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242. If the extremes and one of the means are given, the other mean may be found by dividing the product of the extremes by the given mean. Thus, if the extremes are 3 and 24, and the given mean 6, the other mean is 12; because 24 X 3 = 72; and 72 ; 6 12
243. If the means and one of the extremes are given, the other extreme may be found by dividing the product of the means by the given extreme. Thus, if the means are 8 and 16, and the given extreme 4, the other extreme is 32; because 16 X 8 - 128; and 1.28 ; 4 = 32.
224. Simple Proportion is an equality between two simple ratios.
Simple Proportion is sometimes called the Rule of Three, from three terms being given to find a fourth.
245. To state and solve questions in Simple Proportion.
Ex. 1. If 7lb. of sugar cost 56 cents, what will 36lb. cost ?
Ans. $ 2.88.
Since 7lb. have the same ratio to Extreme. Mean.
36lb. as 56 cents, the cost of the 7 lb. : 3 6 lb. : :5 6 cts. former, have to the cost of the latter, 36
we have the first three terms of a
proportion given, namely, one of the 3 3 6
extremes and the two means. Now, 1 68
to arrange the given numbers in the
order of a proportion, or state the 7) 2 0.1 6
question, we make 56 cents the third $ 2.8 8 Extreme.
term, because it is of the same kind,
and has the same ratio to the required answer, or fourth term, as the first has to the second. From the nature of the question, since the answer will be more than 56 cents, or the third term, the second term must be larger than the first; we make the 36 the second term, and the 7 the first
, and then the product of the means divided by the given extreme, gives the required extreme. (Art. 243.)
242. If the extremes and one of the means are giren, how can the other mean be found ? — 243. When the means and one of the extremes are given, how can the other extreme be found ? — 244. What is simple proportion ? How many terms are given in questions in simple proportion ?
BY ANALYSIS. --- If 7lb. cost 56 cents, 1 pound will cost of 56 cents, or 8 cents. Then, if ilb. cost 8 cents, 361b. will cost 36 times 8 cents, or $ 2.88.
Ex. 2. If 76 barrels of flour cost $ 456, what will 12 barrels
Ans. $ 72. OPERATION.
We state this question by making
$ 456 the third term, because it is of 7 6:12:: 4 5 6
the same kind of the required term. 12
Then, since the answer must be less
than $ 456, because 12 barrels will 76) 54 72 ($ 72
cost less than 76 barrels, we make 12 5 3 2
barrels, the smaller of the two terms, 152
the second term, and 76 barrels the 152
first term, and proceed as before.
BY ANALYSIS. — If 76 barrels cost $ 456, 1 barrel will cost to of $456, or $6. Then, if i barrel cost $ 6, 12 barrels will cost 12 times $ 6, or $ 72.
Ex. 3. If 3 men can dig a well in 20 days, how long will it take 12 men ?
Ans. 5 days. Since the required answer is days, we make men. days. 20 days the third term. And as 12 men will 1 2 : 3 :: 20 dig the well in less time than 3 men, the 3 answer must be less than 20 days. There
fore we make the 3 the second term, and the 1 2) 60
12 the first, and proceed as in the other ex
5 days. amples.
BY ANALYSIS. — If 3 men can dig the well in 20 days, it will take one man 3 times 20 days, or 60 days. Again, If one man can dig the well in 60 days, 12 men can dig it in t of 60 days, or 5 days.
RULE. Write for the third term that number which is of the same kind as the required fourth term.
Of the other two numbers, write the larger for the second term, and the less for the first, when the answer should exceed the third term; but write the less for the second term, and the larger for the first, when the answer should be less than the third term.
Multiply the second and third terms together, and divide their product by the first.
245. What is meant by stating the question? Which of the terms given in the example do you make the third ?' Why? Which the second? Why? Which the first? Why? After the question is stated, how do you obtain the answer ?
NOTE 1. - When the first and second terms consist of different denomi. nations, they must be reduced to the same denomination ; and when the third term is a compound number, it must be reduced to the lowest denomi. nation mentioned in it. The answer will be the same denomination as the third term.
NOTE 2. To shorter the operations, factors common to the dividend and divisor may be cancelled.
NOTE 3. — It is often a convenient method, to divide the third term by the ratio of the first term to the second.
Ex. 4. If 16 bushels of wheat are worth $ 24, what are 96 hushels worth ?
Ans. $ 144. OPERATION BY CANCELLATION. We first state the question as dibu. bu.
rected in the rule, and then write 16 : 96::24
the second and third terms above 6
a horizontal line, with the sign of $2 4 x 96
multiplication between them, for a $144
dividend, and the first term below the line for a divisor, and cancel the common factors.
Again, we place the $ 24, which
is of the same kind of the required 6
answer, above a line for a dividend ; $ 24 x 96
and then say, Since 16 bushels are $ 1 44 worth $ 24, i bushel is worth to of 16
$ 24, and express the division by
placing the 16 below the line for å divisor. Now, since 96 bushels are worth 96 times as much as 1 bushel, we express the multiplication by placing the 96 above the line, and then cancel as before.
BY ANALYSIS AND CANCELLATION.
