Business Mathematics: Ratio, Proportion, and Percent - Ratio

A ratio is a way of comparing two or more quantities. If we have five red balls and three green balls, we say that the ratio of red balls to green balls is 5 to 3. Often this is abbreviated 5:3. If, in 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 4/6. It is often convenient to simplify the ratio by reducing its fractional value to 2/3, 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, 60/25. Since 60/25 = 12/5 the ratio may be expressed 12:5. Since 12/5 = 2.4, we can also express the ratio as 2.4:1.
In finance, it is common to allocate funds according to a specific 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 8:5:3. 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, $40,000. 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 5:3 = 10:6 because 5/3 = 10/6.

Example 4
Determine x, given x:3 = 4:5.
We write this in fractional form: x = 12. Then divide both sides of the equation by 5 and x = 2.4.

Example 5
How many pounds of peanuts should be added to 50 lb of cashews if their weight ratio is to be 3:2?
Let x be the number of pounds of peanuts to be added. It follows that x : 50 = 3:2. In fractional form this is written 2x = 150. Dividing by 2 yields x = 75 lb of peanuts.

Solved Problems

2.1   Reduce the ratios (a) 250:75, (b) 69:15, (c) 1.2 to 3.6.
Solution


2.2   Find the ratio of lengths 3 ft to 6 in.
Solution
Since 3 ft = 36 in, the ratio must be expressed 36:6 or 6:1.

2.3   Determine the ratio of weights 3 lb to 24 oz.
Solution
Since 3 lb = 48 oz, the ratio is 48:24 or 2:1.

2.4   A business spends $180,000 on advertising, $120,000 on research and development, and $150,000 on office rent. Find the ratios between these expenses.
Solution


2.5  Allocate $1,500 in the ratio 3:2.
Solution
3 + 2 = 5, so divide $1,500 into 5 parts of $300 each:
300 × 3 = 900
300 × 2 = 600
so the allocations are $900 and $600.

2.6   Allocate $11,250 in the ratio 3:5:7.
Solution
3 + 5 + 7 = 15, so divide 11,250 by 15 to get 750:
750 × 3 = 2,250
750 × 5 = 3,750
750 × 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 finish 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, $3,000, and the horse that shows receives 1 part, $1,500.

2.9   Solve for x: x:5 = 18:30.
Solution
We first express the ratio in terms of fractions:

Then we cross-multiply to obtain 30x = 90. Finally, divide botIn 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 75 pancakes made. How many cups of milk are needed to make 525 pancakes?
Solution
Represent the number of cups of milk required by x. Then 2:75 = x:525. In terms of fractions, this reads x = 1,050. Finally, divide by 75 and we get x = 14 cups of milk.

2.11   The ratio of carnations to daisies in a floral display is required to be 7:5. How many carnations are needed if 300 daisies will be used?
Solution
Let x represent the number of carnations needed. The problem requires that x:300 = 7:5. Expressing this ratio as a fraction gives x = 2,100 so x = 420.

2.12   On an investment of $50,000 Joe Brown receives $65,000 after one year. If Joe Green invested $80,000 in the same venture, how much should he be receiving?
Solution
The amount received after one year is proportional to the amount invested. Let x represent the amount of money Mr. Green will receive. For convenience, we shall represent all numbers as thousands of dollars:

Mr. Green will receive $104,000.