increase."
Wrong base. Increase is measured against the old value 800, so . The student divided by the new value.
"15% of ₹240 is 36%."
Wrong unit. of a quantity is itself a quantity — ₹36, not a percent. The % sign in the question doesn't force a % in the answer.
"Sales fell from 300 to 250, a fall of 20% (since 250→300 was 20%)."
The base changed. The fall is measured against 300 now: . Same 50-unit jump, bigger base, smaller percent.
"A grew 50% and B grew 50%, so A+B grew 50%."
This one is actually valid only because both grew by the same rate — . But if the two rates differ, the combined rate is a base-weighted average, not the plain average — so distrust the shortcut.
"To reverse a 25% increase, just take 25% off."
Wrong. A 25% rise multiplies by 1.25; to undo it you divide by 1.25, i.e. take off , a 20% decrease. Reversing a rise of needs a smaller drop.
"Interest of 8% per year for 5 years is 40% total."
True only for simple interest. For compound interest it's , i.e. 46.9% (see Simple and Compound Interest). The claim silently assumes no compounding.
"He's 20% taller than her, so she's 20% shorter than him."
Wrong. If he is her height, she is his, i.e. 16.7% shorter. Reversing the comparison changes the base.
"−30% of 200 doesn't make sense — percentages can't be negative."
Wrong. of ; the percent is simply a signed multiplier and the result carries the sign. This is exactly how a fall or a loss is written in Profit and Loss.
Why questions
Why can a price rise 50% but only ever fall by at most 100%?
Rising is unbounded — you can multiply by any factor above 1. Falling removes parts of the original whole, and once you've removed the whole (100%) there's nothing left; you can't remove more than exists.
Why do we always divide by the original value in percentage change, not the new one?
Because we're asking "how big is the change relative to where we started?" The starting value is the reference frame; using the new value would answer a different, inconsistent question.
Why does "+10% then −10%" leave you below the start rather than exactly back?
The two percentages use different bases. The −10% is applied to the raised value , which is bigger, so it subtracts more than the +10% added: (same mechanism as Figure 1).
Why can we multiply successive multipliers (like ) but not add the raw percentages?
Each percentage acts on the running total, so its effect is proportional to the current amount — that's multiplication. Adding percents ignores that each step reshuffles the base (this multiplicative chaining, drawn in Figure 2, is exactly Exponential Growth and Decay).
Why is 100 a convenient denominator for comparing fractions?
Fixing the denominator gives every fraction the same measuring stick, so and become instantly comparable without finding a common denominator.
Why is profit% based on cost price and not selling price (usually)?
Profit is the return on what you spent, so the natural base is the cost you laid out. Using selling price answers a different question (margin), and mixing them causes the classic errors in Profit and Loss.
Why is a 20% increase reversed by only a ~16.7% decrease, not another 20%?
The reversing decrease acts on the larger post-increase amount, so a smaller fraction of it equals the same absolute gap. , a 16.7% cut.
Edge cases
What is 0% of any quantity?
Always zero — you're taking , no matter how large is.
What is 100% of a quantity?
The quantity itself — . "100% of it" means the whole thing.
What percent is 0 of 50?
. Zero is zero parts of any nonzero whole.
What percent is 50 of 0?
Undefined — the base is zero, so has no value. You can't express something as a fraction of nothing.
What percent is 0 of 0?
Indeterminate — this is the form , which is different from "50 of 0". Here there's no contradiction (0 is 0 parts), but also no unique answer: any percentage of 0 is 0, so the ratio isn't pinned down. Treat it as undefined-by-indeterminacy, not undefined-by-division-by-a-nonzero-numerator.
What percent is −50 of 100?
. A negative part of a positive whole gives a negative percentage — the sign says the part points the opposite way (e.g. a debit against a positive balance).
What is −30% of 200?
. The negative percent is a signed multiplier; the answer is a negative quantity, which is how a decrease or loss is recorded.
What is the percentage change from 0 to 40?
Undefined (or "infinite/undefined growth") — the formula divides by the old value 0. Percentage change needs a nonzero starting point.
Can a percentage be greater than 100%?
Yes — just means , e.g. sales at 250% of last year are 2.5 times as large. Only shares of a single whole are capped at 100%.
If A = B, what percent is A of B, and what's the percentage change?
A is 100% of B (they're equal), and the change from B to A is 0%. Equality is the fixed point of both formulas.
What does a "negative percentage change" tell you, and can the amount itself be negative?
The negative sign flags a decrease (new < old); the magnitude is still a size. The underlying quantity needn't be negative — only the change is (though for a signed quantity like temperature, both can be).
Recall One-line survival kit
Every trap here is a base question. Read "increase/decrease of what?", "% of what?", "compared to which starting value?" — pin the base first, and almost none of these can fool you. Which single question defeats most percentage traps? ::: "What is the base?" — identify the whole/original before computing anything.
Connections
- Percentages — finding %, % of a quantity, % increase - decrease — the parent this bank drills.
- Fractions and Decimals — a percent is a fraction over 100; signed parts give signed percents.
- Ratio and Proportion — every "% of" is a ratio scaled to 100.
- Simple and Compound Interest — the compounding traps (Figure 2) live here.
- Profit and Loss — cost-price-vs-selling-price base confusion, and signed loss percentages.
- Exponential Growth and Decay — why successive multipliers multiply, not add (Figure 2).