A high-frequency Problem-Solving type (4.5% of the bank), built on one conversion: a percent change is a multiplier. Covers turning any increase or decrease into a number you multiply by, why successive changes multiply rather than add, percent of versus percent greater than, reversing a change by dividing rather than subtracting, and the percent-of-what question that decides most of these — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 11 of 19
4.5% of the question bank — 76 of 1675 questions
Objectives
Percentages are the arithmetic students are most confident about and lose the most marks on. The reason is that percentages are taught additively — add ten per cent, subtract twenty — and the test is built around situations where adding gives the wrong answer. Converting every change into a multiplier fixes essentially all of it.
One line carries the deck: up r per cent is times (1 plus r), down r per cent is times (1 minus r). Everything else follows from it.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 76 tagged questions of this type in the site's bank
Section
Section 1
Concept
Percentages are the arithmetic students are most confident about and lose the most marks on. The reason is that percentages are taught additively — add ten per cent, subtract twenty — and the test is built around situations where adding gives the wrong answer. Converting every change into a multiplier fixes essentially all of it.
You will see it phrased in these ways:
The single most important question to ask on any of these is per cent of what? Almost every wrong answer on this type comes from taking a percentage of the wrong base.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Percentages: the tell, the move, and the trap
The 0.96 in the green panel is the whole type in one number: a rise and an equal fall leave you four per cent down, every time, whatever you started with.
Prediction
The conversion has to be automatic before anything else works.
Predict first
A price falls by 35 per cent. What do you multiply by?
Correct: 0.65
Why: A fall of 35 per cent leaves 65 per cent behind, so the multiplier is 0.65. Multiplying by 0.35 would keep only 35 per cent, which is a 65 per cent crash — the same error the other way round. Multiplying by 1.35 models an increase. And a multiplier is never negative in these problems, because the quantity does not change sign. The rule to say aloud: the multiplier is what SURVIVES, not what disappears.
Concept
Four question shapes, all answered from the multiplier idea.
| variant | what it asks | the move |
|---|---|---|
| Apply a change | the new value after an increase or decrease | multiply by 1 plus or minus the rate |
| Find the percent | what percent of A is B, or how much greater | divide by the base, then read as a percentage |
| Successive changes | the result of two or more changes in a row | multiply the multipliers together |
| Reverse a change | the original value, given the new one | divide by the multiplier |
The last two are where the marks are. Both are handled badly by additive thinking and easily by multipliers.
Definition probe
Per cent of what is the question that decides the answer.
Sort into buckets
In each phrase, what is the percentage taken OF?
Discrimination
Six pairs of changes. Which end where they began?
Sort into buckets
Back to the starting value, or not?
Warm-up
Try it on instinct first, then check.
Discussion prompt
A jacket costs 200 dollars. Its price rises 30 per cent, then falls 30 per cent. What is the final price?
Hint: Compute each step rather than reasoning about the percentages.
Answer:
182 dollars, not 200. The rise gives 200 times 1.30, which is 260. The fall gives 260 times 0.70, which is 182.
Why: the 30 per cent rise was taken of 200, but the 30 per cent fall was taken of 260 — a bigger number, so it removed more than the rise added.
The multiplier version says it in one line: 1.30 times 0.70 is 0.91, so the final price is 91 per cent of the original, a 9 per cent fall.
That 0.91 does not depend on the starting price at all. Any item rising then falling 30 per cent ends at 91 per cent of where it began.
Pattern
Three steps, and the first is a question rather than a calculation.
Ask per cent of what, and underline the base in the stem.
Why: The base is whatever follows the word of or than. Almost every wrong answer on this type comes from using the wrong base.
Convert each change into a multiplier: 1 plus the rate for a rise, 1 minus the rate for a fall.
Why: A multiplier makes successive changes multiply and makes reversal a division, which is where additive thinking fails.
Multiply the multipliers together, then apply the result once.
Why: Combining first and applying once avoids rounding at intermediate stages and shows the overall effect as a single number.
Step 3 gives you something extra: the combined multiplier is the answer to the overall percentage change, which is often what the question actually wanted.
