Solve percent problems in Desmos

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Difficulty: Intermediate
Estimated time: 55 minutes
Domains: Problem Solving and Data Analysis
Techniques: PercentTranslationPercent-ofPercent-changeReverse-percentSuccessive-changes

What you’ll learn

Identify the base, the quantity a percent is taken of.
Enter percent-of statements directly in Desmos.
Translate increases, decreases, discounts, taxes, and tips into the final percent of the base.
Solve reverse-percent questions by placing the unknown in the base position.
Chain successive percent changes without adding their percentages.
Calculate an unknown percent or percent change from the correct reference value.
Distinguish 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 percentage questions often hide a short equation inside a sentence. Once you identify what the percent is of, Desmos can keep the relationship visible and solve the arithmetic without a separate decimal conversion.

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?

A
$162$
B
$243$
C
$250$
D
$656$

Fast Desmos solution

Let $x$ be the previous survey's count. Type 162% of x = 405. Desmos graphs a vertical solution line at $x=250$ . Therefore, the answer is C.

The calculator does not decide which quantity belongs after of. That modeling decision is the mathematics. In this example, $162\%$ is taken of the previous survey's count, so $x$ belongs after of.

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The percent relationship becomes a one-variable equation with solution x equals 250.

When should you use Desmos?

Identify the base first, then choose the shortest reliable method.

Enter a literal relationship when…

one quantity is stated as a percent of another;
the original amount is unknown after an increase, decrease, discount, tax, or tip;
awkward values make decimal conversion or division easy to mistype; or
successive percent changes apply to different intermediate amounts.

Use hand reasoning when…

the percent is an easy benchmark such as $10\%$ , $25\%$ , or $50\%$ ;
a simple fraction makes the result immediate; or
the question asks for a comparison that needs no calculation.

Before typing, ask: What quantity comes immediately after the word of? That quantity is the base.

Check your understanding:

One problem asks for $25\%$ of $68$ . Another says that a quantity became $117$ after a $35\%$ decrease and asks for the original quantity. Which method is stronger for each problem?

1. Put the base after “of”

A percent describes a part of a base:

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

Desmos lets you enter a percent relationship directly with % of. For example, 18% of 250 returns

45.

If the base is unknown, leave $x$ in that position. The statement $84$ is $35\%$ of a number becomes 35% of x = 84. The vertical solution line is at $x=240$ .

Statement	Desmos entry
$p\%$ 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% while keeping the result $84$ . Predict whether the unknown base will be greater or less than $240$ , then check your prediction in the calculator.

Check your understanding:

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

Common mistake:

Reversing the part and the base. In 35% of x = 84, the unknown whole is $x$ because $84$ is the part produced by taking $35\%$ of that whole.

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A known base gives a calculation; an unknown base gives a one-variable equation.

2. Convert more and less into the final percent

The phrases greater than, more than, less than, increase, and decrease compare a final amount with its base. They do not give the final percent directly.

If a quantity increases by $p\%$ , the final amount is

(100+p)\%

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

(100-p)\%

of the base.

Wording	Final amount
$p\%$ 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
tax or tip of $p\%$	$(100+p)\%$ of the pre-tax or pre-tip amount

Suppose a company's revenue this year is $18\%$ greater than last year's revenue and equals $708,000. Let $x$ represent last year's revenue, in thousands of dollars. An $18\%$ increase produces $118\%$ of the original. Enter 118% of x = 708. The solution is $x=600$ , so last year's revenue was $600,000.

Common mistake:

Translating " $18\%$ greater than" as " $18\%$ of." If the new value were only $18\%$ of the original, it would be much smaller. An $18\%$ increase produces $118\%$ of the original.

Check your understanding:

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

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An 18 percent increase means the final amount is 118 percent of the original.

3. Chain successive changes in order

Successive percentages apply to different bases. The second change acts on the result of the first change, not on the original amount.

Suppose a game has a listed price of $80, receives a $25\%$ discount, and then has $6\%$ sales tax applied to the discounted price. The discount leaves $75\%$ of the listed price. The tax makes the final price $106\%$ of the discounted price. Enter 106% of (75% of 80). Desmos returns

\boxed{63.60}.

