Solving Proportions and Ratio Word Problems

A proportion says two ratios are equal, such as 3/4 = x/20. To solve it, multiply across the equals sign (cross-multiply) and divide. For 3/4 = x/20, that gives 4x = 60, so x = 15. This works for recipes, maps, prices, and speed, and the details below cover what to do when a problem is worded differently.

1. Ratios and unit rates

A ratio compares two quantities. The Common Core grade 6 standards give the example of a recipe with 3 cups of flour to 4 cups of sugar, which is 3/4 cup of flour for each cup of sugar (6.RP.A.2). That "per one" number is the unit rate.

The same standard gives another case: $75 for 15 hamburgers is a rate of $5 per hamburger. Divide the total by the count: 75 / 15 = 5.

2. Method 1: find the unit rate first

A 12-pack of juice boxes costs $9.00. How much do 20 boxes cost at the same price?

1. Unit rate: 9.00 / 12 = $0.75 per box
2. Multiply: 20 x 0.75 = $15.00

This method works well when you need to compare deals or when the numbers do not divide evenly.

3. Method 2: cross-multiply

A recipe uses 3 cups of flour for every 4 cups of sugar. If you use 10 cups of sugar, how much flour do you need?

1. Set up equal ratios, keeping flour on top: 3/4 = f/10
2. Cross-multiply: 4 x f = 3 x 10, so 4f = 30
3. Divide: f = 7.5 cups

Check: 7.5 / 10 = 0.75, and 3 / 4 = 0.75. The ratios match.

The one rule is consistency. If flour is on top on the left, flour must be on top on the right. Mixing the order, such as 3/4 = 10/f, gives f = 13.33, which is wrong.

4. Map scale problems

A map says 1 inch represents 25 miles. Two towns are 3.5 inches apart on the map.

1 / 25 = 3.5 / d, so d = 25 x 3.5 = 87.5 miles.

The inches on the map stay on one side of each fraction and the miles on the other.

5. Converting units with a ratio

The grade 6 standards include using ratio reasoning to convert measurement units (6.RP.A.3.d). To change 60 miles per hour into feet per second, use 5,280 feet per mile and 3,600 seconds per hour:

60 x 5,280 / 3,600 = 316,800 / 3,600 = 88 feet per second.

Each conversion factor equals 1 (5,280 feet per 1 mile), so multiplying by it changes the units but not the amount.

6. Part-to-part and part-to-whole

A bag has red and blue marbles in a ratio of 3 to 5, and 40 marbles total.

The ratio 3:5 compares red to blue (part to part). To find each count, first add the parts: 3 + 5 = 8. Red is 3/8 of the total and blue is 5/8.

• Red: 3/8 x 40 = 15
• Blue: 5/8 x 40 = 25

Check: 15 + 25 = 40, and 15:25 simplifies to 3:5.

A common error is to solve 3/5 = x/40, which treats 40 as the number of blue marbles.

7. Is it really proportional?

The grade 7 standards ask students to decide whether two quantities are proportional by testing for equivalent ratios in a table, or by checking whether the graph is a straight line through the origin (7.RP.A.2.a).

Table A (hours, miles)	Ratio	Table B (hours, cost)	Ratio
2, 5	2.5	1, 3	3.00
4, 10	2.5	2, 5	2.50
6, 15	2.5	3, 7	2.33

Table A has a constant ratio of 2.5, so it is proportional. Table B has a different ratio in each row, so it is not. Table B looks like a flat fee plus an hourly charge. At zero hours the cost would be $1, not $0, so the line does not pass through the origin.

8. Writing the equation

For a proportional relationship, the grade 7 standards give the form t = pn, where t is the total cost, n is the number of items, and p is the price per item (7.RP.A.2.c).

If 3 notebooks cost $7.50, then p = 7.50 / 3 = $2.50, and the equation is t = 2.50n. Eight notebooks cost 2.50 x 8 = $20.00. The unit rate is the constant of proportionality (7.RP.A.2.b).

9. Common mistakes

• Flipping one fraction. Keep the same quantity in the same position on both sides.
• Using the whole as a part. Check whether the ratio is part-to-part or part-to-whole.
• Forgetting units. Write the unit with each number so you see when it cancels.
• Assuming every pattern is proportional. Test a table before using a proportion.

10. Key takeaways

• A unit rate is a ratio written "per one": divide the total by the count.
• To solve a/b = c/d, cross-multiply and divide.
• Keep matching quantities in matching positions.
• For part-to-part ratios, add the parts to get the total, then take fractions of the whole.
• A proportional table has the same ratio in every row, and its graph passes through the origin.