Renewing Acquaintance with Ratios

1. What is the ratio of 5 to 10? of 5 to 20? of 20 to 15? of 16 to 24?

2. What is the ratio of 100 to 1000? of 4 bu. to 8 bu.? of 6 feet to 10 yards? of 5 days to 10 weeks? of 6 dimes to 2 dollars?

Hint: Do not forget that only like numbers can form a ratio, that is, the quantities must be of the same kind. Change yards to feet, weeks to days, and so on.

3. What number has the same ratio to 12 that 5 has

to 10?

4. What number has the same ratio to 3 that 20 has to 5? 5. What number has the same ratio to 8 that 30 has to 40?

6. How much is 6 times the ratio of 5 to 3?

7. How much is 10 times the ratio of 8 to 5?

8. John copied a large map to the scale of 1 to 4. (This means that each line on John's map was as long as the corresponding line on the large map.) On the large map the distance between two cities was 1 inches. How far apart will they be on John's map? The dimensions of the large map are 32 by 20 inches. What will be the dimensions of John's map?

9. Draw on your paper a line 3 inches long, then construct lines having the following ratios to the first line. (a) 3; (b) 1; (c) į; (d) ;; (e) }; (f) }; (g) 5.

10. Is the ratio of 4 to 6 the same as the ratio of 20 to 30? 11. Is the ratio of 10 to 6 the same as the ratio of 15 to 9? 12. Is the ratio of 8 to 12 the same as the ratio of 18 to 24? 13. The ratio is the same as the ratio of what to 100? the ratio of what to 100?

14.

15.

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equals the ratio of what to one hundred?

17. If 1 inches on a certain map represents 60 miles, how many miles are represented by a line on that map 6 inches long?

18. Two suits of clothes cost $24 and $36. What is the ratio of their costs?

19. A vertical pole is 6 feet high. of a tree to that of the pole is 9 tree?

The ratio of the height to 1. How high is the

20. A man earning $50 a week spends $35 a week. What is the ratio of his spendings to his earnings? What is the ratio of his savings to his earnings? What is the sum of the two ratios?

Proportion. A proportion is a statement of the fact that two ratios are equal. For example, 9: 12 = 15: 20. This means that the ratio of 9 to 12 is the same as the ratio of 15 to 20. This is true since both ratios equal . A proportion like this is usually read "9 is to 12 as 15 is to 20," but it would mean exactly the same to say nine-twelfths equals fifteen-twentieths.

The first and last terms of a proportion are called extremes; the second and third terms are called means. In the above proportion 9 and 20 are extremes and 12 and 15 are means. You will notice that 9 X 20 = 180 and 12 X 15 = 180. In a proportion the product of the means always equals the product of the extremes.

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A proportion is frequently written as an equality of fractions; 18. In such a proportion the first numerator (extreme) times the second denominator (extreme) equals the first denominator (mean) times the second numerator (mean). This principle furnishes a convenient method of testing the correctness of a proportion.

Testing Proportions by Estimating

Test the following statements to see whether or not they are true. First estimate whether or not they seem to you true, then test by the above principle.

four quantities making up a proportion is lacking we may denote it by some letter such as x. We may then write an equation and solve it by means of the above principle.

This enables us to solve problems like the following.

Example. If five cows eat 600 pounds of feed in a week, how much will eight cows eat?

Solution. Making a proportion of the problem, we should say: "Five cows are to eight cows as 600 pounds are to x, the unknown quantity." Writing this as an equation, we

Find the value of x in the following proportions:

9. If Mrs. Jones gets 60 eggs a day from 80 hens, how many eggs should she get from 120 hens?

10. If John walked 16 miles in 4 hours, how far could he walk at the same rate in 7 hours?

11. Three bags of cement were needed to build a walk 2 rods long. How many bags are required to build a similar walk 6 rods long? How long a walk can be built with 12 bags of cement?

12. In a day 3 one-man tractors plowed a 20-acre field. How many such tractors would be needed to plow an 80-acre field in a day?

13. A 20-pound turkey cost Mr. Petersen $11.00. How much would he have had to pay for a 28-pound turkey at the same rate?

14. Thirty teachers are needed for an enrollment of 600 pupils in a Junior High School. With the same ratio how many pupils can 52 teachers serve?