Ratios and Proportions

Arithmetic is the math of counting, sharing, and comparing amounts. It covers the four operations plus fractions, decimals, percents, and ratios.

Ratios and Proportions: the central idea

A ratio compares quantities in a stated order. A proportion claims that two ratios express the same multiplicative relationship.

Words you need

Ratio

A ratio is a comparison of two quantities by division, written 3:4, 3 to 4, or $\tfrac34$.

Proportion

A proportion is an equation stating that two ratios are equal, such as $\tfrac{3}{12}=\tfrac{5}{20}$.

Unit rate

A unit rate is a rate whose second quantity is 1, such as 30 miles per 1 gallon or 1.50 dollars per 1 pound.

Cross products

Cross products are the two diagonal products in $\tfrac{a}{b}=\tfrac{c}{d}$, namely $ad$ and $bc$, and they are equal exactly when the proportion is true.

Scale factor

A scale factor is the number every part of a ratio is multiplied by to make an equal ratio, so multiplying 3:4 by 5 gives 15:20.

Part-to-whole ratio

A part-to-whole ratio compares one group with the total rather than with another group, so a class with 3 boys for every 4 girls has a boys-to-students ratio of 3:7.

What to know before this lesson

Understand fractions, multiplication and division, units, and equivalent representations.

If one of those prerequisites is uncertain, use the Arithmetic subject guide to locate the earlier concept before memorizing a procedure.

Ratios and proportions: a worked example

Every step, with the arithmetic

1. Step 1 - Write the known ratio with its units210 miles to 7 gallons
2. Step 2 - Find the unit rate by dividing$210\div7=30$ miles per gallon
3. Step 3 - Set up the proportion for 12 gallons$\dfrac{210\text{ miles}}{7\text{ gallons}}=\dfrac{x\text{ miles}}{12\text{ gallons}}$
4. Step 4 - Cross multiply$7x=210\times12=2520$
5. Step 5 - Divide to isolate $x$$x=2520\div7=360$ miles
6. Step 6 - Check with the unit rate instead$30\times12=360$ miles, the same answer by a different road
7. Step 7 - Check that the cross products match$210\times12=2520$ and $7\times360=2520$

Three notebooks cost $12$, so the unit rate is $12/3=4$ dollars per notebook. Five notebooks cost $5(4)=20$ dollars.

Everyday ratios turned into unit rates

The step-by-step method for ratios and proportions

1. Write both ratios with matching quantities and units in corresponding positions.
2. Simplify to a unit rate or solve $a/b=c/d$ using valid cross products.
3. Attach units and test whether scaling one ratio by one factor reproduces the other.

How to check your answer

Check $3/12=5/20$ by cross multiplication: both cross products equal $60$.

Use rounding, inverse operations, and benchmark fractions such as one-half or one-quarter to verify the sign and size of the answer.

A mistake that changes the mathematics

Cross multiplication is a consequence of equal fractions, not a rule for adding or comparing unrelated quantities.

Where you will actually use this

Finding the better buy

A 12-ounce bottle for 3.60 dollars is $3.60\div12=0.30$ dollars per ounce, while a 20-ounce bottle for 5.40 dollars is $5.40\div20=0.27$ dollars per ounce. The larger bottle is cheaper per ounce, so unit rates settle the question the shelf price cannot.

Mixing concrete or paint

A mix calling for 1 part cement to 3 parts sand means the ratio 1:3 no matter how much you make. For 12 shovels of sand you need $12\div3=4$ shovels of cement, and the batch is 16 shovels in total.

Measuring a tree from its shadow

A 6-foot person casting a 4-foot shadow while a tree casts a 26-foot shadow gives $\tfrac{6}{4}=\tfrac{h}{26}$. Cross multiplying gives $4h=156$, so the tree is 39 feet tall. Similar triangles are what make this proportion legal.

How ratios and proportions connects to the rest of arithmetic

• Fractions — A ratio written as $\tfrac34$ is a fraction, so simplifying a ratio and simplifying a fraction are literally the same operation.
• Percentages — A percent is a ratio with 100 in second place, so 15 percent is the ratio 15:100 and every percent problem can be solved as a proportion.
• Geometric transformations — A dilation multiplies every length by a scale factor, which is exactly the ratio of a new figure's side to the matching old side.

Try a transfer problem

A map uses $1$ centimeter for $8$ kilometers. Find the map distance for $52$ kilometers and explain why the units cancel.

Show the worked answer

The map ratio is 1 centimeter to 8 kilometers. Set up the proportion $\dfrac{1\text{ cm}}{8\text{ km}}=\dfrac{x\text{ cm}}{52\text{ km}}$ and cross multiply: $8x=52$, so $x=6.5$ centimeters. The unit rate route is faster: 52 kilometers divided by 8 kilometers per centimeter is 6.5 centimeters. The units cancel because the rate is written as centimeters per kilometer. Multiplying $52\text{ km}\times\tfrac{1\text{ cm}}{8\text{ km}}$ cancels kilometers on the top and bottom and leaves centimeters. If the ratio had been flipped by mistake you would get $52\times8=416$, and a map distance of over 4 meters is the clue that the setup was upside down.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to arithmetic practice questions.

Questions about ratios and proportions

Does the order of the numbers in a ratio matter?

Yes, very much. A 3:4 ratio of boys to girls is not the same as 4:3, which would mean more boys than girls. Always write the label with the number, as in 3 boys to 4 girls, so the order cannot get scrambled halfway through a problem.

When is cross multiplying allowed?

Only when two ratios are set equal to each other, as in $\tfrac{a}{b}=\tfrac{c}{d}$. It works because multiplying both sides by $bd$ clears the denominators. It is not a way to add, subtract, or simplify fractions, and using it on $\tfrac12+\tfrac13$ produces nonsense.

What is the difference between a ratio and a rate?

A rate compares quantities with different units, like 30 miles per gallon or 45 words per minute. A ratio often compares the same kind of thing, like 3 boys to 4 girls, so the units cancel and the ratio has no unit at all. Every rate is a ratio, but not every ratio is a rate.

If a mixture uses a 2:3 ratio, how do I find each amount?

Add the parts to find the total number of shares. A 2:3 ratio has $2+3=5$ shares, so in a 25-liter batch each share is $25\div5=5$ liters. That gives $2\times5=10$ liters of the first ingredient and $3\times5=15$ liters of the second, and the two do add back to 25.