Solving Proportional and Percent Problems
Students represent proportional relationships with tables, equations, and unit rates before using them to solve multistep percent problems.
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Recognizing Proportional Relationships
Two quantities are proportional when their ratio remains constant. This constant ratio is called the constant of proportionality. A table showing 2 notebooks for $5, 4 notebooks for $10, and 6 notebooks for $15 is proportional because cost divided by number of notebooks is always 2.5. The constant of proportionality is 2.5 dollars per notebook. A proportional relationship can be written as y = kx, where k is the constant of proportionality. For the notebook example, the equation is y = 2.5x, with x representing notebooks and y representing cost. A table is not proportional if the ratios change. For instance, points (2, 5) and (4, 11) do not share one constant because 5 ÷ 2 is not equal to 11 ÷ 4.
Unit Rates and Equivalent Ratios
A unit rate compares a quantity to one unit of another quantity. Unit rates make it easier to compare prices, speeds, and other proportional situations. Suppose a cyclist travels 42 miles in 3.5 hours. The unit rate is 42 ÷ 3.5 = 12 miles per hour. Therefore, the relationship can be modeled by d = 12t, where d is distance and t is time. Equivalent ratios can also be used to find missing values. If 5 pounds of apples cost $8, then the cost per pound is 8 ÷ 5 = $1.60. Seven pounds would cost 7 × 1.60 = $11.20. Units should be included throughout the work because they show whether to multiply or divide and help explain what the final number represents.
Discounts, Markups, and Tax
A percent is a ratio compared with 100. To calculate a percent of a quantity, convert the percent to a decimal and multiply. Suppose a jacket originally costs $64 and is discounted by 25 percent. The discount is 0.25 × 64 = $16, so the sale price is $64 − $16 = $48. If a 7 percent sales tax is then added, calculate the tax using the sale price: 0.07 × 48 = $3.36. The final cost is $48 + $3.36 = $51.36. The discount and tax should not be combined because they apply to different amounts. A discount multiplier can also be used. Paying 75 percent after a 25 percent discount gives 0.75 × 64 = $48.
Percent Increase and Decrease
Percent change compares the amount of change with the original value. Subtract to find the change, divide by the original value, and convert the result to a percent. Suppose attendance at a school event increases from 240 people to 300 people. The increase is 300 − 240 = 60. Dividing by the original attendance gives 60 ÷ 240 = 0.25, so the percent increase is 25 percent. For a decrease, the process is the same. If a town’s water use falls from 500,000 gallons to 425,000 gallons, the decrease is 75,000 gallons. The ratio 75,000 ÷ 500,000 equals 0.15, so water use decreased by 15 percent. Always divide by the original amount, not the new amount, because percent change measures change relative to the starting value.