Ratios, Rates, and Proportions: Free SAT Math Lesson
"Ratios, rates, proportional relationships, and units" is the first skill listed in College Board's official Problem-Solving and Data Analysis domain, and it's one of the most consistently tested topics on the Digital SAT. This free, complete lesson covers everything: what a ratio is and how to simplify one, the crucial difference between a part-to-part ratio and a part-to-whole ratio, the standard rate formulas for unit price, gas mileage, speed, and density, dividing a quantity according to a given ratio, the definition of a proportion with extremes and means, the cross-products property, checking whether two ratios form a proportion, solving proportions for a variable, and setting up proportions from word problems. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
1. What Is a Ratio?
A ratio is a comparison of two quantities by division. The ratio of a to b can be written three different ways: a to b, a : b, or ab. If two quantities are in the ratio a to b, the actual amounts can be represented as ax and bx, where x is a positive number, this lets you scale the ratio up or down while keeping the same proportion.
Worked Example: The ratio of red marbles to blue marbles in a bag is 5 : 3. If there are 15 red marbles, how many blue marbles are there?
Since 15 red marbles corresponds to 5 parts, each part represents 15 ÷ 5 = 3 marbles. Blue marbles, at 3 parts, total 3 × 3 = 9.
1. A recipe calls for 2 cups of flour for every 3 cups of sugar. Which ratio represents flour to sugar?
Explanation: "Flour to sugar" means flour is listed first, so the ratio is 2 : 3.
2. Simplify the ratio 18 to 24 to lowest terms.
Explanation: The greatest common factor of 18 and 24 is 6. Dividing both by 6 gives 3 : 4.
3. The ratio of red marbles to blue marbles in a bag is 5 : 3. If there are 15 red marbles, how many blue marbles are there?
Explanation: This is the worked example above: 15 red marbles means each of the 5 parts equals 3, so blue marbles (3 parts) total 9.
4. Two numbers are in the ratio 4 : 7. If the smaller number is 4x, which expression represents the larger number?
Explanation: If the two quantities are in the ratio 4 : 7, they can be represented as 4x and 7x for the same value of x. Since the smaller number is 4x, the larger is 7x.
2. Part-to-Part vs. Part-to-Whole Ratios
This distinction trips up more students than any other ratio concept, and it's worth slowing down for. A ratio like "boys to girls is 3 : 5" is a part-to-part ratio, it compares one part of a group to another part. But the fraction of the group that is boys is not 35, it's 38, since the total group has 3 + 5 = 8 parts. Comparing a part to the entire total is a part-to-whole ratio, and mixing the two up is one of the most common ratio errors on the SAT.
Worked Example: In a class, the ratio of boys to girls is 3 : 5. If there are 24 students total, how many are boys?
The ratio has 3 + 5 = 8 total parts. Each part represents 24 ÷ 8 = 3 students. Boys, at 3 parts, total 3 × 3 = 9 students.
1. A fruit basket has apples and oranges in the ratio 2 : 3. What fraction of the fruit is apples?
Explanation: The total number of parts is 2 + 3 = 5. Apples make up 2 of those 5 parts, so the fraction is 25, not 23.
2. In a bag of 40 marbles, red and blue marbles are in the ratio 3 : 5. How many red marbles are there?
Explanation: The ratio has 3 + 5 = 8 total parts. Each part represents 40 ÷ 8 = 5 marbles. Red marbles, at 3 parts, total 3 × 5 = 15.
3. The ratio of cats to dogs at a shelter is 4 : 7. What is the ratio of cats to the total number of animals?
Explanation: The total number of parts is 4 + 7 = 11. The ratio of cats (a part) to the total (the whole) is 4 : 11.
3. Rates and Unit Rates
A rate is a ratio of two measurements with different units, like $2.59 per gallon or 60 miles per hour. A unit rate is a rate with a denominator of 1. Several common unit rates appear repeatedly on the SAT:
| Quantity | Formula |
|---|---|
| Unit price | Price of packageNumber of units in the package |
| Gas mileage | Number of miles traveledNumber of gallons of gas used |
| Speed | Number of miles traveledNumber of hours |
| Density | MassVolume |
Worked Example: A car travels 276 miles on 8 gallons of gas. What is the car's gas mileage?
276 mi8 gal = 34.5 miles per gallon
1. A 15-ounce box of cereal costs $4.50. What is the unit price, per ounce?
Explanation: Unit price = $4.5015 oz = $0.30 per ounce.
2. A car travels 276 miles on 8 gallons of gas. What is the car's gas mileage?
Explanation: This is the worked example above: 276 ÷ 8 = 34.5 miles per gallon.
