When can I cross-multiply and when can I not?

Only when your equation is exactly one fraction equal to one fraction. If there are extra terms, like x/4 + 3 = 5/2, first isolate the fraction or combine everything into a single fraction on each side. Cross-multiplying across an addition sign gives a wrong equation.

Why can't I just add percents when there are two changes in a row?

Because the second percent is taken from a different base. A 20\% discount followed by another 20\% discount has multiplier 0.80 · 0.80 = 0.64, which is a 36\% total discount, not 40\%. The second discount applies to the already-reduced price, so it removes fewer dollars.

How do I know which number goes in the denominator of a percent change?

Always the original, starting, or "before" value — the amount you are comparing against. Reading the sentence for the word that signals time order ("rose from 40 to 55", "was 200, now 170") tells you which value came first.

Is the percent equation different from setting up a proportion?

No, they are the same relationship in two forms. partwhole = p/100 becomes part = p/100 × whole after cross-multiplying and dividing. Use whichever form you find faster; both give identical answers.

ALG1-2.4

Proportions & Percent Problems

Learn to set up proportions, solve them by cross-multiplication, and write equations for percent of, percent change, and increase or decrease problems in Algebra 1.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Proportions & Percent Problems, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

A proportion is just an equation that says two ratios are equal, and percent problems are proportions in disguise — every percent is a ratio out of 100. Once you see that connection, tipping at a restaurant, computing sales tax, finding a discount, and scaling a recipe all collapse into the same two or three equations you already know how to solve.

In this lesson you will build proportions from word problems with the units lined up correctly, solve them by cross-multiplication, and then translate percent language — "what percent of", "increased by", "marked down" — into equations you can solve. You will also learn the multiplier shortcut that makes percent increase and decrease fast, and how to work backwards from a final amount to the original amount, which is where most students slip.

Proportions and Why Cross-Multiplication Works

A ratio compares two quantities; a proportion is a statement that two ratios are equal, like

\frac{3}{4} = \frac{x}{20}

. Cross-multiplication is not a magic trick — it is one multiplication step applied to both sides. Starting from

\frac{a}{b} = \frac{c}{d}

multiply both sides by

bd

. On the left the

b

cancels and on the right the

d

cancels, leaving

ad = bc

. That is the whole justification, and it is why cross-multiplication is legal only when you have a single fraction equal to a single fraction. If your equation looks like

\frac{x}{5} + 2 = \frac{7}{3}

, you cannot cross-multiply until the left side is one fraction.

So to solve

\frac{3}{4} = \frac{x}{20}

: cross-multiply to get

3 \cdot 20 = 4x

, so

60 = 4x

and

x = 15

.

One more caution about the variable's position. In

\frac{8}{x} = \frac{12}{9}

the unknown is in a denominator. Cross-multiplying gives

8 \cdot 9 = 12x

, so

72 = 12x

and

x = 6

. Students often panic and try to "flip" only one side. You may flip both sides at once —

\frac{x}{8} = \frac{9}{12}

is an equivalent equation — but never just one.

Always check your answer by substituting back and comparing the two ratios as decimals. Here

\frac{8}{6} \approx 1.33

and

\frac{12}{9} \approx 1.33

, so the solution holds.

Building a Proportion from a Word Problem

The setup, not the algebra, is where proportion problems go wrong. The rule: matching units must occupy matching positions. If miles are on top in the first ratio, miles must be on top in the second.

Suppose a car travels 156 miles on 6 gallons of gas, and you want to know how far it goes on 11 gallons. Write

\frac{156 \text{ miles}}{6 \text{ gallons}} = \frac{x \text{ miles}}{11 \text{ gallons}}

Cross-multiply:

156 \cdot 11 = 6x

, so

1716 = 6x

and

x = 286

miles.

A useful habit is to write the units next to the numbers while you set up, then drop them once the equation is built. If you end up with miles over gallons on one side and gallons over miles on the other, your answer will be wildly off, and the units check catches it before you waste time.

Wording in the problem	Proportion setup
3 pounds cost 7 dollars; find cost of 10 pounds	$\frac{3}{7} = \frac{10}{c}$
Scale 1 inch to 24 feet; drawing is 5.5 inches	$\frac{1}{24} = \frac{5.5}{f}$
4 of 25 students are absent; how many of 300	$\frac{4}{25} = \frac{a}{300}$

Notice each row keeps the same quantity type in the numerator of both ratios. Also watch for problems that change units mid-sentence: if a rate is given per hour but the question asks about 90 minutes, convert first —

90

minutes is

1.5

hours — or the proportion will be comparing unlike things.