EXAMPLES FOR PRACTICE.
5. What cost 9 gallons of molasses, if 63 gallons cost $ 14.49 ?
Ans. $ 2.07. 6. What cost 97 acres of land, if 19 acres can be obtained for $ 337.25 ?
Ans. $ 1721.75. 7. If a man travel 319 miles in 11 days, how far will he travel in 47 days?
Ans. 1363 miles. 8. When $ 120 are paid for 15 barrels of mackerel, what will be the cost of 79 barrels ?
Ans. $ 632.
245. What is the rule for simple proportion? How should the pupil perform the questions? How do you state the question and arrange the terms for cancellation? What do you cancel? How do you arrange the terms for cancellation by analysis ?
9. If 9 horses eat a load of hay in 12 days, how many horses can eat the same quantity in 3 days ?
Ans. 36 horses. 10. When $ 5.88 are paid for 7 gallons of oil, what cost 27 gallons ?
Ans. $ 22.68. 11. When $ 10.80 are paid for 9lb. of tea, what cost 147lb ?
Ans. $ 176.40. 12. What cost 27 tons of coal, when 9 tons can be purchased for $ 85.95 ?
Ans. $ 257.85. 13. If 15 tons of lead cost $ 105, what cost 765 tons ?
Ans. $ 5355.00. 14. If 16hhd. of molasses cost $ 320, what cost 176hhd. ?
$ 3520.00. 15. If 15cwt. 3qr. 171b. of sugar cost $ 124.67, what cost 76cwt. 2qr. 191b. ?
Ans. $ 600.56+ 16. If the cars on the Boston and Maine Railroad go one mile in 2 minutes and 8 seconds, how long will they be in passing from Haverhill to Boston, the distance being 32 miles ?
Ans. lh. 8min. 16sec. 17. If a man can travel 3m. 7fur. 18rd. in one hour, how far can he travel in 9h. 45min. 19 sec. ? Ans. 38m. 2fur. 32rd.+.,
18. A fox is 96 rods before a greyhound, and while the fox is running 15 rods the greyhound runs 21 rods; how far must the dog run before he can catch the fox ?
Ans. 336 rods. 19. If 5 men can reap a field in 12 hours, how long would it take them if 4 men were added to their number?
Ans. 6 hours. 20. Ten men engage to build a house in 63 days, but 3 of their number being taken sick, how long will it take the rest to complete the house?
Ans. 90 days. 21. If a 4 cent loaf weighs 5oz. when flour is $ 5 per barrel, what should it weigh when flour is $ 7.50 per barrel ?
Ans. 34oz. 22. If 7 men can mow a field in 10 days when the days are 14 hours long, how long will it take the same men to mow the field when the days are 13 hours long ? Ans 101% days.
23. If 291b. of butter will purchase 40lb. of cheese, how many pounds of butter will buy 7915. of cheese?
Ans. 57 lb. 24. If of a yard cost of a dollar, what will 13 of a yard cost ?
25. If 47 yards of cloth cost $ 27, what will 194 yards Cost?
Ans. $ 11.50. yd. yd.
39 2 26. If for 411 yards of velvet there be received 114 yards of calico, how many yards of velvet will be sufficient to purchase 100 yards of calico?
Ans. 39 yards. 27. A certain piece of labor was to have been performed by 144 men in 36 days, but a number of them having been sent away, the work was performed in 48 days; required the number of men discharged.
Ans. 36 men. 28. James can mow a certain field in 6 days, John can mow it in 8 days; how long will it take John and James both to mow it ?
Ans. 34 days. 29. A. Atwood can hoe a certain field in 10 days, but with the assistance of his son Jerry he can hoe it in 7 days, and he and his son Jacob can hoe it in 6 days; how long would it take Jerry and Jacob to hoe it together?
Ans. 92 days. 30. Bought a horse for $ 75; for what must I sell him to gain 10 per cent. ?
1.00 : 1.10 :: $75 : $ 82.50, Ans. 31. Bought 40 yards of cloth at $ 5.00 per yard; for what must I sell the whole amount to gain 15 per cent. ?
Ans. $ 230.00. 32. My chaise cost $ 175, but, having been injured, I am willing to sell it at a loss of 30 per cent. ; what should I receive ?
Ans. $ 122.50. 33. Bought a cargo of flour on speculation, at $5.00 per barrel, and sold it at $ 6.00 per barrel; what did I gain per cent. ?
Ans. 20 per cent. 34. Bought a hogshead of molasses for $ 15.00, but, it not proving as good as I expected, I sell it for $ 12 ; what do I lose
Ans. 20 per cent. 35. Bought a hogshead of molasses for $ 27.50, at 25 cents per gallon ; how much did it contain ?
Ans. 110 gallons. 36. A certain farm was sold for $ 1728, it being $ 15.75 per acre; what was the quantity of land ? Ans. 109A. 2R. 344p.
37. A certain cistern has 3 cocks; the first will empty it in 2 hours, the second in 3 hours, and the third in 4 hours ; in what time would they all empty the cistern together?
Ans. 55 23 minutes.
per cent. ?