SAT percent problems: multipliers and changes

What the questions look like

Most percent questions are only one or two sentences long. The last line might say:

“What number is 40% greater than 85?”
“69 is 92% of what number?”
“What percentage of 650 is 117?”
“The number of members increased from 30 to 81. What was the percent increase?”
“The final price is $k$ times the original price. What is the value of $k$ ?”

Harder ones link two or three letters: “The number $a$ is 45% less than the number $b$ . The number $c$ is 30% more than $a$ . The number $c$ is how many times $b$ ?”

For every one of them, you find the amount the percent is taken of and turn the wording into a multiplier. After that it's one or two multiplications.

Desmos works as a calculator here, but it can't translate the wording for you. On a one-step question like 8% of 650, you'll finish faster in your head.

A percent of a number

Percent means hundredths, so 8% is 8 hundredths, or 0.08. To turn any percent into a decimal, move the point two places left.

“ $p\%$ of a number” means $\frac{p}{100}$ times the number. Write the decimal before you multiply.

Right: 8% of 650 is $0.08 \times 650 = 52$ .
Wrong: 8% of 650 is $0.8 \times 650 = 520$ .

0.8 is 80%. Moving the point only one place makes the answer 10 times too big.

The same move works for tiny and huge percents. 0.3% is 0.003, 0.045% is 0.00045, and 365% is 3.65.

Trap: A decimal place lost

For 0.28% of 2,400, a choice of 672 comes from using 0.28, and 67.2 from using 0.028. The right answer, 6.72, uses 0.0028.

What to do: Write the decimal down before you touch the calculator, two places left every time. If your answer is 10 or 100 times a choice, check this first.

More than and less than

Here's the same percent and the same number in three wordings.

Wording	Multiply by	Result
40% of 85	$0.40$	34
40% greater than 85	$1+0.40=1.40$	119
40% less than 85	$1-0.40=0.60$	51

“Greater than” keeps all of the 85 and adds 40% of it on top, so you add 1 to the decimal. “Less than” takes 40% away and leaves 60%. “More than” and “increased by” work like “greater than.” “Decreased by” and “off” work like “less than.” And “40% as much as 85” is just 40% of 85.

An increase of $p\%$ multiplies by $1+\frac{p}{100}$ . A decrease of $p\%$ multiplies by $1-\frac{p}{100}$ .

Right: 40% greater than 85 is $85 \times 1.40 = 119$ .
Wrong: 40% greater than 85 is $85 \times 0.40 = 34$ .

34 is only the part that gets added. A number 40% greater than 85 has to be bigger than 85.

This still works when the percent is over 100. 220% greater than 45 is $45 \times (1+2.2) = 45 \times 3.2 = 144$ .

Trap: “Of” read as “greater than”

For “220% greater than 45,” 99 is 220% of 45. It leaves out the 45 you started with.

What to do: Read the words around the percent before you calculate. For “greater than,” “more than” or “increased by,” add 1 to the decimal.

Trap: The part taken off instead of the part left

A $72 jacket is 25% off. A choice of $18 is the discount, $72 \times 0.25$ . The price you pay is $72 \times 0.75 = 54$ dollars.

What to do: “Off” and “less than” leave $100-p$ percent. Write $1-\frac{p}{100}$ before you calculate.

Worked example

What number is $25\%$ greater than $76$ ?

A
$19$
B
$57$
C
$77.9$
D
$95$

Show the answer

A
$19$
This computes $25\%$ of $76$ , since $0.25(76)=19$ , but it does not add the increase to $76$ .
B
$57$
This treats “ $25\%$ greater than” as a $25\%$ decrease and computes $75\%$ of $76$ : $0.75(76)=57$ .
C
$77.9$
This treats $25\%$ as $2.5\%$ , using the multiplier $1.025$ . That gives $76(1.025)=77.9$ .
D
$95$
Correct answer
A number that is $25\%$ greater than $76$ equals $100\%+25\%=125\%$ of $76$ . Compute $76(1.25)=95$ .

Solution

Convert $25\%$ to $0.25$ . The increase is $76(0.25)=19$ . Add the increase to the original number: $76+19=95$ . Equivalently, multiply by $1+0.25=1.25$ : $76(1.25)=95$ .

Report a problem

Percent of what?

What percentage of 650 is 117?