Section
Section 2
Concept
An increase of r per cent means multiplying by 1 plus r; a decrease of r per cent means multiplying by 1 minus r.
| the change | the multiplier | check on 100 |
|---|---|---|
| up 12 per cent | 1.12 | 100 becomes 112 |
| down 12 per cent | 0.88 | 100 becomes 88 |
| up 100 per cent | 2 | 100 becomes 200 |
| down 100 per cent | 0 | 100 becomes 0 |
| up 5 per cent | 1.05 | 100 becomes 105 |
| down 5 per cent | 0.95 | 100 becomes 95 |
Testing on 100 is the free check. If your multiplier does not send 100 to the obviously right number, it is wrong.
Concept
Two changes in a row give the product of their multipliers, never the sum of their percentages.
That last point is worth knowing, because questions sometimes ask whether applying a discount before or after tax matters. It does not.
Prediction
What survives, not what is lost.
Predict first
A population falls by 8 per cent. What is the multiplier?
Correct: 0.92
Why: Losing 8 per cent leaves 92 per cent, so the multiplier is 0.92. Checking on 100: it becomes 92, which is right. Using 0.08 would leave only 8 per cent, a 92 per cent collapse, and it is the single most common error on decrease questions. The rule to say aloud is that the multiplier is what survives.
Concept
Percent of gives the whole new value; percent greater than gives only the change.
| phrase | means | if the base is 80 |
|---|---|---|
| 25 per cent of the base | 0.25 times the base | 20 |
| 25 per cent greater than the base | 1.25 times the base | 100 |
| 25 per cent less than the base | 0.75 times the base | 60 |
| the base is 25 per cent of it | the base divided by 0.25 | 320 |
Read the sentence twice on these. The difference between of and greater than is the difference between 20 and 100.
Prediction
Multiply the multipliers.
Predict first
A price rises 10 per cent, then rises 10 per cent again. What is the overall change?
Correct: up 21 per cent
Why: The combined multiplier is 1.10 times 1.10, which is 1.21, so the overall rise is 21 per cent. Checking on 100: it becomes 110, then 121. The additive answer of 20 per cent misses the extra 1 per cent, which is the second rise applied to the first rise — small here, and much larger over more periods.
Concept
To find a percentage change, divide the difference by the value you started from.
\[ \text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100 \]
The word from tells you the base: a change FROM 80 TO 100 divides by 80. Reading the direction fixes the denominator.
Concept
If a value has already changed, recover the original by dividing by the multiplier.
Reversal questions are frequent and are almost always answered additively by students. Dividing by the multiplier is the whole method.
Prediction
Divide, do not subtract.
Predict first
After a 25 per cent discount an item costs 90 dollars. What was the original price?
Correct: 120
Why: The sale price is 75 per cent of the original, so the original is 90 divided by 0.75, which is 120. Checking: 25 per cent of 120 is 30, and 120 minus 30 is 90. Adding 25 per cent of 90 gives 112.50, which is the additive error — it takes the discount off the wrong base, since the discount was 25 per cent of 120, not of 90.
Concept
To return to the starting value, the two multipliers must multiply to exactly 1.
The asymmetry is the point: the percentage needed to undo a change is never the same as the percentage of the change itself.
Concept
A rise from 20 per cent to 25 per cent is 5 percentage points, but a 25 per cent increase.
Read whether the question says points or per cent. It is a single word and it changes the answer completely.
Prediction
One word changes the answer.
Predict first
A rate rises from 40 per cent to 50 per cent. By what per cent did it increase?
Correct: 25 per cent
Why: The change is 10 divided by the original 40, which is 0.25, so a 25 per cent increase. The answer 10 is the rise in percentage POINTS, which is a different and equally valid quantity — but it answers a different question. Both numbers are always offered, and the deciding word is whether the stem says points or per cent.
Two truths and a lie
Three of these percentage statements are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: A 50 per cent rise is times 1.5, and its reciprocal is two thirds, which is a fall of about 33.3 per cent. A 50 per cent fall would take you to 0.75 of the original, a quarter below where you began. The general fact is that undoing a rise of r always requires a smaller percentage fall, because the fall is applied to a larger base.
Check
Combine the multipliers first.
