104. Compound proportion. An expression of the equality of the products of ratios is called a compound proportion.

2 5 I 3

For example, of of. This may also be written thus:

3

=

4

105. Inverse ratio. The ratio 2:3 is called the inverse of the ratio 3:2.

Problems formerly solved by compound proportion are now more frequently solved by analysis (see Smith's Intermediate Arithmetic, p. 169). Where it is necessary to distinguish the ordinary proportion of four terms from compound proportion, the former is called simple proportion. Compound proportion may be omitted without interfering with the subsequent work.

106. Illustrative problems. 1. In the compound proportion here given, find the value of x.

1. Writing this in the more familiar form of fractions, we have

2. If we divide these equals by, or multiply by, they will still be equal. Therefore

3. Since we now have of x, x equals 6 times as much.

by being substituted

Such results should always be checked

for z in the compound proportion. Thus, of § = 1 of ¦¦ of }, for each reduces to 10.

2. If 4 loads of hay will last 6 horses 8 weeks, 8 loads will last how many horses 3 weeks?

1. Considering only the number of loads in relation to the number of horses, the ratio of the latter equals the ratio of the former, for twice as many horses will need twice as many loads, and so on. Therefore

x horses: 6 horses

=

8 loads: 4 loads, or x : 6 8:4.

=

2. Considering only the number of weeks in relation to the number of horses, the ratio of the latter equals just the inverse ratio of the former; for if we have twice as many horses, the hay will last only half as long, the number of weeks diminishing in the same ratio that the number of horses increases. Therefore

x horses: 6 horses 8 weeks: 3 weeks, or x: 6 = 8:3.

3. Finally, considering both the loads and the time, we have

107. We may simplify the actual work by noticing that, as in simple proportion,

In any proportion the product of the means equals the product of the extremes.

Hence, in the above example, after reasoning out the form of the proportion, we have simply to write

WRITTEN EXERCISE

1. If 4 bbl. of flour last 6 persons 12 mo., how many barrels will last 15 persons 18 mo.?

2. If 75 hens eat 16 bu. of feed in 12 wk., how bushels will 30 hens eat in 5 wk.?

many

3. If a farm 130 rd. by 80 rd. costs $1750, how much will one 118 rd. by 50 rd. cost at the same rate?

4. If the total cost of feeding 75 hens for 7 wk. is $10.50, what will it cost to feed 40 hens for 9 wk.?

5. If 4 five-cent loaves of bread last 6 persons 2 da., how many ten-cent loaves will last 5 persons a week?

6. If $800 yields $324 interest in 6 yr. 9 mo., how much interest will $500 yield in 4 yr. 5 mo. at the same rate?

7. In a sewing guild 18 girls make 40 garments in 4 hr. At this rate, how many garments can 30 girls make in 3 hr.?

8. If the value of the eggs laid by 75 hens in 12 wk. is $67.50, what is the value of the eggs laid by a third as many hens in 5 wk.?

9. If 5 boys, working 8 hr. a day, can build a cabin in the woods in 12 da., how many days will it take 15 boys working 2 hr. less a day?

10. If 5 girls in a sewing class make 20 aprons in 3 da., how many girls will it take to make 32 aprons in 4 da.? to make 16 aprons in 6 da.?

11. A stone wall is to be constructed around a rectangular piece of land 2000 ft. by 1075 ft. Ten men, working 8 hr. a day, build 890 ft. in 3 da. The number of men is then diminished 50%, and the remainder work 9 hr. a day. How long will it take to complete the wall?

12. If 25 children use 10 boxes of crayons, 200 in a box, in 4 mo., how long will 6 boxes, 100 in a box, last 40 children?

13. If 2 pumps, working 10 hr. a day for 4 da., can raise 120 T. of water, how many tons can 3 pumps raise in 5 da. of 8 hr. each?

14. If 5 men, working 14 da., 6 hr. a day, can excavate a cellar, how long will it take if there are 7 more men, and they work 1 hr. longer each day?

15. If it will take 5 boys 14 da., working 6 hr. a day, to clear a stony field for a ball ground, how long will it take 12 boys, working 8 hr. a day?

16. Two men rented a pasture for $83. The first puts in 6 horses for 4 mo., the second puts in 5 horses for 31⁄2 mo. What is the share to be paid by each?

17. If it takes 6 boys 15 da., working 4 hr. a day, to build a playhouse, how many boys should be able to do the work in 12 da., working 3 hr. a day?

18. If 9 girls in a manual training class made 12 baskets in 4 da., working 45 min. a day, how many baskets can 15 girls make in 6 da., working an hour a day?

19. If a railroad charges $9 for transporting 3 T. of goods 360 mi., how much should be charged for transporting 7 T. of goods 280 mi. at the same rate?

20. If 5 girls, earning money for Christmas gifts, dressed 14 dolls in 4 da., working 6 hr. a day, how many dolls could 8 girls dress in 3 da. of 5 hr. each, at the same rate?

21. Of 2 boys raising eggs for the market, one has 40 hens and the other 75. If the 40 hens lay 1440 eggs in 12 wk., how many will the 75 hens lay in half that time, at the same rate?

FARM PROBLEMS

WRITTEN EXERCISE

Estimate the answers, putting the estimates in writing for comparison with the correct answers. Some estimates should be exact.

1. A meadow is 19 rd. long and 94 rd. wide. What is its area in acres?

2. A square bin has a capacity of 2352 cu. ft. The bin is 51 ft. deep. What is its length? (Given that 441=21 × 21.) 3. If 1 qt. of corn will plant 230 hills, what will the corn cost for 2760 hills @ 25% a quart?

4. If 1 qt. of beans will plant 150 hills, what will the beans cost for 1950 hills @ 35g a quart?

5. What is the cost of seed for 7500 celery plants, allowing oz. to 1000 plants, @ 22 an ounce?

6. What is the cost of seed for 3500 cauliflower plants, allowing 1 oz. to 1000 plants, @ $2.30 an ounce ?

7. Find the cost of oats required to sow 15 acres, allowing 24 bu. to the acre, the oats being worth 90g a bushel. 8. Find the cost of beans required to plant 5 acres, allowing 12 qt. to the acre, the beans being worth $1.50 a peck.

9. Find the cost of peas required to plant 5 acres, allowing 1 bu. to the acre, the peas being worth $3.50 a bushel.

10. Find the cost of corn required to plant 20 acres, allowing 8 qt. to the acre, the corn being worth $2 a bushel.

11. Find the cost of potatoes required to plant 12 acres, allowing 10 bu. to the acre, the potatoes being worth $1.15 a bushel.