Proportion Calculator

Proportion calculator at a glance#

What is a proportion?#

A proportion is an equation stating that two ratios are equal, written as a/b = c/d. The fractions 3/24 and 5/40 form a proportion because both reduce to 1/8.

How do you solve a proportion?#

To solve a proportion, cross multiply and then divide. In a/b = c/d, the cross products a × d and b × c are always equal. For 3/24 = x/40, you get 24 × x = 3 × 40 = 120, and x = 120 ÷ 24 = 5. The calculator above follows the same principle when you leave one box of the right-hand fraction empty.

What is the missing number in 8/56 = ?/49?#

The missing number is 7. Cross multiply to get 56 × ? = 8 × 49 = 392, then divide: 392 ÷ 56 = 7. Both fractions reduce to 1/7, which confirms it.

What a proportion says#

Imagine a cookie recipe that uses 3 cups of flour for 24 cookies, and you want to bake 40 cookies. The ratio of flour to cookies is 3 to 24. A batch of 40 cookies requires maintaining the same ratio, so you write the flour needed over 40 and set the two ratios equal: 3/24 = x/40. This equation is a proportion, read as "3 is to 24 as x is to 40."

OpenStax Prealgebra 2e defines a proportion as an equation of the form a/b = c/d, where neither b nor d is zero, and it states that the cross products of a proportion are equal. This single rule is all you need to find a missing value.

How to solve a proportion by cross multiplying#

1. Write the proportion with matching units in matching places. For instance, cups go on top and cookies on the bottom on both sides: 3/24 = x/40.
2. Cross multiply by multiplying each numerator by the opposite denominator: 24 × x = 3 × 40, so 24x = 120.
3. Divide both sides by the number next to x: x = 120 ÷ 24 = 5.
4. Check the answer: 5/40 reduces to 1/8, the same as 3/24, so the recipe needs 5 cups of flour.

Cross multiplying is effective because it eliminates the fractions. By multiplying both sides of a/b = c/d by b × d, the denominators cancel, leaving a × d = b × c.

How to use the proportion calculator#

The calculator has two fractions side by side with an equals sign between them. Enter the known ratio in the left fraction and the value you have on the right.

• To find a missing value, enter 3 over 24 on the left, leave the top right box empty, and enter 40 below it. Press Calculate, and the empty box will fill with 5.
• The unknown must be in the right-hand fraction. If your problem reads x/12 = 3/4, swap the sides and enter 3/4 = x/12. An equation reads the same in both directions, and the answer is 9.
• To test a proportion, fill all four boxes. The calculator will tell you if the statement is true or false.
• Leave both right-hand boxes empty, and it returns an equivalent fraction, the simplest form when the left fraction reduces.

Every box requires a positive whole number. If your ratio includes a decimal, like 1.5 cups for 12 cookies, multiply both terms by 10 first and enter 15/120. The answer may still be a decimal: 3/4 = 5/x results in x = 6.666..., which you can round as needed.

Two other ways to solve the same proportion#

Find the unit rate#

Divide to find the value of one unit, then multiply. 24 cookies from 3 cups is 24 ÷ 3 = 8 cookies per cup. 40 cookies require 40 ÷ 8 = 5 cups. A double number line aligns both scales, so every matching mark pair is a valid proportion.

Use the scale factor#

Determine how much larger the new batch is, then scale the other term accordingly. 40 cookies represent 40/24 = 5/3 of the original batch, so the flour is 3 × 5/3 = 5 cups. This method is fastest when numbers divide easily, like 4/6 = 10/x, where 10 is 2.5 times 4, and x = 6 × 2.5 = 15.

Setting up word problems#

Most errors come from setting up a proportion incorrectly, not from arithmetic mistakes. Ensure the same quantity occupies the same position on both sides. If cups are on top on the left, ensure they're on top on the right.

Mixing them up still runs the math, but answers a different question. 3/24 = 40/x places cookies on top on the right, yielding x = 320, which doesn't fit a recipe needing only 3 cups for 24 cookies. OpenStax includes checking answer reasonableness in its steps for proportion word problems, and that check catches this mistake.

Where proportions are used#

Recipes and batch sizes#

Scaling any recipe involves a proportion, one ingredient at a time. The flour example works the same way for sugar, eggs, or baking time per tray.

Map scales#

If 1 inch on a map equals 50 miles, 7 inches represent x miles in 1/50 = 7/x. Cross multiply to find x = 50 × 7 = 350 miles.

Unit prices#

If 4 cans cost 6 dollars, then 10 cans cost x dollars in 4/6 = 10/x. Cross multiply: 4 × x = 6 × 10 = 60, resulting in x = 15 dollars.

Heights and shadows#

Objects and their shadows form equal ratios at the same time of day. A 6 foot pole casting a 4 foot shadow next to a tree with a 22 foot shadow yields 6/4 = x/22, so the tree is x = 6 × 22 ÷ 4 = 33 feet tall, approximately 10 m.

Percentages#

A percentage is a ratio out of 100, making a percent question a proportion. To find 15% of 80, solve 15/100 = x/80: 100 × x = 15 × 80 = 1,200, so x = 12. The percentage calculator handles these directly.

Proportions solved step by step#

These are common practice proportions, including ones people search for word for word. The middle column shows the cross products leading to each answer.

Sources#

• OpenStax, Prealgebra 2e, section 6.5: Solve Proportions and their Applications

Frequently asked questions#

What is the difference between a ratio and a proportion?#

A ratio or rate compares two quantities, like 3 cups of flour to 24 cookies. A proportion is an equation stating that two ratios are equal, such as 3/24 = 5/40. To simplify or scale a single ratio, use the ratio calculator.

How do you know if two ratios form a proportion?#

Cross multiply and compare. 6/9 and 8/12 form a proportion because 6 × 12 = 72, and 9 × 8 = 72. 4/6 and 6/8 do not, because 4 × 8 = 32 while 6 × 6 = 36. You can also reduce both fractions, and the simplifying fractions calculator shows that 6/9 and 8/12 both come to 2/3.

Can the missing value be on the left side?#

Yes, it can. Swap the sides before entering the numbers, as the calculator solves for a blank in the right-hand fraction. x/12 = 3/4 becomes 3/4 = x/12, so x = 9.

Can a proportion have decimals or fractions?#

Yes, it can. By hand, cross multiply as usual. In the calculator, multiply the terms of the decimal ratio by 10 or 100 until both are whole numbers, as this doesn't change the ratio. 2.5/10 becomes 25/100.

Does this calculator solve inverse proportions?#

No, it solves direct proportions, where two quantities increase together and their ratio remains constant. In an inverse proportion, one quantity increases as the other decreases, and their product remains constant: 4 workers taking 6 hours and 8 workers taking 3 hours both total 24 worker hours. Set the products equal instead of the ratios.

What is the golden proportion?#

The golden proportion divides a length so that the whole is to the longer part as the longer part is to the shorter part. This ratio is the golden ratio, about 1.618. A 100 cm line divided this way has a longer part of about 61.8 cm. It involves an irrational number, so calculations are made with decimals rather than whole number boxes.