Check your understanding
A price increases by 50 per cent and then decreases by 40 per cent. What is the overall change?
Answer: A
Why: The combined multiplier is 1.50 times 0.60, which is 0.90, so the price ends at 90 per cent of where it began — a 10 per cent decrease. Checking on 100: it rises to 150, then falls by 60 to 90.
Note that B is the additive answer and it has the wrong SIGN as well as the wrong size here, which makes this a particularly clean test of the method.
Section
Section 3
Worked example
A laptop costs 640 dollars and is discounted by 15 per cent. What is the sale price?
Figure (svg): Bars comparing the original price with the discounted price
Convert the discount to a multiplier: down 15 per cent leaves 85 per cent, so multiply by 0.85.
Why: The multiplier is what remains after the reduction, not the reduction itself.
Compute 640 times 0.85.
Why: One multiplication replaces finding the discount and subtracting it.
That gives 544 dollars.
Why: Eighty-five per cent of 640 is 544.
The two-step route works and takes twice as long. The multiplier becomes essential once there is more than one change.
Verify: check the other way: 15 per cent of 640 is 96, and 640 minus 96 is 544.
Why: Both routes agree, confirming the multiplier was the right way round.
Answer: 544 dollars
Worked example
A share price rises 25 per cent in one year, then falls 20 per cent the next. What is the overall change?
Figure (svg): Bars showing a rise to 125 and a fall back to 100
Convert both: up 25 per cent is 1.25, and down 20 per cent is 0.80.
Why: Each change becomes a multiplier before anything is combined.
Multiply them: 1.25 times 0.80 is exactly 1.00.
Why: Successive changes multiply, and these two happen to be reciprocals.
A combined multiplier of 1 means no overall change.
Why: The price ends exactly where it started.
This is the case that surprises people: a 25 per cent rise is undone by a 20 per cent fall, not a 25 per cent one. The percentages differ because the bases differ.
Verify: check on 100: it rises to 125, then falls by 25 to 100.
Why: Twenty per cent of 125 is 25, so the fall exactly undoes the rise.
Answer: no overall change
Worked example
A town's population grew from 12,500 to 14,000. What was the percent increase?
Figure (svg): A number line marking the old and new populations in thousands
Find the change: 14,000 minus 12,500, which is 1,500.
Why: The difference is the numerator of the percent-change formula.
Divide by the ORIGINAL, 12,500: 1,500 over 12,500 is 0.12.
Why: Percent change is always relative to the starting value.
Convert to a percentage: 12 per cent.
Why: Multiplying the decimal by 100 expresses it as a percentage.
Dividing by 14,000 instead gives about 10.7 per cent, which is a genuine quantity — the change as a fraction of the NEW value — and the wrong answer here.
Verify: check forwards: 12,500 times 1.12 is 14,000.
Why: Applying the computed increase reproduces the new value exactly.
Answer: a 12 per cent increase
Step zero
Before any arithmetic.
Discussion prompt
A question says a price was reduced by 30 per cent and is now 84 dollars, then asks for the original price. What is the first question you ask yourself, and what does it rule out?
Hint: The percentage was taken of something.
Answer:
Ask: 30 per cent of WHAT? The answer is the original price, not the current one.
That rules out adding 30 per cent to 84. Thirty per cent of 84 is 25.20, giving 109.20 — but that discount would have been taken of 84, which was never the base.
The correct reading: 84 is 70 per cent of the original, so the original is 84 divided by 0.70, which is 120.
The forward check confirms it: 30 per cent of 120 is 36, and 120 minus 36 is 84.
The whole difficulty of reversal questions is that the base is the number you do not yet have, which is exactly why dividing rather than subtracting is the method.
Worked example
After a 12 per cent increase, a salary is 47,040 dollars. What was it before?
Figure (svg): A table comparing subtracting twelve per cent with dividing by the multiplier
The new salary is 1.12 times the old one.
Why: A 12 per cent increase multiplies the original by 1.12.
So the old salary is 47,040 divided by 1.12.
Why: Reversing a multiplication is a division, not a subtraction.
That gives 42,000 dollars.
Why: The division undoes the increase exactly.
