Solving Ratio and Percent Problems
Students use tables, double number lines, and equations to solve proportional reasoning and percent problems.
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Build Ratio Tables
A ratio table organizes pairs of quantities that have the same relationship. To make equivalent ratios, multiply or divide both quantities by the same number. Suppose 4 tickets cost $12. Put 4 and 12 in the first column. Dividing both numbers by 4 shows that 1 ticket costs $3, so the unit rate is $3 per ticket. From the unit rate, multiply both quantities by 6 to find that 6 tickets cost $18. You could also multiply the original ratio by 2 to find that 8 tickets cost $24. Each column represents an equivalent ratio: 4 to 12, 1 to 3, 6 to 18, and 8 to 24. Never change only one quantity, because that would change the relationship.
Use Double Number Lines
A double number line shows two related quantities on parallel lines. Matching points line up vertically, making equivalent ratios easy to see. Suppose a hiker travels 3 miles in 24 minutes at a constant rate. Place 0 miles above 0 minutes and 3 miles above 24 minutes. To find the time for 5 miles, first find the unit rate. Divide both values by 3: 1 mile takes 8 minutes. Then multiply both values by 5: 5 miles take 40 minutes. Mark 5 directly above 40 on the number lines. The equal spacing matters. Each increase of 1 mile matches an increase of 8 minutes. Double number lines are especially useful when you want to see how quantities grow together.
Write Ratio Equations
A ratio equation, also called a proportion, states that two ratios are equal. Keep corresponding quantities in the same positions. Suppose 5 notebooks cost $15, and you want to find the cost x of 8 notebooks. Write 15/5 = x/8 because each numerator is a cost and each denominator is a number of notebooks. The first ratio equals $3 per notebook. To solve using cross products, multiply 15 × 8 and 5 × x. This gives 120 = 5x. Divide both sides by 5 to get x = 24. Therefore, 8 notebooks cost $24. Check by finding the unit rate: 24 ÷ 8 = 3, which matches 15 ÷ 5 = 3.
Interpret Percent as Per Hundred
Percent means per hundred. The symbol % tells how many parts are being counted out of 100 equal parts. For example, 37% means 37 out of 100. It can be written as the ratio 37:100, the fraction 37/100, or the decimal 0.37. On a 10-by-10 grid, each small square represents 1% because there are 100 equal squares. Shading 37 squares models 37%. A percent can also describe a quantity that is not actually divided into 100 pieces. For example, 15 out of 60 is 25% because the equivalent ratio 15/60 simplifies to 1/4, and 1/4 is equal to 25/100. Rewriting a ratio with a denominator of 100 helps identify its percent.
Solve Percent Problems
Percent problems connect a part, a whole, and a percent. First identify which quantities are known. Suppose 30% of 50 library books are mystery books. The whole is 50, the percent is 30%, and the part is unknown. Change 30% to the decimal 0.30, then multiply: 0.30 × 50 = 15. Therefore, 15 books are mystery books. You can also write the proportion 30/100 = x/50. Cross products give 100x = 1,500, so x = 15. A quick estimate supports the answer: 10% of 50 is 5, so 30%, which is three groups of 10%, is 15. Always label the answer and check that it is reasonable. Because 30% is less than half, the part should be less than 25.