3. A driver traveled 20212 miles in 412 hours. What was his speed, in miles per hour?
Explanation: Speed = 202.5 mi4.5 hr = 45 miles per hour.
4. A volume of 25 cm³ of a metal has a mass of 197.5 grams. What is the metal's density?
Explanation: Density = massvolume = 197.5 g25 cm³ = 7.9 g/cm³.
5. Which is the better unit price: a 12-pack of soda for $5.40, or an 18-pack for $7.65?
Explanation: The 12-pack costs $5.40 ÷ 12 = $0.45 per can. The 18-pack costs $7.65 ÷ 18 = $0.425 per can. Since $0.425 < $0.45, the 18-pack is the better deal.
4. Dividing a Quantity in a Given Ratio
A common SAT setup gives a total amount, like a sum of money, a total angle measure, or a total profit, split among parts in a given ratio. Represent each part using the ax, bx, ... notation, set their sum equal to the total, and solve for x.
Worked Example: The three angles of a triangle are in the ratio 2 : 3 : 4. What is the measure of each angle?
The angle sum of a triangle is 180°.
2x + 3x + 4x = 180
9x = 180, so x = 20
The three angles measure 2(20) = 40°, 3(20) = 60°, and 4(20) = 80°.
1. The measures of the three angles of a triangle are in the ratio 1 : 2 : 3. What is the measure of the largest angle?
Explanation: x + 2x + 3x = 180, so 6x = 180, and x = 30. The largest angle is 3(30) = 90°.
2. A sum of $600 is divided between two people in the ratio 2 : 3. How much does each person receive?
Explanation: 2x + 3x = 600, so 5x = 600, and x = 120. The two shares are 2(120) = $240 and 3(120) = $360.
3. Three business partners divide a profit of $84,000 in the ratio 3 : 4 : 5. What is the largest share?
Explanation: 3x + 4x + 5x = 84,000, so 12x = 84,000, and x = 7,000. The largest share is 5(7,000) = $35,000.
4. A rectangle's length and width are in the ratio 5 : 2. If the perimeter is 84 units, what is the length?
Explanation: Perimeter = 2(length + width) = 84, so length + width = 42. With length and width in the ratio 5 : 2, there are 7 total parts, so each part is 42 ÷ 7 = 6. The length, at 5 parts, is 5(6) = 30.
5. What Is a Proportion?
A proportion is an equation stating that two ratios are equal. The proportion ab = cd is read "a is to b as c is to d." The numbers a and d are called the extremes of the proportion, and the numbers b and c are called the means.
In a proportion, the product of the extremes equals the product of the means, this is called the cross-products property. If ab = cd, then ad = bc. This single property is what makes proportions so useful, it turns a comparison of fractions into a simple equation you can solve.
1. In the proportion 37 = 921, which two numbers are the means?
Explanation: In ab = cd, the means are b and c, the two "inner" numbers. Here, a = 3, b = 7, c = 9, d = 21, so the means are 7 and 9.
2. If ab = cd, which equation represents the cross products?
Explanation: The cross-products property states that the product of the extremes (a and d) equals the product of the means (b and c): ad = bc.
3. In the proportion ab = cd, which two quantities are called the extremes?
Explanation: The extremes are the two "outer" numbers in the proportion, a and d.
6. Checking Whether Ratios Form a Proportion
To determine whether two ratios form a proportion, find the cross products. If the cross products are equal, the ratios form a proportion. If not, they don't.
Worked Example: Determine whether 0.62.4 and 1.56 form a proportion.
0.6 × 6 = 3.6 and 2.4 × 1.5 = 3.6
The cross products are equal, so the ratios form a proportion.
1. Do the ratios 812 and 69 form a proportion?
Explanation: 8 × 9 = 72 and 12 × 6 = 72. The cross products are equal, so the ratios form a proportion.
2. Do the ratios 1435 and 49 form a proportion?
Explanation: 14 × 9 = 126 and 35 × 4 = 140. Since 126 ≠ 140, the ratios do not form a proportion.
3. Do the ratios 0.753 and 2.510 form a proportion?
Explanation: 0.75 × 10 = 7.5 and 3 × 2.5 = 7.5. The cross products are equal, so the ratios form a proportion.
4. Which pair of ratios does NOT form a proportion?
Explanation: For 58 and 1015: 5 × 15 = 75, but 8 × 10 = 80. Since these are unequal, this pair does not form a proportion. Every other pair has equal cross products.
7. Solving Proportions for a Variable
To solve a proportion that contains a variable, cross-multiply to clear the fractions, then solve the resulting equation using the same steps you'd use for any linear equation.
Worked Example: Solve 58 = 15x + 2.