Taking 12 per cent off 47,040 gives 41,395.20, which fails the forward check. That check takes five seconds and settles the question.
Verify: check forwards: 42,000 times 1.12 is 47,040.
Why: The forward check is the only reliable way to confirm a reversal.
Answer: 42,000 dollars
Worked example
A is 60. B is 40 per cent greater than A. C is 40 per cent of A. Find B and C.
Figure (svg): Bars contrasting forty per cent greater than sixty with forty per cent of sixty
B is 40 per cent GREATER, so the multiplier is 1.40: B equals 60 times 1.40.
Why: Greater than means the change is added on top of the base.
That gives B equals 84.
Why: One hundred and forty per cent of 60 is 84.
C is 40 per cent OF, so the multiplier is 0.40: C equals 60 times 0.40, which is 24.
Why: Of on its own means a portion of the base, not an addition to it.
The verify step exposes the relationship: the amount ADDED in a percent-greater question is exactly the percent-of value. That is why the two answers are so easy to confuse.
Verify: check the difference: B minus A is 24, which is 40 per cent of 60 — and that is exactly C.
Why: The amount B exceeds A by is C, which confirms both readings are consistent.
Answer: B = 84 and C = 24
Faded example
From memory. Four lines that carry the type.
Fill in the blanks
An increase of r per cent means multiplying by 1 plus r. A decrease of r per cent means multiplying by 1 minus r. Two changes in a row are combined by multiplying the multipliers. And to undo a change, you divide by its multiplier.
Why: The third and fourth lines are where additive thinking fails. Adding percentages is wrong because the second change is taken of a different base, and subtracting to reverse is wrong for exactly the same reason. Both errors have the same root: forgetting to ask per cent of what.
Worked example
A shop marks an item up 60 per cent from cost, then offers 25 per cent off. If the cost was 40 dollars, what is the final price, and what is the overall change from cost?
Figure (svg): Bars showing the markup to sixty-four and the discount down to forty-eight
Convert both changes: up 60 per cent is 1.60, and down 25 per cent is 0.75.
Why: Each change becomes a multiplier before combining.
Combine: 1.60 times 0.75 is 1.20.
Why: Successive changes multiply, and combining first avoids intermediate rounding.
Apply once: 40 times 1.20 is 48 dollars, which is a 20 per cent increase over cost.
Why: The combined multiplier gives both the final price and the overall percentage change.
The combined multiplier answered two questions at once. That is the practical reason to combine before applying rather than after.
Verify: check step by step: 40 times 1.60 is 64, and 64 times 0.75 is 48.
Why: The stepwise route reproduces the same final figure.
Answer: 48 dollars, a 20 per cent increase over cost
Fill the middle
Multiply, then read the overall change.
Fill in the blanks
Up 20 then down 20 gives a combined multiplier of 0.96, which is an overall 4 per cent fall. Up 25 then down 20 gives 1, which is no change overall.
Why: The contrast between these two rows is the whole lesson. Equal percentages up and down leave you down; it takes a SMALLER percentage fall to undo a rise, because the fall applies to a larger base. Only reciprocal multipliers cancel exactly.
Estimation
A rough answer catches a wrong-base error.
Predict first
An item costing 79.99 dollars is discounted 30 per cent. Roughly what is the sale price?
Correct: about 56
Why: Round the price to 80 and take 70 per cent of it: 0.7 times 80 is 56. The exact answer is 55.99. The answer 24 uses 0.30 as the multiplier, keeping only 30 per cent — the classic reversal of the multiplier — and 104 applies an increase instead. Having the estimate first means each of those lands visibly far from expectation.
Check
Divide by the multiplier, then check forwards.
Check your understanding
After a 35 per cent discount, a coat costs 143 dollars. What was the original price?
Answer: A
Why: The sale price is 65 per cent of the original, so the original is 143 divided by 0.65, which is 220. Checking forwards: 35 per cent of 220 is 77, and 220 minus 77 is 143.
The forward check separates all four choices in about ten seconds and requires no memory of which operation reverses which.
Section
Section 4
Trap
The trap. A price rises 20 per cent then falls 20 per cent, and you conclude it is back where it started.
