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327. From the preceding illustrations and principles, we deduce the following general
RULE FOR SIMPLE PROPORTION.
I. Place that number for the third term, which is of the same kind as the answer or number required.
II. Then, if by the nature of the question the answer must be greater than this third term, place the greater of the other two numbers for the second term; but if it is to be less, place the less of the other two numbers for the second term, and the other for the first.
III. Finally, multiplying the second and third terms together, divide the product by the first, and the quotient will be the answer in the same denomination as the third term.
Proof.-— Multiply the first term and the answer or fourth term together, and if the product is equal to the product of the second and third terms, the work is right. (Art. 324.)
Obs. 1. If the first and second terms are compound numbers, reduce them to the lowest denomination mentioned in either, before the multiplication or division is performed.
When the third term contains different denominations, it must also be reduced to the lowest denomination mentioned in it.
2. The process of arranging the terms of a question for solution, that is, putting it into the form of a proportion, is called stating the question.
3. It has been shown that all questions in Simple Proportion may easily be solved by Analysis. (Art
. 296.) After solving the following examples by proportion, it will be an excellent exercise for the pupil to solve each by analysis.
4. If 6 yards of broadcloth cost 30 dollars, how much will 20 yards cost?
5. If 8 bls. of flour cost $40, what will 15 bls. cost ? 6. If 16 lbs. of tea cost $12, what will 41 lbs. cost?
Quest.–327. What is the rule for Simple Proportion? Having stated the question, how is the answer found? Of what denomination is the answer? How is Simple Proportion proved? Obs. If the first and second terms contain different denominations, how proceed? When the third term contains different denominations, what is to be done? What is meant by stating the question ?
7. If 12 acres of land produce 240 bushels of wheat, how much will 57 acres produce ?
8. If a man travels 400 miles in 15 days, how far can he travel in 9 days?
9. If 63 barrels of beef cost $504, how much will 7 barrels cost? Statement. 63: 7:: 504 is to the answer.
That is, the product of 504 X 7=63 will be the answer. (Art. 227.)
But this process may be shortened by canceling the factors common to the dividend and divisor. (Art. 91.a.) Thus, the proportion 63: 7:: 504, by canceling the common factor 7, becomes
63 : 5 :: 504; and 504:9=56. Ans. $56.
9 Or thus, since 504x7 is to be divided by 63, the an
504X7 swer is equal to
Now, canceling the factor 7,
63 which is common both to the numerator and denomina
504X7 504 tor, (Art. 136,) the fraction becomes
or 56, 63
9 which is the same result as before.
PROOF. 63 X 56=504 X 7. Hence, 328. If the first term has factors common to either of the other two terms of a proportion, the operation may be shortened by canceling these factors; then proceed as before. (Arts. 91.a, 136.)
Note.- The question should be stated before canceling the common factors.
10. If 16 sheep cost $48, how much will 6 cost ? Statement. 16s. : 6s. :: $48 is to the answer.
48X6 Then by cancelation, =3X6, or $18 Ans.
16 11. If 6 men can build a wall in 36 days, how long will it take 18 men to build it?
Quest.-328. When the first term has factors common to either of the other two terms, how may the operation be shortened?
12. If 10 quintals of fish cost $35, how much will 17 quintals cost ?
13. If a ship has water sufficient to last a crew of 25 men for 8 months, how long will it last 15 men ?
14. If 12 lbs. of sugar cost $1, how much will 84 lbs. cost?
15. If 15 lbs. of lard cost $1.15, how much will 80 lbs. cost?
16. If of an acre of land cost £}, how much will of an acre cost?
A. A. f
5 7 Solution.
is to the answer.
8 8 7 Hence
3 8 x x
Ans. 8 g
5 17. If of a hogshead of molasses cost $28, how much will 16 hogsheads cost?
