I keep tripping over this when I try to do it fast in my head. When something goes from 50 to 60, is the percent increase 10/50 = 20% or 10/60 ≈ 16.7%? My brain wants to divide by the number I’m staring at (the new one), but the book answers don’t match that.
Another one: 80 to 96. I did 16/96 ≈ 16.7% and felt smug, but the answer key says 20% (which would be 16/80). Both feel like they have a story that makes sense. If I think “10% of 50 is 5, so 10 is two tens, so 20%,” that works. But then for 96, “10% of 96 is 9.6, and 16 is a bit less than two of those,” which points me the other way. Pick a lane, brain.
Analogy that’s probably not helping: if I add two pancakes to a stack, is that “two out of the old stack” or “two out of the new stack”? Because my fork only sees the new stack.
Can someone explain, simply, what the correct base is for percent increase and give me a quick rule-of-thumb so I stop second-guessing it? Also, side-quest: if I’m told “it increased by 20% to 96,” which way round do I undo it to get the original? I keep mixing up whether that’s 96 ÷ 0.8 or 96 ÷ 1.2.
My current (probably half-right) shortcut: difference over whichever base gives me cleaner mental math, then I round. But that seems to bite me on the 80→96 type problems.
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3 Responses
I trip on this too, but percent increase is always change ÷ old: 50→60 is 10/50 = 20% and 80→96 is 16/80 = 20%; to undo “increased by 20% to 96,” think new = old×1.2, so old = 96/1.2.
Rule of thumb: change/original for percent change, and divide by (1 + rate) to go backwards (nice refresher: https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-ratios-prop-topic/cc-6th-percent-word-problems/v/percent-word-problems-1); does that framing beat the pancake analogy for you?
Use the original as the base: percent increase = (new − old)/old, so 50→60 and 80→96 are both 20%, and to undo “up 20% to 96” do 96 ÷ 1.2 = 80.
When I first learned this I kept dividing by the new number for decreases because my eye fixed on the final stack, and I still lean on that shortcut sometimes even though it isn’t the official rule.
I love the pancake picture. The standard lane is: percent change = change divided by the original. So 50 to 60 is (60−50)/50 = 10/50 = 20%, and 80 to 96 is 16/80 = 20%. If you divide by the new number instead, you’re really answering a different question: “What fraction of the final stack was the extra bit?” That’s 10/60 ≈ 16.7% and 16/96 ≈ 16.7%. Both stories feel natural, but the first one is the convention for percent increase. My quick rule-of-thumb: start at the starting point. If you began at 80 and ended at 96, base it on 80. (Confession: I still sometimes catch myself doing change/new when the new number is friendlier, especially for small changes, because the two answers are close-just not identical!)
For the undoing: think in multipliers. “Increased by 20% to 96” means original × 1.2 = 96, so original = 96 ÷ 1.2 = 80. “Decreased by 20% to 96” means original × 0.8 = 96, so original = 96 ÷ 0.8 = 120. A tiny mental sticky note that sometimes steers me wrong: I’ll think “took off 20%, so divide by 1.2 to put it back,” which is actually mixing up the increase case with the decrease case-works in my head for about two seconds, then bites me. Multipliers keep it straight: up by p% = ×(1+p), down by p% = ×(1−p). Does it help if you try a decrease example like 150 down to 120 and say out loud which multiplier you’d use-and why?