The amount right after “of” is the base, the whole you're comparing to. Here that's 650, so you divide by it.

Divide the part by the base.
$117 \div 650 = 0.18$
Multiply by 100 to turn it into a percent. 117 is 18% of 650.
$0.18 \times 100 = 18$

Every percent has a base, the amount it's taken of. For “what percentage of $B$ is $A$ ,” divide $A$ by $B$ .

Right: $117 \div 650 \times 100 = 18\%$
Wrong: $650 \div 117 \times 100 \approx 555.6\%$

Turned upside down, the fraction tells you what percent of 117 the 650 is. The question asks for a percentage of 650.

Find a percent

\text{percent}=\frac{\text{part}}{\text{base}}\times100

The base isn't always the bigger number, so a percent can be far above 100%. If one amount is 240 times another, it's $240 \times 100 = 24{,}000\%$ of it. A huge answer isn't a mistake by itself. Check that you divided by the base, then keep it.

Percent increase and decrease

The number of members increased from 30 to 81. What was the percent increase?

Find the change.
$81-30=51$
Divide by the starting amount, 30.
$51 \div 30 = 1.7$
Multiply by 100. It's a 170% increase.
$1.7 \times 100 = 170$

For a percent change, the starting amount is the base.

Right: $51 \div 30 \times 100 = 170\%$
Wrong: $51 \div 81 \times 100 \approx 63.0\%$

A percent change compares the change with where you started, and that's 30.

Percent change

\text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}\times100

A decrease works the same way. From 80 down to 62 the change is 18, and $18 \div 80 = 0.225$ , so it's a 22.5% decrease.

The same question can come as “81 is $p\%$ greater than 30. What is the value of $p$ ?” That's the percent increase again, so $p=170$ .

Trap: The start counted in the increase

For “81 is $p\%$ greater than 30,” 270 is $81 \div 30 \times 100$ . That says 81 is 270% of 30, which counts the starting 30 as part of the increase.

What to do: “Greater than” asks for the change alone. If you worked out the whole new amount as a percent of the old one, subtract 100.

From a multiplier back to a percent

Some questions hand you the multiplier and ask for the percent.

Multiplier	Minus 1	Change
1.32	0.32	32% increase
2.6	1.6	160% increase
0.28	$-0.72$	72% decrease

Subtract 1 from the multiplier to get the change. A negative result is a decrease.

Right: $0.28x$ is $x$ decreased by 72%.
Wrong: $0.28x$ is $x$ decreased by 28%.

$0.28x$ keeps 28% of $x$ . The part taken away is $100-28=72$ percent.

Questions also ask it the other way around. If a town's population grows by 9% and the new population is $k$ times the old one, then $k=1.09$ . After a 145% increase, $k=2.45$ .

Work back to the start

69 is 92% of what number?

Write the sentence as an and solve it.

“Of” means times, and $n$ is the number you want.
$69=0.92n$
Divide both sides by 0.92.
$69 \div 0.92 = n$
Check: $0.92 \times 75 = 69$ .
$n=75$

When you know the part and the percent, divide to find the whole.

Right: $69 \div 0.92 = 75$
Wrong: $69 \times 0.92 = 63.48$

69 is already the part. Since 92% is less than 100%, the whole has to be bigger than 69.

The same move undoes a change. A price is $66 after a 20% increase. What was it before?

A 20% increase multiplies the old price $p$ by 1.20.
$66=1.20p$
Divide both sides by 1.20.
$p=66 \div 1.20=55$

To undo a change, divide by its multiplier.

Right: $66 \div 1.20 = 55$
Wrong: $66 \times 0.80 = 52.80$

The increase was 20% of the old $55, which is $11. Taking 20% of the new $66 removes $13.20, a different amount.

You undo a decrease the same way. After a 27.5% decrease, a water level is 13.05 meters. What was it before?

A 27.5% decrease leaves 72.5%.
$1-0.275=0.725$
The old level $L$ times the multiplier gives the new level.
$13.05=0.725L$
Divide both sides by 0.725.
$L=13.05 \div 0.725=18$

Read the last line again before you answer. If it asks how far the level dropped, subtract to get $18-13.05=4.95$ meters.

Trap: The 1 left out

Increasing $x$ by 300% gives 96. A choice of 32 divides 96 by 3, but a 300% increase multiplies by $1+3=4$ , so $x=96 \div 4=24$ .