The same structure handles a percent of a subgroup. If $60\%$ of a volunteer group works in the morning and $25\%$ of the morning volunteers lead a team, then the morning team leaders make up

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

Using a whole group of $100$ gives $25\%$ of $60$ , or $15$ . Therefore, $15\%$ of the whole group consists of morning team leaders.

Try it yourself:

Predict what percent of the original remains after a $20\%$ increase followed by a $20\%$ decrease. Then enter 80% of (120% of 100) and compare the result with your prediction.

Check your understanding:

A membership increases by $25\%$ and then decreases by $20\%$ . What percent of the original membership remains?

Common mistake:

Adding or subtracting successive percentages. A $30\%$ discount followed by $8\%$ tax is not a $22\%$ decrease because the tax is calculated from the discounted price.

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Each percent acts on the amount produced immediately before it.

4. Calculate the percent from the correct reference value

Sometimes the percent itself is unknown. In that case, calculate the ratio first.

Find what percent one quantity is of another

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

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

For example, $36$ students out of $80$ chose the library. The percentage is

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

Find a percent change

For a change from an old value to a new value, use the old value as the base:

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

This signed formula gives a positive result for an increase and a negative result for a decrease. When you name the change as a percent decrease, report the positive magnitude.

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

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

The positive result means a $25\%$ increase.

Direction matters. From $100 down to $80, the percent decrease is

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

From $80 up to $100, the increase is $25\%$ . The same difference of $20 produces different percentages because the base changes.

A change between two percentages can also be described in percentage points. If an approval rate rises from $40\%$ to $52\%$ , it rises by $12$ percentage points. Relative to the original $40\%$ , the percent increase is

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

Check your understanding:

A survey response rate rises from $25\%$ to $35\%$ . By how many percentage points did it rise, and what was the percent increase relative to the original rate?

Common mistake:

Dividing by the new value in a percent-change question. The change is compared with the starting value, so the old value belongs in the denominator.

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The denominator is the base: the whole for a part-to-whole percent and the old value for percent change.

5. 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

Choose a convenient base

The problem gives only relative percentages, so choose a wholesale cost of $100. The sale price is $20\%$ greater than the wholesale cost, so it is $120\%$ of $100:

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

The sale price is $120 in this convenient model.

Step 2

Work backward through the discount

A $25\%$ discount leaves $75\%$ of the list price. Let $x$ be the list price and enter 75% of x = 120. Desmos shows $x=160$ . In the convenient model, the list price is $160.

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The sale price is 75 percent of the unknown list price.

Step 3

Answer the requested comparison

The list price exceeds the $100 wholesale cost by $160-100=60$ . Relative to the wholesale cost, that is

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

The answer is C.

Common mistake:

Calling the list price $160\%$ greater than the wholesale cost. It is $160\%$ of the wholesale cost, so it is $60\%$ greater than the wholesale cost.

Finish the solution

The remaining percents and nested relationship are entered. Complete the equation and solve for the original price.

Reverse a discount and 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.
An $8\%$ tax makes the final price 108% of (70% of x).
The nested expression is entered on line 2. Set it equal to the final price and solve for $x$ .

Finish it

Your answer

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

Practice problems

SAT practice problems

Identify the base before typing. Keep of relationships literal, convert increases and decreases to the final percent, and preserve the order of successive changes.

Calculate a direct percentage

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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Enter the stated percent-of relationship directly.

Recover a previous amount

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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Translate the increase into the final percent of the previous 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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Use 100 participants as the whole and take 35 percent of the digital-notes subgroup.

Solve equal successive percent 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

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Use x for the unknown positive percent and inspect both algebraic solutions.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Put the base, the quantity a percent is taken of, immediately after of.
Enter p% of B = A when one quantity is $p\%$ of another.
A $p\%$ increase produces $(100+p)\%$ of the original; a $p\%$ decrease leaves $(100-p)\%$ .
In a reverse-percent question, place $x$ in the original or base position.
Successive percentages apply to successive amounts, so nest the relationships instead of adding the percentages.
When the percent is unknown, divide the part or change by the correct base and multiply by $100$ .
A difference between percentages is measured in percentage points; a percent change uses the original percentage as its base.
Desmos handles the arithmetic and equation solving, but you must identify the correct base and answer the exact comparison requested.

Next lesson

Equivalent expressions by graph overlap

Verify whether two algebraic forms trace the same graph and represent the same values.

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Percentages