5(x + 2) = 8(15) (Cross products)
5x + 10 = 120 (Distribute)
5x = 110 (Subtract 10 from each side)
x = 22 (Divide each side by 5)
1. Solve: 35 = x20
Explanation: Cross-multiply: 3(20) = 5x, so 60 = 5x, and x = 12.
2. Solve: 7x = 219
Explanation: Cross-multiply: 7(9) = 21x, so 63 = 21x, and x = 3.
3. Solve: 4x − 3 = 810
Explanation: Cross-multiply: 4(10) = 8(x − 3), so 40 = 8x − 24. Add 24 to each side: 64 = 8x, and x = 8.
4. Solve: x + 16 = 53
Explanation: Cross-multiply: 3(x + 1) = 6(5), so 3x + 3 = 30. Subtract 3: 3x = 27, and x = 9.
5. Solve: 92x = 38
Explanation: Cross-multiply: 9(8) = 3(2x), so 72 = 6x, and x = 12.
8. Setting Up Proportions from Word Problems
Many SAT word problems give you one known ratio and ask you to scale it up or down. The cleanest way to set these up is to write two ratios with matching units in matching positions, top and bottom, then cross-multiply and solve.
Worked Example: A car requires 6 gallons of gasoline to travel 132 miles. How much gasoline, to the nearest gallon, will it need for a 350-mile trip?
Let g be the number of gallons needed.
miles → 1326 = 350g ← miles
gallons → ← gallons
132g = 6(350) = 2,100
g = 2,100132 ≈ 15.9
Rounded to the nearest gallon, the car needs about 16 gallons.
1. A recipe uses 3 eggs for every 8 cups of flour. How many eggs are needed for 24 cups of flour?
Explanation: 38 = x24. Cross-multiply: 8x = 3(24) = 72, so x = 9.
2. A map has a scale of 1 inch = 15 miles. If two cities are 4.5 inches apart on the map, what is the actual distance between them?
Explanation: 115 = 4.5x. Cross-multiply: x = 15(4.5) = 67.5 miles.
3. A factory produces 450 units in 5 hours. At this rate, how many units will it produce in 8 hours?
Explanation: 4505 = x8. Cross-multiply: 5x = 450(8) = 3,600, so x = 720.
4. A car requires 7 gallons of gasoline to travel 154 miles. How much gasoline, to the nearest gallon, will it need for a 400-mile trip?
Explanation: 1547 = 400g. Cross-multiply: 154g = 7(400) = 2,800, so g = 2,800154 ≈ 18.18, which rounds to 18 gallons.
Common Mistakes to Avoid
- Treating a part-to-part ratio as a part-to-whole fraction. If a ratio is "3 to 5," the fraction of the total that the first quantity represents is 38, not 35. Always add the parts of the ratio to find the whole before computing a part-to-whole fraction.
- Writing a ratio in the wrong order. "The ratio of x to y" always means x first, y second. Reading a word problem quickly can lead to flipping the order by accident.
- Confusing which quantity goes in the numerator of a rate. Gas mileage is miles per gallon (miles on top), not gallons per mile. Reversing a rate's numerator and denominator is a common and costly error.
- Skipping the step of finding x when dividing a quantity in a ratio. Don't distribute the total directly across the ratio numbers, always solve for the value of one part first (x), then multiply back to find each share.
- Mismatching units when setting up a proportion. Whatever unit is on top of the first ratio must be on top of the second ratio too. Writing miles over gallons on one side and gallons over miles on the other will give a completely wrong answer.
- Forgetting to distribute when a cross product creates a binomial. If cross-multiplying gives you something like 5(x + 2), you must distribute the 5 across both terms before continuing to solve.
- Rounding too early in a multi-step word problem. Keep exact values (or extra decimal places) through the intermediate steps, and only round at the very end, exactly as the question asks.
Frequently Asked Questions
Is a ratio the same thing as a fraction?
They're closely related but not identical. A ratio compares two quantities and can describe a part-to-part relationship, like 3 boys for every 5 girls. A fraction specifically represents a part out of a whole. Every part-to-whole ratio can be written as a fraction, but a part-to-part ratio cannot be read directly as a fraction of the total without first finding the sum of the parts.
Is a proportion just a ratio written differently?
A proportion is an equation that sets two ratios equal to each other, so it builds on ratios rather than being a different concept entirely. Once you can express a relationship as a ratio, writing a proportion just means saying that ratio stays the same as the quantities scale up or down.
Should I use cross-multiplication or a different method to solve a proportion?
Cross-multiplication is the fastest and most reliable method for a proportion with a variable. It's really just a shortcut for multiplying both sides by a common denominator to clear the fractions, so it always gives the same result.
Where can I practice more problems like these?
The Ratios, Rates, and Proportional Relationships quizzes in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.