Desmos for the SAT · Technique 10 of 23

Solve Percent Problems with Multipliers

The percent sign works. The hard part is deciding which multiplier the sentence is describing.

SolvesPercent of a numberPercent increase and decreaseSuccessive discountsFinding the original price

Watch the lesson

The whole move worked on one real SAT question, narrated.

0:00 / 0:00

A discount, a tax and a final price worked backward: build the multiplier chain, then click the crossing.

Percentage questions are rarely hard arithmetic. They are translation problems: turn the sentence into a multiplier, and the arithmetic that is left is something Desmos finishes instantly. Almost every point lost here is lost in the translation, not the calculation.

Desmos handles percents directly, and the keypad helps: type 25% and Desmos fills in the word of for you, so 25%80 becomes 25% of 80 and returns 20. Percents chain the same way, so 85%80%80 becomes 85% of 80% of 80 and returns 54.4, and they graph, so y = 75% of x draws a line. There is no typing trap here at all. Both forms are worked below on the same question, so you can see they land on the same number and pick whichever you read faster.

Two harder families live elsewhere on purpose. A percent per year hidden inside an exponent belongs with exponential models.

The translation rules are short. "Of" means multiply, so 15% of 480 is 0.15 * 480. A discount of 30% means you pay the other 70%, so multiply by 0.7, not by 0.3. An increase of 6% means multiply by 1.06. Get those three right and the rest of this page is typing.

Nova

Percentage questions are translation, not arithmetic. Once the words are a multiplier, Desmos finishes the job in one line. The typing is not the problem: type 30% and Desmos writes the of for you first.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps tracking two rounds of percentage arithmetic by hand.

The question

An item costs $80. A sale takes 20% off, then a promotion takes a further 15% off the sale price.

What is the final price?

The algebra way

Find 20% of 80: that is 16.
Subtract: 80 - 16 = 64.
Find 15% of 64: that is 9.60.
Subtract again: 64 - 9.60 = 54.40. Four steps, two of them subtractions you have to remember to do.

The Desmos way

Turn each discount into what you pay: 0.8 and 0.85.
Type 80 * 0.8 * 0.85.
Read 54.4.

The payoff: One line instead of four steps, and no intermediate number to copy down wrong. The thinking moves to where it belongs: deciding that 20% off means 0.8.

Quick reference

The sentence, and what you actually type

The question says	The multiplier	Why
15% of 480	`0.15 * 480`	"Of" means multiply. Nothing else happens.
30% discount	`* 0.7`	You pay the part that is left, not the part you save.
6% tax added	`* 1.06`	The original, plus 6% of it, is 106% of it.
20% off, then another 15% off	`* 0.8 * 0.85`	The second cut comes off the reduced price, so they multiply.
Percent change from old to new	`(new - old) / old`	Divide by where you started, not where you ended.
Final price known, original wanted	`y = 0.795x` and `y = 79.50`, or `79.5/0.795` on one line	Graph the chain against the final price and click the crossing, or divide the final price by the chain.

Step by step

The whole move, all 6 steps

Convert every percentage in the question to a decimal multiplier before you type anything. The percent sign works in Desmos too, but the decimal form is the one that chains cleanly and is easy to check.
Translate each change into a multiplier: p% off becomes (1 - p/100), p% added becomes (1 + p/100).
For a straight calculation, type the whole chain on one line, one multiplier per discount or increase. Percentages applied one after another multiply; they never add.
For percent change, type (new - old) / old. The number you divide by is whatever you started from, which is the one the question is testing.
To work backward to an original price, put the chain on a graph: y = (the multipliers) * x on one line and y = (the final price) on the next. Click inside either equation box to reveal the gray dots, then click the crossing and read the x.
Multiply a decimal answer by 100 to report it as a percent, or click the fraction toggle if the choices are fractions.

The question

A store is having a sale where all items are 20% off. During a special promotion, an additional 15% is taken off the sale price. The original price of an item was $80.

What is the final price after both discounts?

A$52.00
B$54.40
C$48.00
D$56.00

Step 1 of 5

The question: an item costs $80, a sale takes 20% off, and a promotion takes another 15% off the sale price. Start by writing the first discount as a multiplier. 20% off means you pay 80%, so that is 0.8.

This is the step worth slowing down on. People multiply by 0.2 here, which is the amount you save, not the amount you pay.

Now try it on a real SAT question

Head or scratch paper. 10% of anything is the decimal point moved one place. This is the fastest question type on the test, and typing it takes longer than doing it in your head. Recognize the ones that are already finished before you reach for anything.

Isabel grows potatoes in her garden. This year, she harvested 760 potatoes and saved 10% of them to plant next year.

How many of the harvested potatoes did Isabel save to plant next year?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

A discount is the part you pay, not the part you save: 30% off is a multiplier of 0.7. Typing 0.3 gives you the saving and it is usually an answer choice.
Successive percentages multiply and never add. 20% off then 15% off is 0.8 times 0.85, which is 0.68, not 0.65.
Percent change depends on the base. Going from 80 to 100 is a 25% increase, while going from 100 to 80 is a 20% decrease, and the same 20 is involved in both.
For plain mental arithmetic like 10% or 50% of a number, typing it is slower than thinking it.
Desmos answers in decimals, so multiply by 100 yourself before matching a percentage against the choices.
The percent sign is not the trap. Typing 30% fills in of automatically, and 30%80 returns 24. The trap is which multiplier the sentence describes, which is what the rest of this list is about.

Nova

10% of a number, or 50% of a number, is faster in your head than in any calculator. Save Desmos for the chains, the ones that run backward from a final price, and the percent-change questions where picking the wrong base is the whole trap.

All Desmos techniques

Put this move into real practice

"Solve Percent Problems with Multipliers" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.