Or a price rises 10 per cent twice and you call it a 20 per cent rise.
Percentages do not add across successive changes, because each one is taken of a different base.
The fix. Convert each change into a multiplier and multiply them together.
Up 20 then down 20 is 1.20 times 0.80, which is 0.96 — a four per cent fall. Up 10 twice is 1.21, a 21 per cent rise.
Trap
The trap. A quantity falls 15 per cent and you multiply by 0.15.
That keeps only 15 per cent, which is an 85 per cent collapse. The multiplier for a 15 per cent fall is 0.85.
The same error in reverse writes 1.15 for a fall, applying an increase instead.
The fix. The multiplier is what SURVIVES, not what disappears.
Test it on 100: a 15 per cent fall must send 100 to 85, and only 0.85 does that.
Error analysis
A student recovering an original price. The arithmetic is right and the base is wrong.
Annotate
On: \( \text{After 20 percent off, price is 60.} \;\Rightarrow\; 60 + 0.20 \times 60 = 72 \)
On any reversal question, apply the change forwards to your answer. If it does not reproduce the number you were given, the base was wrong.
Trap
The trap. After a 20 per cent discount an item costs 60 dollars, and you add 20 per cent of 60 to get 72.
But the discount was 20 per cent of the ORIGINAL, not of the sale price. The original is 60 divided by 0.80, which is 75.
Checking 72 forwards gives 72 times 0.80, which is 57.60 — not 60, so it fails.
The fix. Reversing a multiplication is a division.
Write the relationship forwards first — new equals old times the multiplier — then rearrange.
Elimination
A price rises 40 per cent and then falls 40 per cent.
Eliminate the wrong options
What is the overall effect? Three choices can be ruled out by reasoning about the bases.
Survives elimination: b
Why: The combined multiplier is 1.40 times 0.60, which is 0.84, so the price ends 16 per cent below where it started. Two of the three eliminations needed no arithmetic: a rise then an equal fall always ends below the start, and percentages never simply add. That reasoning alone leaves only one plausible choice.
Trap
The trap. A rate rises from 20 per cent to 30 per cent, and you call it a 10 per cent increase.
It is a rise of 10 percentage POINTS, but a 50 per cent increase, because 10 divided by the original 20 is 0.5.
Both numbers are correct answers to different questions, and both are offered.
The fix. Read whether the stem says points or per cent.
Points is a subtraction; per cent is a division by the original.
Counterexample
A rule that feels symmetric and is not.
Discussion prompt
A student says: if a price falls by p per cent, then raising it by p per cent restores it. Give a counterexample and explain the asymmetry.
Hint: Try a large percentage, where the effect is obvious.
Answer:
Counterexample: a 50 per cent fall then a 50 per cent rise. A price of 100 falls to 50, then rises by 25 to 75. It is a quarter below where it started.
The asymmetry: the fall was 50 per cent of 100, but the rise was 50 per cent of 50 — half as much, in absolute terms.
In multipliers: 0.5 times 1.5 is 0.75, not 1. They are not reciprocals.
What actually restores it: the reciprocal of 0.5 is 2, so you need a 100 per cent rise to undo a 50 per cent fall.
The general rule: to undo a fall of p, you need a rise of MORE than p; to undo a rise of p, you need a fall of LESS than p. The base changes, so the percentage must too.
Edge cases
Multipliers handle almost everything. Where do you have to be careful?
Discussion prompt
When does combining multipliers need extra care, and when does the method not apply at all?
Hint: Think about what the percentages are taken of.
Answer:
Multipliers combine freely when each change applies to the whole quantity — a discount, then a tax, then another discount, all on the running total. Order does not matter.
Care is needed when a percentage applies to only PART of the total. A tax charged only on some items is not a multiplier on the whole bill, and cannot be combined that way.
The method does not apply when a change is a fixed amount rather than a percentage. A 5 dollar coupon is a subtraction, not a multiplier, and mixing it with a percentage means order DOES matter.
That last case is worth watching: a 20 per cent discount then a 5 dollar coupon gives a different total from the coupon first, because the percentage applies to a different base each way.
