may, with equal propriety, place the $4 as the first term. Thus, $4 2d term: 2 lb. 6 lb..

The operation of finding the 2d term in this arrangement, is like that of finding the 4th in the preceding.

It is more convenient to arrange the terms so that the required term shall be the fourth; then,

Since a ratio can subsist only between quantities of the same denomination, Art. 191, Illus. 1,

The third term of a proportion must be of the same denomination as that in which the answer is required.

2. If 3 men can dig a cellar in 10 days, in how many days will 5 men dig it?

SOLUTION. Since the answer is to be days, write 10 days for the 3D term. The result will be found by multiplying the 3D term by the 2D, and dividing by the 1st (Art. 201); and,

Since it will require 5 men a less number of days than 3 men, place the less number of men for the 2D term, and the greater for the 1st.

OPERATION. m. m. da. 5:3:10

3

5)30

Ans. 6.

3. If 3 men can dig a cellar in 10 days, how many men will dig it in 6 days?

SOLUTION. Since the answer is to be men, write 3 men for the third term. Since it will require a greater number of men in 6 days than in 10 days, place the greater number of days for the second term, and the less for the first.

*4. If 3 yd. cloth cost $8, what cost 6 yd.? *5. If 5 bl. flour cost $30, what cost 3 bl.?

STATEMENT. da. da. m. 6:10:3

Ans. 5 men.

Ans. $16.

Ans. $18.

GENERAL RULE FOR SIMPLE PROPORTION.

1. Write for the third term that number which is of the denomination required in the answer.

2. If, from the nature of the question, the answer should be greater than the third term, place the greater of the other two numbers for the second term, and the less for the first; but,

If the answer should be less than the third term, place the less number for the second term, and the greater for the first.

3. Multiply the third term by the 'second, and divide the product by the first; the quotient will be the answer in the denomination of the third term.

NOTES.-1. If the first and second terms contain different denominations, reduce them to the same; if the third term consists of more than one denomination, reduce it to the lowest given.

2. Multiplying the 2d and 3d terms together, and dividing by the 1st, is really multiplying the 3d by the ratio of the 1st to the 2d. It is most convenient, especially when the ratio is a whole number, to express it in its simplest form before multiplying.

3. After dividing, if there be a remainder, reduce it to the next lower denomination, and divide again, and so on.

OPERATION.

lb. oz. lb. oz. $

3 12:11 4:: 3.50

6. If 3lb. 12oz. tea cost $3.50, what cost 11 lb. 4oz.? SOLUTION.-State the question; and, to express the 1st and 2d terms in the same denomination, reduce both to oz. Then multiply the third term ($3.50) by 180, and divide the product by 60. But it is shorter to multiply at once

by 3, the ratio of 60 to 180.

Oz.

Oz. $

601803.50

1 : 3

3

Ans. $10.50

7. If 2lb. 8oz. of tea cost $2, what quantity can you buy for $5? Reduce the 3d term to ounces; the answer will then be in ounces.

Ans. 6 lb. 4 oz.

8. If 4 hats cost $14, what cost 10 hats? 9. If 3 caps cost 69 cents, what cost 11? 10. If 4 yd. cloth cost $7, what cost 9? 11. If 8 yd. cloth cost $32, what cost 12?

Ans. $35. Ans. $2.53 Ans. $15.75

Ans. $48.

REVIEW.-203. In stating a question in simple proportion, what number is put for the 3d term, Rule? How are the other two numbers arranged? How is the answer obtained? Of what denomination is the answer?

203. NOTE 1. If the first and second terms contain different denominations, what is required? What if the third term contains more than one denomination? What other explanations in Notes

and 3?

12. If 12 yd. cloth cost $48, what cost 8 yd.? Ans. $32. 13. If $32 purchase 8 yards of cloth, how many yards will $48 buy? Ans. 12 yd.

14. If $48 purchase 12 yd. of cloth, how many yards can be bought for $327

Ans. 8yd.

15. A man receives $152 for 19 months' work how much should he have for 4 mon. work?

SUGGESTION. Since the product of the second and third terms is to be divided by the first, therefore, (Art. 71,) when the first term contains one or more factors common to either of the other terms, shorten the operation by canceling the common factors.

OPERATION.

mon. mon. $ 19:4152

8

152×4

16. If 8 men perform a piece of work what time can 12 men perform it?

19

$

32 Ans.

in 24 days, in Ans. 16 days.

17. If 60 men perform a piece of work in 8 days, how many men will perform it in 2 days?

Ans. 240.

18. If 15 oz. of pepper cost 25 cts., what cost 6 lb. ?

SUGGESTION.-Instead of multiplying the 6 lb. by 16, to reduce it to ounces, indicate the multiplication, and cancel the factors common to the divisor and dividend. In this manner, operations may be shortened.

Oz.

15

OPERATION.

cts.

lb.

