addition, we have two blue balls, we can express the ratio of red to green to blue as 5:3:2. If units are involved, they are excluded from the ratio. However, it is important that all quantities in a given problem be represented using identical units of measure. EXAMPLE 1 Find the ratio of weights of 2 lb of flour to 9 oz of sugar. Since 2 lb of flour contain 32 oz, we express the ratio as 32:9. If only two quantities are to be compared, the ratio may be conveniently represented as a fraction. For example, a ratio of 4:6 may be written as 416. It is often convenient to simplify the ratio by reducing its fractional value to 218, as discussed in Chap. 1. EXAMPLE 2 Determine the ratio of 1 hour to 25 minutes. Since 1 hour contains 60 minutes, we can write the ratio as 60:25 or, as a fraction, 60l25. Since 60f25 = 125 the ratio may be expressed 12:5. Since 1215 = 2.4, we can also express the ratio as 2.4:1. In nance, it is common to allocate funds according to a specic ratio. This can be conveniently done by dividing the total amount into \"parts.\" EXAMPLE 3 Suppose that $80,000 is to be allocated for advertising, research, and investment in the ratio 8:523. How much money will be allocated for each? Since 8 + 5 + 3 = 16, we divide $80,000 into 16 parts of $5,000 (80,000 + 16 = 5,000). Advertising gets 8 parts. 540.000. research gets 5 parts, $25,000, and investment gets 3 parts, $15,000. Note that the sum of the allocations must equal the original $80,000, Since ratios behave like fractions, we can use the results of Chap. 1 to analyze them. For example, we know that 523 = 10:6 because 5:3 = 1016. EXAMPLE 4 Determine x, given 12-3 = 4:5. We write this in fractional form: go .. Cross-multiply to get 5x 2 12. Then divide both sides of the equation by 5 and x 2 2.4. EXAMPLE 5 How many pounds of peanuts should be added to 50 lb of Since 3 ft = 36 in, the ratio must be expressed 36:6 or 6:1. cashews if their weight ratio is to be 3:2? Let x be the number of pounds of peanuts to be added. It 2.3 Determine the ratio of weights 3 1b to 24 oz. follows that x : 50 = 3:2. In fractional form this is written $ - SOLUTION Cross-multiply to get 2x = 150. Dividing by 2 yields x = 75 1b of peanuts. Since 3 1b = 48 oz, the ratio is 48:24 or 2:1. 2.4 A business spends $180,000 on advertising, $120,000 on SOLVED PROBLEMS research and development, and $150,000 on office rent. Find the ratios between these expenses. SOLUTION 2.1 Reduce the ratios (a) 250:75, (b) 69:15, (c) 1.2 to 3.6. 180,000: 150,000: 120,000 = 18:15:12 (divide by 10,000) SOLUTION = 6:5:4 (divide by 3) (a) 250 10 75 3 (divide by 25), so the reduced ratio is 10:3. 2.5 Allocate $1,500 in the ratio 3:2. 69 23 SOLUTION (b) 15 5 (divide by 3), so the reduced ratio is 23:5. 1.2 12 1 3 + 2 =5, so divide $1,500 into 5 parts of $300 each: (c) 3.6 36 3. so the reduced ratio is 1:3. 300 x 3 = 900 300 x 2 = 600 2.2 Find the ratio of lengths 3 ft to 6 in. so the allocations are $900 and $600. SOLUTION2.6 Allocate $11,250 in the ratio 3:5:?. SOLUTION 3 + 5 + 7 =15, so divide 11,250 by 15 to get 750: 750 X 3 = 2,250 750 X 5 = 3,750 750 X 7 = 5,250 so the allocations are $2,250, $3,750, and $5,250. 2.7 A $14,000 grant is to be divided between Harvard and Yale in the ratio 4:3. How much money should each university receive? SOLUTION 4 + 3 = 7 so the $14,000 is divided into 7 parts of $2,000 each. Harvard gets 4 parts, $8,000 and Yale gets 3 parts, $6,000. 2.8 The order in which a race horse crosses the nish line determines how much money his owner will win. If a purse of $9,000 is divided among the win, place, and show horses in the ratio 3:2: 1, how much will each horse earn? SOLUTION 3 + 2 + 1 = 6, so the purse of $9,000 is divided into 6 parts of $1,500 each (9,000 + 6 = $1,500). The winner receives 3 parts, $4,500, the place horse receives 2 parts, 83,000, and the horse that shows receives 1 part, $1,500. 2.9 Solve forx: x:5 = 18:30. SOLUTION We rst express the ratio in terms of fractions: i = 1% s 30 Then we cross-multiply to obtain 30x = 90. Finally, divide both terms of fractions, thih sides of the equation by 30 and we get x = 3. 2.10 A pancake recipe calls for 2 cups of milk for every ?5 pancakes made. How many cups of milk are needed to make 525 pancakes? SOLUTION