Solve percent problems in Desmos

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Type the percent the way the question reads, like 65% of x = 117, and Desmos solves it. Increase, decrease, reverse and successive changes, with SAT examples.

Difficulty: Intermediate
Estimated time: 55 minutes
Domains: Problem Solving and Data Analysis

What you’ll learn

Find the base, the amount a percent is taken of, and type % of statements straight into Desmos.
Turn increases, decreases, discounts, taxes and tips into one final percent of the base.
Solve reverse-percent questions by putting $x$ where the base goes.
Chain percent changes that come one after another, without adding the percents.
Find an unknown percent or percent change by dividing by the right base.
Tell a percent change from a change in percentage points.

Prerequisites

Solve one-variable equations0%
Enter a one-variable equation and read its vertical solution line.

Why this matters on the SAT

Turn percent words into one equation

SAT percent questions often hide a short equation inside a sentence. In Desmos, you can type that sentence almost word for word, and it handles the decimals and the solving for you.

SAT example

This year, a wildlife survey counted $405$ cranes. This year's count is $162\%$ of the previous survey's count. How many cranes were counted in the previous survey?

Fast Desmos solution

Let $x$ be the previous survey's count. Then type the sentence the way it reads:

162% of x = 405

Desmos draws a vertical solution line at $x=250$ , so the answer is C.

There's one thing Desmos can't do for you: decide which amount goes after of. That's the real math in the problem. Here, the $162\%$ is taken of the previous survey's count, so $x$ goes right after of. The answer makes sense, too. $162\%$ is more than all of the old count, so the old count has to be smaller than $405$ .

Calculator loads as you approach

The percent sentence becomes an equation with one solution, x equals 250.

When should you use Desmos?

You have two ways in. You can type a literal percent relationship, the question’s sentence written as a Desmos equation, or you can use hand reasoning in your head. What the question gives you usually decides it.

Type a literal percent relationship when…

The question says one amount is a percent of another, like “ $405$ is $162\%$ of the previous count.”
You need the amount from before a change, like a meal that cost $46 after a $15\%$ tip.
The numbers are awkward, like $17.5\%$ of $436$ , and a decimal or a division would be easy to mistype.
Changes stack up, like a discount and then tax on the lower price, so each percent acts on a different amount.

Use hand reasoning when…

The percent is a friendly one like $10\%$ , $25\%$ or $50\%$ . Half of $86$ is $43$ , no calculator needed.
A simple fraction gives the answer right away, like $20\%$ of $45$ , which is one fifth of $45$ , or $9$ .
The question asks you to compare, not calculate, like whether $30\%$ of $70$ or $70\%$ of $30$ is bigger. They’re equal, since each is $30\times70$ divided by $100$ .

Either way, the first question is the same: what is the percent taken of? Desmos saves you the arithmetic, not that decision.

Check your understanding:

Which way would you go on each of these? Problem A asks for $25\%$ of $68$ . Problem B says a quantity became $117$ after a $35\%$ decrease and asks for the original quantity.

1. Put the base after “of”

A percent is always a percent of something. That something is the base, and the percent is a part of it:

p\%\text{ of }B=\frac{p}{100}B.

Desmos understands % of directly, so you never have to turn the percent into a decimal. Type

18% of 250

and Desmos returns

45.

When you don't know the base, put $x$ where it goes. The sentence $84$ is $35\%$ of a number becomes

35% of x = 84

and the vertical solution line lands at $x=240$ .

Here's how the sentences line up with what you type:

Sentence	What you type in Desmos
$p\%$ of $B$	`p% of B`
$A$ is $p\%$ of $B$	`p% of B = A`
$A$ is $p\%$ of an unknown number	`p% of x = A`

Try it yourself:

Change 35% to 42% and keep the $84$ . Will the unknown base be bigger or smaller than $240$ ? Make a guess, then check it in the calculator.

Check your understanding:

$72$ is $120\%$ of what number? Write the Desmos equation and find the number.

Common mistake:

It’s easy to flip the part and the base. Typing 35% of 84 gives $29.4$ , which is $35\%$ of the wrong number. In the sentence, $84$ is the part you get after taking $35\%$ of something. That unknown whole is $x$ , so it goes after of: 35% of x = 84.

Calculator loads as you approach

When you know the base, Desmos calculates. When you don’t, it solves for x.

Turn more and less into one final percent

Words like greater than, more than, less than, increase and decrease describe a change from the base. They don't give you the final percent. You build it yourself: start at $100\%$ , the whole base, and move up or down by the change.