What to do: Write the multiplier with the 1 added, then divide by it. Check: $24 \times 4 = 96$ .

Two changes in a row

A price goes up 20%, then down 25%. The final price is $k$ times the original price. What is $k$ ?

Start with $100 if it helps.

Up 20%.
$100 \times 1.20 = 120$
Down 25%, taken of the new price, 120.
$120 \times 0.75 = 90$
Compare the end with the start, so $k=0.9$ .
$90 \div 100 = 0.9$

Changes in a row multiply, so you can skip the $100 and multiply the two multipliers.

Right: Up 20%, then down 25%: $1.20 \times 0.75 = 0.9$ , a 10% decrease.
Wrong: Up 20%, then down 25%: $20-25=-5$ , a 5% decrease.

The 25% is taken of $120, so it removes $30. That's more than the $20 the first change added.

For the same reason, going up 50% and then down 50% doesn't bring you back. You end at $1.5 \times 0.5 = 0.75$ of the start, 25% lower.

Trap: Two percents added

For up 15% and then up 4%, 1.19 adds the percents. Multiplying gives $1.15 \times 1.04 = 1.196$ , a 19.6% increase.

What to do: Write each multiplier and multiply them. The second change is taken of the new amount.

A percent of a percent multiplies too. Say 60% of a club's members are juniors, and 45% of the juniors play an instrument. Then $0.60 \times 0.45 = 0.27$ , so 27% of all the members are juniors who play an instrument. The 45% is a share of the juniors only, so you take 45% of the 60%. A question might ask about the members outside that group instead. That's $100-27=73$ percent.

Worked example

A nursery’s monthly plant sales decreased by $70\%$ , then increased by $35\%$ , becoming $k$ times the initial sales.

What is the value of $k$ ?

A
$0.405$
B
$0.65$
C
$0.945$
D
$1.35$

Show the answer

A
$0.405$
Correct answer
A $70\%$ decrease leaves $0.30$ times the initial sales. Increasing that amount by $35\%$ gives $k=0.30(1.35)=0.405$ .
B
$0.65$
The value $0.65$ comes from adding the successive percent changes: $1-0.70+0.35=0.65$ . This treats both changes as percentages of the initial sales, even though the $35\%$ increase applies to the reduced sales.
C
$0.945$
The value $0.945$ comes from $0.70(1.35)$ , which confuses the fraction lost with the fraction remaining. A $70\%$ decrease leaves $30\%$ of the sales.
D
$1.35$
The value $1.35$ is the multiplier for the $35\%$ increase alone, so it leaves out the $70\%$ decrease. The sales first fall to $0.30$ times the initial sales, and the increase then multiplies that reduced amount by $1.35$ .

Solution

Let $s$ represent the initial monthly sales. After a $70\%$ decrease, the sales are $s(1-0.70)=0.30s$ .

The $35\%$ increase multiplies the reduced sales by $1+0.35=1.35$ , giving $(0.30s)(1.35)=0.405s$ . Since the final sales equal $ks$ , $k=0.405$ .

Report a problem

Markups and discounts

A store buys a lamp for $6.00 and sells it for 240% of that cost. Later the lamp is marked 65% off the selling price. What is the sale price?

240% of the cost. “Of” means multiply by 2.4.
$6.00 \times 2.4 = 14.40$
65% off the selling price leaves 35% of it.
$14.40 \times 0.35 = 5.04$

Each percent is taken of the amount it names, and that can be an earlier result.

Right: $14.40 \times 0.35 = 5.04$
Wrong: $6.00 \times 0.35 = 2.10$

The 65% off is taken of the selling price, $14.40. Taking it off the $6.00 cost skips the markup.

Tax and tip work the same way. A 6% tax multiplies the price by 1.06.

In Desmos, type the whole thing on one line, 6*2.4*0.35, so nothing gets rounded along the way.

Trap: Money rounded too early

A store marks up an $18.95 shirt by 35%, then sells it at 15% off. The marked-up price is $18.95 \times 1.35 = 25.5825$ . Rounding that to $25.58 before the discount gives $21.74. Kept exact, $25.5825 \times 0.85 = 21.745125$ , which rounds to $21.75.

