A rise and an equal fall never cancel, because the fall is a percentage of a different, larger amount than the rise was. Raise 500 by 20% and it becomes 600; fall by 20% from there and it lands at 480, not 500. Drag both percent changes below and watch exactly where it lands.
Raise 500 by 20% in the tool above and it becomes 600. Fall by 20% from there and it lands at 480 — not back at 500. The rise and the fall are the same 20%, but the fall is 20% of 600, a bigger number than the 500 the rise started from, so it takes away more than the rise added.
That is the whole trap. A percent change is only a percent of the original amount for the first change — every change after that is a percent of whatever the previous step left behind. Two changes that look like they should cancel almost never do, and when they don't cancel, the shortfall always goes the same direction: down.
Every percent change is a multiplication: a 20% rise multiplies by 1.20, and a 20% fall multiplies by 0.80. Chain two changes together and the two multipliers multiply, in either order, to give one combined multiplier for the whole journey: 1.20 × 0.80 = 0.96, so the two-step trip from 500 to 480 is the same as multiplying by 0.96 once.
This is the fact that makes stacked percent-change questions solvable without tracking every intermediate amount: find the combined multiplier first, and the overall percent change is just that multiplier minus one, turned into a percentage. Set the two sliders to +20% and −20% in the tool above and read the combined multiplier: ×0.96, an overall change of −4%.
Set both percent-change sliders to −50 in the tool above. A single 50%-off coupon takes 500 down to 250. Applying a second 50%-off coupon does not finish the job — it takes 250 down to 125, an overall change of −75%, not −100%. The two coupons together are worth 75% off, not free.
The reason is the same one driving every case here: the second coupon is 50% of 250, not 50% of the original 500. Adding the two percentages, −50% and −50%, suggests −100%, but the real overall change is −75% — an error of 25 percentage points, which is exactly (−50 × −50) / 100.
The combined effect of two percent changes, p1 and p2, is not p1 + p2. It is p1 + p2 + (p1 × p2) / 100 — the sum, plus a cross term that is the second change applied to the first change's own effect. That cross term is zero only when one of the two changes is 0%, which is the one case where simple addition happens to be exact.
Two rises compound the other way: set both sliders to +20% in the tool above and the combined change is +44%, not +40% — the missing 4 points are (20 × 20) / 100, the second rise applied to the extra amount the first rise already created. Successive increases always beat simple addition; successive decreases, or a rise and a fall, always fall short of it.
The Digital SAT rarely asks for a single percent change. The more common question chains two changes — a price marked up, then discounted; a population that grows, then shrinks — and asks for the overall percent change, or asks you to find one of the two percentages given the overall result. Both are the same skill: convert every step to a multiplier, multiply them, and convert back.
The wrong-but-tempting move is adding the two percentages, which the tool above shows failing by exactly the cross term every time neither change is 0%. A question that describes a rise and then an equal-sounding fall — or two coupons, or two years of decline — is testing whether you multiply the steps or just add them.
Drag the start slider to set the beginning amount, then set the two percent-change sliders to whatever sequence you want to test. Every readout updates together: the two multipliers, the combined multiplier, the final amount, and the overall percent change, alongside what simply adding the two percentages would have given instead.
The chart's three bars are the amount at each stage, with a dashed line marking where you started, so whether the final bar lands above or below it is visible before you read a single number.
Interactive Margin of Error Calculator: Sample Size vs. Confidence
See exactly how many responses it takes to shrink a margin of error, and why doubling your sample only gets you partway there. Drag the sample size and switch confidence level, and watch the exact interval narrow with the real z-score math shown at every step.
What does a practice SAT score range actually mean?
A score range is the platform admitting how much it does not know. The centre is its best estimate of your scaled score; the width is how far that estimate could be off given the evidence behind it. A single number from a practice test carries the same uncertainty — it just hides it.
How should you read your practice test results?
Read the domain breakdown before the score. The total tells you where you are; the breakdown tells you what to do on Monday. A 690 with one weak domain is about a week of work — a 690 that is evenly weak is closer to a month.
Every interactive tool: the full set. How a practice result turns into a score range: the scoring method.
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Two percent changes in a row
Every percent change is one multiplication, so two changes collapse into a single multiplier. Watch the third bar against the dashed start line: a rise and an equal fall never cancel, because the fall is a percentage of a different amount than the rise was.
500 x 1.20 = 600, and then 600 x 0.80 = 480. The two steps collapse into one multiplier, because multiplication does not care about the order or the stopping point: 1.20 x 0.80 = 0.96, so the whole journey is 500 x 0.96 = 480, an overall change of -4%.
Adding the two percentages gives 0%, and it is wrong by exactly (20 x -20) / 100 = -4 percentage points. That term is the second change applied to the first change, and it only vanishes when one of the changes is 0%. An equal rise and fall is the case worth memorising: x1.20 times x0.80 is x0.96, never x1.00, so 500 comes back as 480 — short by 20, a loss of 4%. The rise was 20% of 500; the fall was 20% of the larger 600.
