Separating a Number in a Given Ratio. Required to separate 25¢ in the ratio of 2 to 3.

Since there are to be 2 parts in one group and 3 parts in 5 parts in all.

the other, there will be

1. Separate 9 in the ratio of 2 to 1; of 1 to 1.

2. What is the ratio of 6 hr. to 24 hr.? of 6 hr. to 3 hr.? 3. In a class of 24 boys and 20 girls what is the ratio of the number of boys to the number of girls?

4. What is the ratio of the length of summer, 3 mo., to the whole year? to the remainder of the year?

5. There were six cloudy days in September, the rest being clear. What was the ratio of cloudy days to clear days?

6. In playing ringtoss Will has a score of 15 and Mollie has one of 18. What is the ratio of Will's score to Mollie's?

7. Divide 100 ft. in the ratio of 4:1; of 7: 3; of 1:9. 8. In a year having 165 cloudy days and 200 clear days what is the ratio of cloudy days to clear days?

9. In November the ratio of clear days to cloudy days was 3:2. How many days were there of each kind?

10. Four boys arrange a three weeks' camping trip. The first boy stays 18 da.; the second comes late and stays 12 da.; the third is in camp the last two weeks; the fourth stays the three weeks. The total expense is $32.50. What is the share of each?

Proportion. An expression of equality between two ratios is called a proportion.

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For example, the ratio $2: $3 is equal to the ratio 10 ft.: 15 ft. Therefore $2 $3 10 ft.: 15 ft. is a proportion. It is read, "$2 is to $3 as 10 ft. is to 15 ft.” It may, of course, be written simply 2:3=10:15, or 18.

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10

Extremes and Means. The first and last terms of a proportion are called the extremes; the second and third, the means. In the proportion

we see that

7 x 15 = 3 x 35.

Similarly, we see that in any proportion

The product of the means is equal to the product of the extremes. We also see that if x: 2135:105,

Similarly, in any proportion the product of the means divided by either extreme is equal to the other extreme, and the product of the extremes divided by either mean is equal to the other mean. Illustrative Problems. 1. If x:7-13:21, find the value of x.

2. If 31.2: 14.4: =x: 3, find the value of x.

USING WHAT YOU HAVE LEARNED

1. A miller uses 18 bu. of wheat for making 4 bbl. of flour. How many barrels of flour can he make from 207 bu. of wheat?

2. If 4 men can do a piece of work in 10 da., how men would it take to do the work in 8 da. ?

many

In all such problems the words "at the same rate are evidently to be understood.

It will take more men to do the work in fewer days. Therefore x must be greater than 4, and the proportion is x : 4 10:8.

3. A certain circle has a circumference of 366.66 in. What is the circumference of a circle the radius of which is to the radius of the first as 3.4: 5.82?

The longer the radius the longer the circumference in the same proportion. Therefore x: 366.66 = 3.4 : 5.82.

If the terms "circumference" and "radius" are not understood, explain them informally.

4. How long will it take a train to go 148 mi., at the rate of 55 mi. in 1 hr. 30 min. ?

5. If 9 men can complete some work in 20 hr., how many men could complete it in 16 hr. more time?

6. If a man can skate 75 yd. in 8 sec., how long would it take him to skate 200 yd.?

7. If a man can skate 220 yd. in 19 sec., how long would it take him to skate 5 mi. if he could maintain this rate?

8. If 750 lb. can be transported 100 mi. for a certain sum, how far should T. be transported for the same sum ?

9. A prize of $90 was divided between two persons in the ratio 4:5. How much did each receive?

PROPORTION

Find the value of x in each of the following proportions :

15. If you are thinking of a number which has to 17 the ratio 35: 119, what is the number?

16. If a boat crew is rowing at the average rate of 1 mi. in 5 min. 30 sec., how long will it take to row 1 mi. 1540 yd.?

17. If a man can swim 45 yd. in 23 sec., how long would it take him to swim 2 mi. if he could maintain this rate?

18. A prize of $60 was divided between two persons in the ratio of to. How much was the share of each?

19. A map is drawn so that 1 in. represents 0.8 mi. on the surface of the earth. How many square miles of land are represented by a portion of the map 3 in. square?

20. When a sum of money is divided equally among seven persons, each receives $16.80. How much would each receive if the same sum were divided equally among eight persons?

21. One man can do a piece of work in 10 da., and another can do the same work in 9 da. If the wages of the first are $5.40 a day, what should be the wages of the second?

of seed for A., how many

22. If a farmer needs 26 lb. pounds does he need for 19.9 A.?

Proportional Numbers. Numbers that are in proportion are called proportional numbers and are said to be proportional. Similar Figures. Figures that have exactly the same shape are called similar figures.

Proportional Sides. The corresponding sides of similar figures are proportional.

For example, the sides XY, YZ, AB, BC, in the two trian

X

Ꮓ

Y A

B

gles here shown, are proportional; that is, XY: YZ= AB: BC.

PROPORTIONAL SIDES

All work oral

1. The two triangles shown above are similar. If XY is twice AB, how does ZX compare with CA?

2. In the triangles above, if the ratio ZX: XY is equal to what is the value of the ratio of CA: AB?

3. In the triangles above, if ZX: XY = 3 : 4, and if XY = 1 in. and AB = { XY, how long is CA?

4. In this figure what proportion exists among AB, AD, AC, and AE? If AB = AD, what is the ratio of AC to AE?

A

E

B D

5. In the same figure, if AD=13 in., AC= 18 in., or ğ in., and AB=19 in., what is the length of AE?

6. In the same figure, if DE represents the height of a man 6 ft. tall, BC the height of a boy, AD the shadow cast by the man, and AB the shadow cast by the boy, show how to find the height of the boy from measuring the shadows.