2 Equations, Inequalities, and Applications

2.6 Ratio, Proportion, and Percent Objectives 1. Write ratios. 2. Solve proportions. 3. Solve applied problems using proportions. 4. Find percents and percentages.

Writing Ratios Ratio A ratio is a comparison of two quantities using a quotient. The ratio of the number a to the number b is written a to b, a : b, or

Example 1 Write a ratio for each word phrase. Write Ratios Example 1 Write a ratio for each word phrase. (a) The ratio of 7 yards to 4 yards is 7 4 7 yd 4 yd = . (b) To find the ratio of 8 feet to 6 yards, first convert 6 yards to feet. 6 yards = 6 • 3 = 18 ft The ratio of 8 feet to 6 yards is thus 4 9 8 ft 18 ft = . 6 yd

Write Ratios Example 2 What size is the best buy? That is, which size has the lowest unit price? PEANUT BUTTER Size Price 18-oz $3.49 28-oz $4.99 40-oz $6.79

Write Ratios Example 2 (continued) Size Price Unit Cost (dollars per ounce) 18-oz $3.49 28-oz $4.99 40-oz $6.79 $3.49 18 = $0.194 $4.99 28 = $0.178 $6.79 40 = $0.170 Best Buy! Because the 40-oz size produces the lowest unit price, it is the best buy.

If , then ad and bc are equal and are called cross products. Solve Proportions A proportion says that two ratios are equal, so it is a special type of equation. We read the proportion a b c d = (b, d ≠ 0). as “a is to b as c is to d.” We can also find the products ad and bc by multiplying diagonally. a b c d = If , then ad and bc are equal and are called cross products.

Check by substituting 34 for x in the proportion. The solution is 34. Solve Proportions Example 3 Solve the proportion. 2 3 x 51 = 2 3 x 51 = 2 • 51 = 3 • x Cross products must be equal. 102 = 3x Multiply. 34 = x Divide by 3. Check by substituting 34 for x in the proportion. The solution is 34.

Solving Proportions Example 4 Solve the equation. = w – 4 3 w + 1 6 = w – 4 3 w + 1 6 6 ( w – 4 ) = 3 ( w + 1 ) Cross products must be equal. 6w – 24 = 3w + 3 Distribute. 6w = 3w + 27 Add 24. 3w = 27 Subtract 3w. w = 9 Divide by 3. Check that the solution is 9.

Solve Applied Problems Using Proportions Example 5 After Edwin pumped 7.0 gal of gasoline, the display showing the price read $15.33. When he finished pumping the gasoline, the display read $35.04. How many gallons did he pump? Price = $15.33 7.0 $35.04 x Price Gallons Gallons 15.33x = 7.0(35.04) Cross products must be equal. 15.33x = 245.28 Multiply. x = 16 Divide. He pumped a total of 16 gal. Check this answer. Notice that the way the proportion is set up uses the fact that the unit price is the same, no matter how the gallons are purchased.

Find Percentages and Percents We can solve a percent problem by writing it as a proportion amount base percent 100 = a b P 100 = or . The amount, or percentage, is compared to the base (the whole amount). Since percent means per 100, we compare the numerical value of the percent to 100.

Find Percentages and Percents Example 6 What is 16% of 750. a b P 100 = a 750 16 100 = 100a = 750(16) Cross products must be equal. 100a = 12,000 Multiply. a = 120 Divide. Thus, 16% of 750 is 120.

Find Percentages and Percents Example 7 A CD with a regular price of $15 is on sale this week at 26% off. Find the amount of the discount and the sale price this week. a b P 100 = a 15 26 100 = 100a = 15(26) Cross products must be equal. 100a = 390 Multiply. a = 3.90 Divide. The amount of the discount on the CD is $3.90, and the sale price is $15.00 – $3.90 = $11.10.

Find Percentages and Percents Example 7 A computer advertisement was listed in the newspaper for $595. The regular price was $850. What percent of the regular price was the savings? a b P 100 = 255 850 P 100 = The savings amounted to $850 – $595 = $255. 100(255) = 850P Cross products must be equal. 25,500 = 850P Multiply. 30 = P Divide. The sale price represented a 30% savings.