The check: ask whether each change scales the whole current amount. If yes, multipliers combine and order is irrelevant. If any change is a fixed amount, work step by step in the stated order.
Check
Identify the base before computing.
Check your understanding
Store A sells a chair for 240 dollars. Store B's price is 25 per cent less than Store A's. Store C's price is 20 per cent more than Store B's. What is Store C's price?
Answer: A
Why: Store B is 240 times 0.75, which is 180. Store C is 20 per cent more than B, so 180 times 1.20, which is 216. Note that C's increase is taken of B's price, not of A's, which is what the wording specifies.
The combined multiplier 0.75 times 1.20 equals 0.90 answers it in one line, and also tells you Store C is 10 per cent cheaper than Store A.
Section
Section 5
Matching
Six phrases, six numbers. No arithmetic beyond the conversion.
Match the pairs
Why: The second and third rows are the pair that gets confused: decreases BY 15 per cent leaves 85 per cent, while IS 15 per cent OF leaves 15 per cent. One word separates a small reduction from an enormous one, and both multipliers are always among the choices.
Sorting
Each is a combined multiplier from two successive changes.
Sort into buckets
What does each combined multiplier mean?
Reading a combined multiplier is a skill in itself: subtract from 1 for a fall, subtract 1 for a rise, and the decimal is the percentage directly.
Comparison
Fill the blanks from memory. One column is how percentages are taught; the other is how they work.
Comparison matrix
| situation | the additive answer | the correct answer |
|---|---|---|
| Up 20 then down 20 | back to the start | 4 per cent below the start |
| Up 10 then up 10 | up 20 per cent | up 21 per cent |
| Undo a 25 per cent rise | fall 25 per cent | fall 20 per cent |
| Reverse a 20 per cent discount on 60 | add 20 per cent to get 72 | divide by 0.8 to get 75 |
Every row in the middle column is what a confident student produces under time pressure, and every one of those numbers appears among the answer choices. The multiplier method exists to keep you out of that column.
Trade off
Fill in what each costs.
Comparison matrix
| method | steps for one change | steps for three changes |
|---|---|---|
| Find the change, then add or subtract | two: compute the part, then combine | six, with a new base each time |
| Multiply by the multiplier | one multiplication | one product, applied once |
| Reverse by subtracting | one step, and wrong | wrong at every stage |
| Reverse by dividing | one division | divide by the combined multiplier |
The multiplier method is not merely more accurate — for anything beyond a single change it is also fewer steps. The additive route gets slower and less reliable at exactly the same rate.
Real world
One minute on why the asymmetry matters.
Discussion prompt
An investment falls 50 per cent one year. What return does it need the next year just to break even, and why does this surprise people?
Answer:
It needs a 100 per cent gain, because the multipliers must be reciprocals: 0.5 times 2 equals 1.
Why it surprises: the loss was 50 per cent of the original, but the recovery is 50 per cent of a much smaller number, so it has to be twice as large in percentage terms.
This is why losses hurt more than equal gains help. Down 30 then up 30 leaves you 9 per cent behind; you need about 43 per cent to recover from a 30 per cent fall.
The same asymmetry appears in retail: marking up 50 per cent and then discounting 50 per cent leaves the shop selling below cost.
And in reporting statistics: a fall from 4 per cent to 2 per cent is 2 percentage points and a 50 per cent decrease, and which is quoted usually depends on which sounds better.
Ranking
Each starts at 100. Order the final values.
Put in order
Why: Computing each: (b) is 0.9 times 0.9, which is 0.81, giving 81. (d) is 0.75 times 1.25, which is 0.9375, giving 93.75. (a) is 1.2 times 0.8, which is 0.96, giving 96. (c) is 1.1 times 1.1, which is 1.21, giving 121. Note that three of the four end BELOW 100 even though two of them involved an increase — which is the asymmetry of this whole type in one exercise.