18. If 24 yds. of broadcloth cost $18, how much will 27 yds. cost?
19. If 6 acres and 40 rods of land cost $125, how much will 25 acres and 120 rods cost?
20. If 15 yds. of silk cost £4, 10s., how much will 75 yds. cost?
21. If a railroad car goes 35 m. in 1 hr. 45 min., how far will it go in 3 days ?
22. If 41 lbs. of chocolate cost 9s., how much will 22; lbs. cost?
23. If 35 lbs. of butter cost $4, how much will 151 lbs. cost?
24. If 84 lbs. of cheese cost $53, how much will 60 lbs. cost?
25. If of a ship is worth $6000, how much is of her worth?
26. If 4į bu. of wheat make 1 barrel of flour, how many barrels will 84 bu, make ?
27. If the interest of $1500 for 12 mo. is $90, what will be the interest of the same sum for 8 mo.?
28. If a tree 20 ft. high cast a shadow 30 ft. long, how long will be the shadow of a tree 50 ft. high?
29. How long will it take a steam ship to sail round the globe, allowing it to be 25000 miles in circumference, if she sails at the rate of 3000 miles in 12 days ?
30. How many acres of land can a man buy for $840, if he pays at the rate of $56 for every 7 acres ?
31. How much will 85 cwt. of iron cost, at the rate of $91 for 13 cwt. ?
32. At the rate of $45 for 6 cwt. of beef, how much can be bought for $980 ?
33. If 9 ounces of silver will make 4 tea spoons, how many spoons will 25 pounds of silver make ?
34. If 15 tons of wool are worth $90000, how much is 5 cwt, worth?
35. If 54 yds. of cloth is worth $27], how much is 504 yds. worth?
36. If 60 men can build a house in 901 days, how long will it take 15 men to build it?
37. A bankrupt owes $25000, and his property is worth $20000 : how much can he pay on a dollar ?
38. At 7s. 6d. per week, how long can a man board, for £24, 10s.?
39. What cost 94 tons of coal, if 141 tons cost £85 ?
40. What cost 291 yds. of cambric, if 13 yds. cost £8, 6s. 31d.?
41. What cost 3 lbs. of raisins, at £6,7s.6d. per 100 lbs.? 42. If 20 sheep cost £37, 12 s., what will 311 cost?
43. At 78. 6d. per ounce, what is the value of a silver pitcher weighing 9 oz. 13 pwt. 8 grs. ?
44. If 405 yards of linen cost £69, 78. 6d., what will 243 yards cost ?
45. A can saw a cord of wood in 6 hours, and B in 9 hours : how long will it take both together to saw a cord ?
46. A cistern has 3 cocks, the first of which will empty it in 10 min.; the second in 15 min.; and the third in 30 min.: how long will it take all of them together to
47. A man and a boy together can mow an acre in 4 hours; the man can mow it alone in 6 hours; how long will it take the boy to mow it ?
329, Compound Proportion is an equality between a compound ratio and a simple one. (Arts. 308, 310.)
Thus, 8:4 3
: : 12 : 3, is a compound proportion. That is, 8 x6:4*3::12:3; for, 8 X 6X3=4x3x12.
Obs. Compound Proportion is often called the Double Rule of Three.
Ex. 1. If 4 men can earn $24 in 6 days, how much can 10 men earn in 8 days.
Suggestion. When stated in the form of a compound proportion, the question will stand thus :
4m.: 8m. 6d. : 10d.
} :: $24 to the answer required. That is, “the product of the antecedents, 4X6, has the same ratio to the product of the consequents, 8X10, as $24 has to the answer.”
Operation. We divide the product of all 24x8x10=1920, the numbers standing in the 2d and 4 x 6=24. and 3d places of the proportion, Now 1920--24=80. by the product of those standing
Ans. 80 dollars. in the first place.
Note.-1. The learner will observe, that it is not the ratio of 4 to 8 alone, nor that of 6 to 10, which is equal to the ratio of 24 to the answer, as it is sometimes stated; but it is the ratio compounded of 4 to 8 and 6 to 10 which is equal to the ratio of 24 to the an
Thus 4X6 : 8X10 :: 24 : 80, the answer. For, 4X6X80=8X10X24 (Art. 324.)
2. A compound proportion, when stated as abovo, is read," the ratio of 4 into 6 is to 8 into 10 as 24 to the answer."
2. If 5 men can mow 20 acres of grass in 4 days, working 10 hours per day, how much can 8 men mow in 5 days, working 12 hours per day? Operation.
State the question, then 8m. 5d.
multiply and divide as be: : 20 to Ans.
fore. 10hr. : 12hr. 8 X5 x 12 x 20=9600; and 5 X 4x10=200. Now 9600 -200=48. Ans. 48 acres.
QUEST.-329. What is compound proportion? Obs. What is compound proportion sometimes called ?
5m. : 4d. :