So if a quantity increases by $p\%$ , the new amount is

(100+p)\%

of the base. If it decreases by $p\%$ , the new amount is

(100-p)\%

of the base.

The question says	The new amount is
$p\%$ of the base	$p\%$ of the base
$p\%$ greater than the base	$(100+p)\%$ of the base
increased by $p\%$	$(100+p)\%$ of the base
$p\%$ less than the base	$(100-p)\%$ of the base
decreased by $p\%$	$(100-p)\%$ of the base
discounted by $p\%$	$(100-p)\%$ of the original price
a tax or tip of $p\%$	$(100+p)\%$ of the amount before the tax or tip

Say a company's revenue this year is $708,000, which is $18\%$ greater than last year's. What was last year's revenue? Let $x$ be last year's revenue in thousands of dollars, so you can type $708$ instead of $708{,}000$ .

An $18\%$ increase takes you from $100\%$ up to

100\%+18\%=118\%

of last year's revenue. So type

118% of x = 708

The solution is $x=600$ , so last year's revenue was $600,000.

Common mistake:

Reading “ $18\%$ greater than” as “ $18\%$ of.” If this year were only $18\%$ of last year, revenue would have shrunk to less than a fifth. “Greater than” means past $100\%$ , so the new revenue is $118\%$ of the old.

Check your understanding:

After a $35\%$ decrease, an account balance is $52. What was the balance before the decrease? Write the equation and solve it.

Calculator loads as you approach

This year’s 708 is 118 percent of last year’s x.

Chain successive changes in order

When percent changes come one after another, each one acts on the amount the last one left behind. The second change never sees the original amount.

Say a game lists for $80. It gets a $25\%$ discount, and then $6\%$ sales tax is added to the discounted price.

The discount leaves $75\%$ of the listed price:

75\%\text{ of }80.

The tax makes the final price $106\%$ of that discounted price, so you wrap the first expression inside the second:

106\%\text{ of }\left(75\%\text{ of }80\right).

Type:

106% of (75% of 80)

Desmos returns

\boxed{63.6},

so the game costs $63.60. Read it from the inside out. The inner part is the discounted price, and the outer part adds tax to it. That's the idea to hold on to: nest the percents, don't add them.

The same nesting works for a group inside a group. Say $60\%$ of a volunteer group works in the morning, and $25\%$ of the morning volunteers lead a team. The morning team leaders are

25\%\text{ of }\left(60\%\text{ of the whole group}\right).

Picture a whole group of $100$ people, so each count is also a percent of the group:

25\%\text{ of }\left(60\%\text{ of }100\right)=15.

So $15\%$ of the whole group are morning team leaders.

Try it yourself:

A price goes up $20\%$ and then down $20\%$ . Guess what percent of the original is left. Then enter 80% of (120% of 100) and compare the result with your guess.

Check your understanding:

A membership grows by $25\%$ and then drops by $20\%$ . What percent of the original membership is left?

Common mistake:

Adding or subtracting percents that come one after another. A $30\%$ discount followed by $8\%$ tax isn’t a $22\%$ decrease, because the tax is figured on the lower, discounted price.

Calculator loads as you approach

Each percent acts on the amount right before it.

Find an unknown percent from the right base

Sometimes the percent is the thing you're looking for. Then you divide first and turn the result into a percent.

Find what percent one amount is of another

If $A$ is the part and $B$ is the base, then

\text{percent}=\frac{A}{B}\times100.

Say $36$ out of $80$ students chose the library. That's

\frac{36}{80}\times100=45\%.

Find a percent change

For a change, the base is the old value, the one you started from:

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

The result has a sign. An increase comes out positive, and a decrease comes out negative. When a question asks for a percent decrease, give the size of the change without the minus sign.

A price that rises from $80 to $100 has percent change

\frac{100-80}{80}\times100=25.

It's positive, so that's a $25\%$ increase.

Now go the other way. From $100 down to $80, the old value is $100$ :

\frac{80-100}{100}\times100=-20.

That's a $20\%$ decrease. It's the same $20 gap both ways, but the percent changes because the base does.

Things get subtle when the amount that changes is itself a percent. Say an approval rate rises from $40\%$ to $52\%$ . Subtract, and you get the change in percentage points:

52-40=12

percentage points. Use the percent-change formula with the old rate as the base, and you get the percent increase:

\frac{52-40}{40}\times100=30\%.