6: 25

15

6 X 16 :: 25

19. If 6 gal. of molasses cost 65 cts., what cost 2hhd.?

Ans. $13.65

20. If 5cwt. 3qr. 10 lb. of sugar cost $21.06, what will 35 cwt. 1 qr. cost? Ans. $126.90

21. If 1 yd. 2 qr. of cloth cost $2.50, what will be the cost of 1 qr. 2 na.? Ans. $0.625

22. If 90 bu. of oats supply 40 horses 6 days, how long will 450 bu. supply them? Ans. 30 da.

23. If 6 men build a wall in 15 days, how many men can build it in 5 days? Ans. 18 men.

24. If 15 bu. of corn pay for 60 bu. of potatoes, how much corn can be had for 140 bu. potatoes? Ans. 35 bu. 25. If 3cwt. 1 qr. of sugar cost $22.60, what will be the cost of 16 cwt. 1 qr.? Ans. $113.

26. If a perpendicular staff, 3 ft. long, cast a shadow 4 ft. 6 in., what is the hight of a steeple, whose shadow measures 180 ft.? Ans. 120 ft. 27. If a man perform a journey in 60 days, traveling 9 hours each day, in how many days can he perform it by traveling 12 hours a day?

Ans. 45 da. 28. A merchant failing, paid 60 cts. on each dollar of his debts. He owed A $2200, and B $1800: what did each receive? Ans. A $1320. B $1080.

29. A merchant having failed, owes A $800.30; B $250; C $375.10; D $500; F $115. His property, worth $612.12, goes to his creditors: how much will this pay on the dollar? Ans. 30 cts.

30. If the 4 cent loaf weigh 9 oz. when flour is $8 a bl., what will it weigh when flour is $6 a bl.?

31. I borrowed $250 for 6 mon. lend $300, to compensate the favor?

how

Ans. 12 oz. long should I Ans. 5 mon.

32. A starts on a journey, and travels 27 miles a day; 7 days after, B starts, and travels the same road, 36 mi. a day in how many da. will B overtake A? Ans. 21 da. 33. If William's services are worth $153 a mon., when he labors 9 hr. a da., what ought he to receive for 4 mon., when he labors 12 hr. a da.? Ans. $91.91

9

34. If 5 lb. of butter cost $g, what cost 3lb.? Ans. $33. 35. If 6 yd. cloth cost $53, what cost 73 yd.? Ans. $653. 36. If bu. wheat cost $3, what cost bu.? Ans. $1%. 13 37. If 13 yd. cloth cost $4, what cost 2 yd.? Ans. $1. 38. If $293 buy 59 yards of cloth, how much will $31 buy? Ans. 62 yd. 39. If .85 of a gallon of wine cost $1.36, what will be the cost of .25 of a gallon?

40. If 61.3 lb. of tea cost $44.9942, cost of 1.07 lb.?

Ans. $0.40 what will be the Ans. $0.78538

220*

41. If of a yard of cloth cost $3, what will, of an Ell English cost? Ans. $18 42. If 3⁄4 of a yd. of velvet cost $43, what cost 173 yd. ? Ans. $178.38!

43. A wheel has 35 cogs; a smaller wheel working in it, 26 cogs in how many revolutions of the larger wheel, will the smaller gain 10 revolutions? Ans. 288.

44. If a grocer, instead of a true gallon, use a measure deficient by 1 gill, what will be the true measure of 100 of these fulse gallons? Ans. 96 gal.

45. If the velocity of sound be 1142 feet per sec., and the number of pulsations in a person 70 per min., what the distance of a cloud, if 20 pulsations are counted between the time of seeing a flash of lightning and hearing the thunder? Ans. 3 mi. 5 fur. 145 yd. 24 ft.

46. The length of a wall, by a measuring line, was 643 ft. 8 in., but, the line was found to be 25 ft. 5.25 in. long, instead of 25 feet, its supposed length: what the true length of the wall? Ans. 654 ft. 11.17 in.

NOTE. A bushel of wheat is 60 lb.; of rye, 56 lb.; corn, 56 lb.; oats, 32 lb.; barley, 48 lb.; timothy seed, 42 lb.; clover seed, 60 lb.; flax seed, 56 lb. See Dry Measure, Note 2, page 83.

The simplest method for the following examples, is by Simple Proportion. The first term will be the number of pounds to the bushel; second, the weight; third, the price.

What will be the cost of

47. 2136 lb. wheat, at 75 cts. per bu.? Ans. $26.70 48. 1225 lb. wheat, at 81 cts. per bu.? Ans. $16.53+ 49. 1000 lb. rye, at 63 cts. per bu.? Ans. $11.25 3000 lb. oats,

50.

at 24 cts: per bu.?

Ans. $22.50

ART. 204. OF CAUSE AND EFFECT.

The rule for Simple Proportion may be presented under another form, based on the obvious truth, that,

When a cause produces a certain Effect, if the Cause be increased or diminished, the Effect will be increased or diminished in the same ratio.