Concept
This type is 4.5 per cent of the section and is almost entirely fixed by one habit, which makes it among the fastest to improve.
| session | what you do | why |
|---|---|---|
| 1 | Convert forty percent changes into multipliers, out loud, until it is automatic. | Everything else in the type depends on this conversion being instant. |
| 2 | Fifteen successive-change questions, combining multipliers before applying anything. | Trains the habit that replaces the additive instinct. |
| 3 | Fifteen reversal questions, checking every answer forwards. | Reversal is the variant most often answered by subtracting. |
| 4 | Mixed set including percent-of versus percent-greater and percentage points. | The wording distinctions, which are read errors rather than method errors. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 76 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, state how to turn a percent change into a multiplier, how to combine two changes, how to reverse one, and why up 20 then down 20 does not return to the start.
Hint: The last part is about which number each percentage is taken of.
Answer:
Convert: up r per cent is times 1 plus r; down r per cent is times 1 minus r. The multiplier is what survives.
Combine: multiply the multipliers. Do not add the percentages.
Reverse: divide by the multiplier, then check forwards.
Why up 20 then down 20 fails to return: the rise was 20 per cent of the original, but the fall was 20 per cent of a larger number, so it removed more than the rise added. In multipliers, 1.2 times 0.8 is 0.96, not 1.
If you gave the reason in terms of the base changing, you have the idea that fixes every trap in this deck.
Explain it
Two minutes, out loud.
Discussion prompt
A friend insists that a 20 per cent rise followed by a 20 per cent fall must return to the original price. How do you convince them?
Answer:
Do not argue — compute it with a friendly number. Start at 100. Up 20 per cent gives 120. Now take 20 per cent off 120.
Ask them: 20 per cent of 120 is what? It is 24, not 20. So the price falls to 96.
Name the reason: the rise was 20 per cent of 100, and the fall was 20 per cent of 120. Different bases, different amounts.
Then give the general form: 1.2 times 0.8 is 0.96, and that holds whatever the starting price. Always four per cent down.
Finish with the useful corollary: to actually get back to 100 from 120 you need a fall of one sixth, about 16.7 per cent. Undoing a rise always takes a smaller percentage than the rise itself.
Commit first
Commit before you check.
Predict first
A price is 90 dollars after a 25 per cent increase. What was it before?
Correct: 72
Why: The new price is 1.25 times the old, so the old price is 90 divided by 1.25, which is 72. Checking forwards: 25 per cent of 72 is 18, and 72 plus 18 is 90. The answer 67.50 subtracts 25 per cent of 90, taking the increase off the wrong base — the classic reversal error, and it fails the forward check at 84.375.
Connect it up
Blank paper.
Draw it
Draw a map of percentages. In the centre write MULTIPLIER, with two arrows out of it: up r per cent goes to 1 plus r, and down r per cent goes to 1 minus r, with the note that the multiplier is what survives. From there draw three branches. First, successive changes: multiply the multipliers, with 1.2 times 0.8 equals 0.96 written out. Second, reversal: divide by the multiplier, with the forward check written beside it. Third, finding a percent change: difference over the ORIGINAL. Around the edge, write the four traps: adding percentages, using the percentage as the multiplier, reversing by subtracting, and confusing percentage points with per cent. At the bottom, in large letters: PER CENT OF WHAT?
Exit ticket
One question before you close the deck.
Predict first
A question gives a price after a discount and asks for the original. What is the safest way to check your answer?
Correct: Apply the discount to your answer and see if you get the given price
Why: The forward check reproduces exactly what the question described, so it confirms both the arithmetic and the base at once. Adding the percentage back is the error itself rather than a check on it. Confirming the answer is larger is necessary but far too weak — the additive wrong answer is also larger. And computing the discount as a percentage of your answer is really the same forward check said less directly.
Recap
One type, one conversion: every percent change is a multiplier.
| never do this | do this instead |
|---|---|
| Add 20 and subtract 20 and expect the start | Multiply 1.20 by 0.80 and get 0.96 |
| Multiply by 0.15 for a 15 per cent fall | Multiply by 0.85 — the multiplier is what survives |
| Add the discount back to reverse it | Divide by the multiplier, then check forwards |
| Divide a change by the new value | Divide by the original |
| Call a rise from 20 to 30 per cent a 10 per cent increase | That is 10 points, but a 50 per cent increase |
| Apply changes one at a time with rounding | Combine the multipliers, then apply once |
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