The Percent Equation: Part, Whole, and Percent

A percent is a ratio whose denominator is

100

, so

\frac{\text{part}}{\text{whole}} = \frac{p}{100}

is a proportion you can cross-multiply. Many students prefer the equivalent one-line form:

\text{part} = \text{percent (as a decimal)} \times \text{whole}

Translating the English is the real skill. The word of signals multiplication, is signals the equals sign, and what signals the variable.

Sentence	Equation	Result
What is 18% of 250?	$x = 0.18(250)$	$x = 45$
45 is what percent of 250?	$45 = p(250)$	$p = 0.18 = 18\%$
45 is 18% of what number?	$45 = 0.18w$	$w = 250$

All three sentences describe the same relationship; only the unknown moves. If you can identify which of the three quantities is missing, you can write the equation immediately.

Two frequent errors are worth naming. First, forgetting to convert the percent:

18\%

0.18

, not

18

. Multiplying by

18

gives an answer a hundred times too big, and a quick sanity check (

18\%

250

must be well under

250

) catches it. Second, mixing up part and whole in "percent of what number" problems. The whole is the quantity that follows the word "of", and it is usually the larger amount. When your computed percent comes out above

100\%

but the part was clearly smaller than the whole, you divided in the wrong order.

Percent Change, Increase, and Decrease

Percent change compares how much a quantity moved to where it started:

\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%

The denominator is always the original amount. A positive result is an increase; a negative result is a decrease. Using the new value in the denominator is the single most common mistake in this topic, and it produces an answer that is close enough to look right but is not.

For computing a new amount after a change, the multiplier method is faster than finding the change and adding it. A

15\%

increase multiplies by

1.15

; a

15\%

decrease multiplies by

0.85

. So a 40 dollar item after a

15\%

markup costs

40(1.15) = 46

dollars, and after a

15\%

discount costs

40(0.85) = 34

dollars.

Situation	Multiplier	Equation form
Increase by 8%	$1.08$	$N = 1.08P$
Decrease by 8%	$0.92$	$N = 0.92P$
Increase by 100%	$2.00$	$N = 2P$
Decrease by 25%, then 25% again	$0.75 \cdot 0.75 = 0.5625$	$N = 0.5625P$

That last row shows why successive percent changes do not simply add: two

25\%

discounts are a

43.75\%

total discount, not

50\%

.

The multiplier form also makes reverse problems easy. If a shirt costs 34 dollars after a

15\%

discount, do not take

15\%

34

. Instead solve

0.85P = 34

, giving

P = 40

dollars. Whenever the problem hands you the amount after a change and asks for the original, write the multiplier equation and solve for the original.

Checking Answers and Avoiding the Usual Traps

Every problem in this lesson ends with an equation you can test, so build the habit of substituting back. In a proportion, plug your value in and compare the two ratios as decimals. In a percent problem, run the arithmetic forward: if you claim the original price was 40 dollars and the discount was

15\%

, verify that

40(0.85)

really equals

34

.

Estimation is your second safety net. Before computing

23\%

860

, note that

25\%

would be about

215

, so your answer should be a bit under that. An answer of

19.78

or of

1978

is immediately suspect — both come from misplacing the decimal when converting the percent.

A few situations that regularly cause trouble:

Tax and tip stack onto the same base. A 60 dollar meal with an

18\%

tip is

60(1.18) = 70.80

dollars; the tip is normally computed on the pre-tax amount unless the problem says otherwise, so read carefully.

A percent increase can exceed

100\%

. If a population goes from

200

650

, the change is

450

and the percent increase is

\frac{450}{200} = 2.25 = 225\%

. Nothing is wrong; the quantity more than tripled.

For a quantity that cannot go below zero — a price, a population, a length — a percent decrease cannot exceed

100\%

, because you cannot remove more than all of something. If such a problem gives you a

130\%

decrease, check whether you divided by the new value instead of the original. Quantities that can go negative, such as a temperature, are the exception:

20

degrees falling to

-6

degrees really is a

130\%

decrease.

Finally, round only at the end. Rounding a percent to two decimals mid-problem and then multiplying by a large number can shift the final answer by several units.

Key terms

Ratio.: A comparison of two quantities by division, written $a:b$ or $\frac{a}{b}$ .
Proportion.: An equation stating that two ratios are equal, such as $\frac{a}{b} = \frac{c}{d}$ .
Cross-multiplication.: The step that turns $\frac{a}{b} = \frac{c}{d}$ into $ad = bc$ ; it comes from multiplying both sides by $bd$ .
Percent.: A ratio with denominator 100. To use a percent in an equation, write it as a decimal by dividing by 100.
Percent equation.: $\text{part} = \text{percent} \times \text{whole}$ , where the whole is the quantity following the word "of".
Percent change.: $\frac{\text{new} - \text{original}}{\text{original}} \times 100\%$ ; positive for an increase, negative for a decrease.
Multiplier.: The single factor that produces the result of a percent change: $1 + r$ for an increase of rate $r$ , $1 - r$ for a decrease.
Unit rate.: A ratio written with a denominator of 1, such as 26 miles per gallon, often used to shortcut a proportion.

