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In reference to the number of terms in their ratios, proportions are Simple or Compound.

282: A Simple Proportion is a statement of the equality of two simple ratios. Thus, 5 8 10 16, or 18, is a simple proportion.

283. A Compound Proportion is a statement of the equality of a simple and a compound ratio, or of 2:3 two compound ratios. Thus, 69: 4: 6, or 2 x 6 :

3 x 94: 6, or

2 × 6
3 x 9

, represents the equality of a

compound and simple ratio; while 1:2

1 x 32 x 6 :: 5 × 2: 10 × 4, or

3:6

1 x 3

2 × 6

resents the equality of two compound ratios.

5: 10

::

or

2 : 4'

5 × 2

=

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SIMPLE PROPORTION.

Simple Proportion is employed to find a fourth proportional, when three are given.

Oral Exercises +

EXAMPLE.—If 9 lb. sugar cost 81%, what cost 12 lb. ? SOLUTION.-If 9 lb. cost 819., 12 lb. will cost more in the ratio of 12 to 9, or 12 = as much.

of 81 = 1084. Therefore, if 9 lb. of sugar cost 814, 12 lb. cost 108, or $1.08.

PROBLEMS.

1. If 6 yd. carpet cost $15, what do 10 yd. cost?

2. If a tree 20 ft. high casts a shadow 8 ft. long, how high is the tree whose shadow is 32 ft. long?

3. 7 bbl. molasses cost $63. What cost 14 bbl.?

4. 2 lb. coffee last 6 persons 8 days. How long will it last 24 persons?

5. Paid $1.20 for 3 lb. butter. How many lb. can be bought for $3.60 ?

6. 5 horses eat a ton of hay in 8 months. How long would it take 2 horses to eat the same amount?

7. If the boarding of 3 men for a week costs $21, how many men can be boarded the same length of time for $63 ?

8. If a 10 loaf of bread weighs 12 oz. when flour is $8 a bbl., how much should it weigh when flour is $16 a bbl.? 9. If 3 lb. sugar cost 33, what will 6 lb. cost? 10. If 4 bu. apples cost $3, what do 8 bu. cost? 11. If 5 T. iron cost $400, what will 8 T. cost? 12. If 6 bu. wheat cost $9, what do 5 bu. cost? 13. If 11 lb. coffee cost $1.65, what do 12 lb. cost?

14. If 12 lb. rice cost 72, what do 11 lb. cost?

15. If 15 bbl. beef contain 3000 lb., how many do 20 bbl. contain?

Written Exercises

EXAMPLE 1.-If 50 bu. wheat cost $130, what will 60 bu. cost at the same rate?

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or 50 60 $130: $( ). Solving by 281, RULE I., we have 8× $130 $156.

=

EXAMPLE 2.-If 10 men can build a boat in 12 days, how long will it take 15 men to do the same work?

men

SOLUTION.

men days days 15 10: 12: (8)

:

12 × 10

15

= 8 da. Ans.

EXPLANATION.-If 10 men build the boat in 12 days, 15 men will build it in less time; in other words, the relation of 15 to 10, or 15, determines the relation of 12 da. to the required time.

284. RULE.-Make that number which is of the same kind as the required answer, the third term.

Then determine from the nature of the question whether the answer is to be greater, or less than the third term.

If greater, write the less of the remaining two terms as the first term of the proportion, and the greater as the second. If less, write the greater of the two remaining terms as the first and the less as the second.

Find the fourth term by 281, RULE I.

As the processes are all multiplication and division, it is evident that the work may frequently be shortened by cancellation.

This and all similar examples may be solved by Analysis. Thus, SOLUTION 1.-If 10 men can build a boat in 12 da., it would take 1 man 10 times 12 da., or 120 da. And if 1 man will build the boat in 120 da., 15 men will build it in of 120 da., or 8 da., Ans.

2. If 10 men build a boat in 12 da., 15 men will build it in the same part of 12 da. that 10 is of 15, or 1g

=

of 12 da., or 8 da., Ans.

PROBLEMS.

1. If 50 bu. wheat cost $75, what cost 40 bu.? Ans. $60. 2. If 20 yd. cloth cost $40, how much do 18 yd. cost?

Ans. $36.

3. If 12 lb. tea cost $10, how much would 25 lb. cost?

4. If 7 caps cost $9, what de 10 cost?

Ans. $205.
Ans. $129.

5. If $63 buy 21 hats, how many will $48 buy?

Ans. 16 hats.

6. If 30 bu. wheat make 6 bbl. flour, how many bu. will be needed to make 8 bbl.?

Ans. 40 bu.

7. If 50 bu. wheat make 10 bbl. flour, how many bbl. of flour will 60 bu. make? Ans. 12 bbl.

8. If a man walks 60 mi. in 4 da., how far would he walk in 6 da., at the same rate? Ans. 90 mi. 9. If a man travels 132 mi. in 6 da., how long will it take him to travel 462 mi., at the same rate? Ans. 21 da.

10. If a man, at the rate of 3 mi. an hr., walks a certain distance in 12 hr., how long would it take him to walk the same distance, at 6 mi. an hr. ? Ans. 6 hr.

11. If a ship's provisions will last 36 men 216 da., how long would they last 60 men? Ans. 129.6 da.

Ans. 63 da.

12. If 24 men can build a wall in 42 da., how long would it take 16 men? 13. If 36 men can build a wall in 42 da., how many men would build it in 63 da. ? Ans. 24 men.

14. If a ship's provisions would last 48 men 162 da., how many men would they last 324 da.?

15. If 40 gal. syrup cost £6 10s. (130s.),

Ans. 24 men.

what cost 50 gal.?

Ans. £8 2s. 6d.

16. If 5 lb. 8 oz. tea cost $8.80, how much do 3 lb. 2 oz. cost?

Ans. $5.

17. If 5 bu. 4 qt. wheat cost $12.30, how much do 7 bu. 3 pk. cost? Ans. $18.60. 18. If a ship has water enough to allow each of her crew of 50 men 1 pt. per day, till the end of her voyage, how much can be allowed if she takes on board 30 more men?

Ans. 34 gi.

19. If $150 gain $12, what sum will gain $1200, in the

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in the solution, comparing gain with gain, and capital with capital.

20. How long will oats last 12 horses, which would last 4 horses 24 wk.? Ans. 8 wk. 21. How long will hay last 15 cows, which would last 8 cows 20 wk.? Ans. 10 wk. 22. How many men can do that work in 15 da., which would take 25 men 90 da. ?

Ans. 150 men.

lb.?

23. If 5 lb. sugar cost 80%, what cost
24. If lb. tea cost 371, what cost 5 lb.?

Ans.

Ans. $7.50.

T. ?

25. If 6 T. iron cost £90 18s. 6d., what cost
26. If bbl. flour costs $18, what costs 4 bbl.?

Ans. $6.75.

27. An upright stick 1 ft. 6 in. high, casts a shadow 3 ft., when a tree casts a shadow 1924 ft. How high was the tree? Ans. 961 ft. 28. If I borrow $150 for 6 mo., how long should the lender have $300 of my money to equalize favors?

Ans. 3 mo.

29. If I borrow $150 for 6 mo., how many dollars should the lender have 5 mo. to equalize favors? Ans. $180. 30. If a set of men can lay the brick of a building in 18 days, working 10 hr. per day, in how many days could they lay them working 14 hr. per day? Ans.

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