Both are correct. They answer different questions, so check which one the question asks for.

Check your understanding:

A survey response rate rises from $25\%$ to $35\%$ . How many percentage points did it rise? What was the percent increase, compared with the original rate?

Common mistake:

Dividing by the new value. A percent change compares the change with where you started, so the old value goes in the denominator. For $80 to $100, dividing the $20 by $100$ gives $20\%$ , but the real increase is $25\%$ .

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The denominator is the base: the whole group for a part, and the old price for a change.

Example: Compare a list price with a wholesale cost

Worked example

A store sells an item at a 25% discount off its list price. The sale price is 20% greater than the wholesale cost. By what percent is the list price greater than the wholesale cost?

A
$40\%$
B
$50\%$
C
$60\%$
D
$75\%$

Step 1

Pick an easy number for the cost

The problem gives only percents, no dollar amounts, so you can choose the wholesale cost yourself. Every price grows or shrinks along with it, so the final percent comes out the same whatever you pick. Take $100, because it makes percents easy to read.

The sale price is $20\%$ greater than the cost, so it's $120\%$ of $100:

120\%\text{ of }100=120.

With this cost, the sale price is $120.

Step 2

Work backward through the discount

A $25\%$ discount leaves $75\%$ of the list price. You don't know the list price, so call it $x$ and type:

75% of x = 120

Desmos shows $x=160$ , so the list price is $160.

Calculator loads as you approach

The sale price, 120, is 75 percent of the unknown list price x.

Step 3

Answer what the question asks

The question wants the percent the list price is above the cost. The list price is $160 and the cost is $100, so the list price is higher by

160-100=60.

Compared with the $100 cost, that's

\frac{60}{100}\times100=60\%

more. The answer is C.

Common mistake:

Saying the list price is $160\%$ greater than the cost. It’s $160\%$ of the cost, which makes it $60\%$ greater than the cost. It’s the “of” versus “greater than” trap again, running the other way.

Finish the solution

Now work backward through two changes at once.

Undo a discount and a tax

Finish the solution

A store discounts an item by $30\%$ . It then adds $8\%$ sales tax to the discounted price. The final price is $75.60.

What was the item's original price, in dollars?

First steps

A $30\%$ discount leaves 70% of x.
The $8\%$ tax acts on that discounted price, so the final price is 108% of (70% of x).
That nested expression is on line 2. Set it equal to the final price and solve for $x$ .

Finish it

Your answer

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Set line 2 equal to the final price, then read x.

Practice problems

Your turn. Try each one before you open its hint, and before you type anything, ask what each percent is taken of.

Find a percent of a number

Practice problem

Last season, a theater sold $1{,}600$ tickets. This season, the number of tickets sold was $125\%$ of last season's number.

How many tickets did the theater sell this season?

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Type the percent sentence just as it reads.

Find the amount before an increase

Practice problem

A recycling program collected $391$ tons of material after a $15\%$ increase from the previous year.

How many tons of material did the program collect in the previous year?

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Turn the increase into one percent of last year’s amount.

Find a percent of a subgroup

Practice problem

In a survey, $60\%$ of the participants prefer digital notes. Of those participants, $35\%$ also use a stylus.

What percentage of all the participants both prefer digital notes and use a stylus?

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Picture 100 participants, then take 35 percent of the digital-notes group.

Solve equal up-and-down changes

Practice problem

An event's attendance increases by $p\%$ from one year to the next. The following year, attendance decreases by the same $p\%$ . The final attendance is $16\%$ less than the original attendance.

What is the positive value of $p$ ?

Your answer

Calculator loads as you approach

Let x be the percent. Desmos shows two solutions, so check which one fits.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A percent is always taken of a base, and the base goes right after of.
When one amount is $p\%$ of another, type p% of B = A.
A $p\%$ increase gives $(100+p)\%$ of the original. A $p\%$ decrease leaves $(100-p)\%$ .
When the original amount is unknown, put $x$ where the base goes and solve.
Changes that come one after another nest. Don't add the percents.
To find an unknown percent, divide the part or the change by the right base and multiply by $100$ .
Subtracting two percents gives percentage points. A percent change still divides by the original percent.
Desmos does the arithmetic and the solving. You pick the base and answer exactly what the question asks.

Next lesson

Build ratio, rate, and unit chains in Desmos

Organize multi-step ratio, rate, scale, mixture, and unit calculations as labeled expressions.

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1,207 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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