Worked example

A jacket is on sale. After a 30 percent markdown, the sale price is 87.50 dollars. (a) What was the original price? (b) A sales tax of 6 percent is applied to the sale price. What is the total the customer pays? (c) By what percent is the total the customer pays below the original price?

Part (a). The markdown is a percent decrease, so the multiplier is

1 - 0.30 = 0.70

. Let

P

be the original price. The relationship is

0.70P = 87.50

Divide both sides by

0.70

P = \frac{87.50}{0.70} = 125

. The original price was 125 dollars.

Check it forward:

125(0.70) = 87.50

. Correct. Notice what happens if you instead take

30\%

87.50

and add it on:

87.50(1.30) = 113.75

, which is not

125

. Taking a percent of the wrong base is the classic error in reverse percent problems.

Part (b). Tax is a percent increase applied to the sale price, so the multiplier is

1.06

T = 87.50(1.06) = 92.75

The customer pays 92.75 dollars.

Part (c). Percent change always divides by the original amount, which here is the 125 dollar list price.

\frac{92.75 - 125}{125} = \frac{-32.25}{125} = -0.258

Multiply by

100\%

: the total is

25.8\%

below the original price. The negative sign means a decrease, so state the answer as a

25.8\%

decrease.

Sanity check: the customer got

30\%

off but then paid tax, so the net savings should be somewhat less than

30\%

. A result of

25.8\%

fits. If you had gotten something above

30\%

, you would know a step went wrong.

Practice questions

A store raises the price of a backpack from 32 dollars to 40 dollars. What is the percent increase?

Answer: 25%

The change is

40 - 32 = 8

dollars, and percent change divides by the original amount:

\frac{8}{32} = 0.25 = 25\%

. The common wrong answer of

20\%

comes from dividing by the new price,

\frac{8}{40} = 0.20

. The answer

8\%

comes from reporting the raw change in dollars as if it were a percent. Check forward:

32(1.25) = 40

A recipe that serves 6 people calls for 15 ounces of tomato sauce. Set up and solve a proportion to find how many ounces are needed to serve 16 people. Show why your setup keeps the units aligned.

Answer: 40 ounces

Put ounces on top in both ratios and people on the bottom in both:

\frac{15}{6} = \frac{x}{16}

. Cross-multiplying gives

15 \cdot 16 = 6x

, so

240 = 6x

and

x = 40

ounces. The units check tells you the setup is right — each ratio is ounces per person, and

\frac{15}{6} = 2.5

ounces per person, so

16

people need

16(2.5) = 40

ounces, matching the proportion. A setup like

\frac{15}{6} = \frac{16}{x}

would put people over ounces on the right and give a nonsense answer of

6.4

After a 20 percent price increase, a concert ticket sells for 78 dollars. What was the price before the increase?

62.40 dollars
58.00 dollars
65.00 dollars
93.60 dollars

Answer: 65.00 dollars

20\%

increase has multiplier

1.20

, so if

P

is the old price,

1.20P = 78

, giving

P = \frac{78}{1.20} = 65

dollars. Check:

65(1.20) = 78

. The choice of 62.40 dollars comes from subtracting

20\%

78

, which uses the wrong base —

20\%

of the new price is not the same as

20\%

of the old price. The choice of 93.60 dollars comes from increasing

78

again instead of undoing the increase.

FAQ

When can I cross-multiply and when can I not?: Only when your equation is exactly one fraction equal to one fraction. If there are extra terms, like $\frac{x}{4} + 3 = \frac{5}{2}$ , first isolate the fraction or combine everything into a single fraction on each side. Cross-multiplying across an addition sign gives a wrong equation.
Why can't I just add percents when there are two changes in a row?: Because the second percent is taken from a different base. A $20\%$ discount followed by another $20\%$ discount has multiplier $0.80 \cdot 0.80 = 0.64$ , which is a $36\%$ total discount, not $40\%$ . The second discount applies to the already-reduced price, so it removes fewer dollars.
How do I know which number goes in the denominator of a percent change?: Always the original, starting, or "before" value — the amount you are comparing against. Reading the sentence for the word that signals time order ("rose from 40 to 55", "was 200, now 170") tells you which value came first.
Is the percent equation different from setting up a proportion?: No, they are the same relationship in two forms. $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$ becomes $\text{part} = \frac{p}{100} \times \text{whole}$ after cross-multiplying and dividing. Use whichever form you find faster; both give identical answers.

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