What to do: Keep the whole chain on one calculator line and round once, at the end.

Chains of letters

0.0546 is 0.35% of the number $b$ . The number $b$ is 40% of $c$ . What is the value of $c$ ?

Write each sentence as its own equation, then start from the number you know.

0.35% is 0.0035, and “of” means times.
$0.0546=0.0035b$
Divide both sides by 0.0035.
$b=0.0546 \div 0.0035=15.6$
Put $b=15.6$ into the second sentence.
$15.6=0.40c$
Divide both sides by 0.40.
$c=15.6 \div 0.40=39$

Then check the answer against each sentence. 40% of 39 is 15.6, and 0.35% of 15.6 is 0.0546.

Trap: A link turned around

6.24 is $15.6 \times 0.40$ , which reads “ $b$ is 40% of $c$ ” as “ $c$ is 40% of $b$ .” A choice can get the first link right and still turn this one around.

What to do: Write each sentence as an equation with the “of” amount on the right. When that amount is the one you need, divide.

A chain on one line

Type 0.0546/0.0035.
Press the right arrow to leave the bottom of the fraction.
Type /0.4. The line shows 39.

If you skip the right arrow, the /0.4 lands in the bottom of the first fraction and Desmos shows 6.24, the wrong answer from the trap above.

Not sure which way to divide? Type 0.0546=0.0035x. Desmos draws a vertical line at the answer, so click where it crosses the horizontal to read 15.6, the value of $b$ . The solving by graphing lesson shows how.

Some chains give no starting number at all: “The number $a$ is 45% less than the number $b$ . The number $c$ is 30% more than $a$ . The number $c$ is how many times $b$ ?” Pick $b=100$ and follow it through.

45% less than $b$ leaves 55% of it.
$a=100 \times 0.55=55$
30% more than $a$ means times 1.30.
$c=55 \times 1.30=71.5$
Compare $c$ with $b=100$ .
$71.5 \div 100=0.715$

So $c$ is 0.715 times $b$ . Multiplying the two multipliers, $0.55 \times 1.30$ , gives the same 0.715.

Worked example

$a$ is $0.4\%$ of $13$ , and $13$ is $30\%$ less than $c$ .

What is the value of $ac$ , rounded to the nearest hundredth?

Show the answer

Accepted answers: 0.97, .97, 0.970, .970, .9700

Solution

Convert $0.4\%$ to a decimal: $0.4\div100=0.004$ . Then $a=0.004(13)=0.052$ . Since $13$ is $30\%$ less than $c$ , it equals $70\%$ of $c$ . Thus $13=0.70c$ , so $c=\frac{13}{0.70}=\frac{130}{7}$ . Multiply without rounding the intermediate values: $ac=0.052\left(\frac{130}{7}\right)=0.965714\ldots$ . Rounded to the nearest hundredth, the product is $0.97$ . The wrong answer $0.47$ comes from reversing the relationship and calculating $c=13(0.70)$ . The quantity $13$ is $70\%$ of $c$ , so you must divide $13$ by $0.70$ to find $c$ . The wrong answer $2.25$ comes from reading “ $30\%$ less than $c$ ” as “ $30\%$ of $c$ ” and calculating $c=13\div0.30$ . “ $30\%$ less than” leaves $70\%$ of the original quantity. The wrong answer $96.57$ comes from treating $0.4\%$ as $0.4$ instead of $0.004$ , making $a$ , and therefore the product $ac$ , 100 times too large.

Note that $0.97$ and $.97$ are examples of ways to enter a correct answer.

Report a problem

Entering the answer

On a , type the exact value even when it isn't a whole number, like 5.04 or .715. Leave out the $ and % signs and any commas, so 24,000% goes in as 24000. A fraction works too if it fits in five characters, such as 275/2 for 137.5. When the exact answer is a fraction, the small round fraction button on the result row in Desmos shows it as one.

The grid-in lesson covers what fits in the box.

Practice

Read each percent's wording and ask “percent of what?” before you calculate.

Try it

5 questions on this skill, with every choice explained.

Start the drill

Next steps

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Related lessons

Percentages

What the questions look like

A percent of a number

More than and less than

Percent of what?

Percent increase and decrease

From a multiplier back to a percent

Work back to the start

Two changes in a row

Markups and discounts

Chains of letters

Entering the answer

Practice