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Second Model.
16.427

14

10000 ) 229.978

.0229978 Note. - It will be seen that by both solutions, we multiply by the numerator, 14, and divide by the denominator, 10000; i. e., we multiply by 14, and remove the point four places towards the left. The only difference between them is, that in the first we remove the point before we multiply, while in the second we remove it afterwards. Some of the above written work might have been avoided by multiplying by 14, and changing the position of the point at the time of writing the product, thus :

16.427

.0014

.0229978 In like manner we may get .07 of a number by multiplying by 7, and removing the point two places towards the left; we may get 5.2 * of a number by multiplying by 52, and removing the point one place towards the left ; &c. What is the value of each of the following ? 9. .03 of 2589

14. .006 of .006 10. .005 of 3.7479

15. 2.5 of 2.5 11. .28 of 437.96

16. .372 of 5.46 12. 4.2 of 67.93 +

17. .0004 of 200 13. .25 of .25

18. .006 of .006

132. To multiply by a Fraction. 1. What is the product of 865 multiplied by 4 ?

Suggestion. — The product of 865 multiplied by 4 equals 4 times 865, which is the same as of 865, and may be found by methods be. fore explained.

Note. — This follows from the definition of multiplication - A process by which we ascertain how much any given number will amount to, if taken as many times as there are units in some other given number. In multiplying by more than a unit, then, more than once the multiplicand is taken, while in multiplying by less than a unit, less than once the mul

* It will be more convenient, for purposes of explanation, to regard this as an improper fraction than as a mixed number.

+ The number of which a fractional part is taken may be regarded either as a mixed number or an improper fraction.

tiplicand (i. e., some fractional part of it) is taken ; the product being always the same part of the multiplicand that the multiplier is of unity.

When, therefore, the multiplier is greater than unity, the product will be greater than the multiplicand; and when the multiplier is less than unity, the product will be less than the multiplicand. In multiplying by }, { of once the multiplicand is taken ; in multiplying by }, { of once the multiplicand is taken ; &c.

This conclusion is in exact accordance with the principle, that the larger the multiplier the larger the product, and the sinaller the multiplier the smaller the product, so long as the multiplicand remains tho same; or, to speak more definitely, that the product of a number multiplied by 1 of another, is equal to 1 of its product multiplied by the whole of the other ; that the product of a number multiplied by of another, is equal to of its product multiplied by the whole of the other; &c. Thus, 12 multiplied by 6 = 72.

12 multiplied by t of 6, or 3, = 36, which is I of 72.
12 multiplied by $of 6, or 2, = 24, which is } of 72.

12 multiplied by of 6, or 4, 48, which is of 72; &c. So, since 12 multiplied by 1 : 12,

12 multiplied by 1 of 1, or 1, must equal 3 of 12, or 6. 12 multiplied by } of 1, or }, must equal } of 12, or 4.

12 multiplied by of 1, or , must equal of 12, or 8; &c. Note. — By the foregoing explanations, it appears that the change of denomination mentioned in the note under 74, (h.) is caused by the fact that to multiply by a fraction requires a division as well as a multiplication.

What is the product
2. Of times 8649 ? 5. Of is times 6964 ?
3. Of 4 times 6743 ?

6. Of .7 times 6397 ?
4. Of _f times 16872 ? 7. Of .06 times 59.37 ?
8. What is the product of 3 times 2948 ?
Suggestion. 33 times 2948 3 times 2948 + f of 2948.

WRITTEN WORK.

First Form. a= 2948

9 ) 2948 = à 33

327f = of a 3 X a=b= 8844

5 f of a=c= 16377

1637} = ģ of a b+c=35 ta= 10481} = Ans.

[blocks in formation]

*b+c+d=3ž ta= 104813= Ans. Note. — The second of the above forms is preferable to the first, inasmuch as it presents to the eye, in a compact and convenient form, all the numbers which it is necessary to use, and avoids all side work. The references by means of letters may be omitted as soon as the processes and explanations are familiar.

What is the product —

9. Of 81 times 6794 ? 12. Of 3 times 85476 ? 10. Of 26ş times 2579 ? 13. Of 254 times 2764? 11. Of 437 times 45687 ? 14. Of 698 times 29679 ?

133. Practical Problems.

1. How much will } of an acre of land cost at $478.36

per acre ?

Solution. — If i acre costs $478.36, of an acre will cost s of $478.36, which, found in the usual way, is $418.561. First Form.

Second Form. 8 ) $478.36

a= $478.36 = cost of 1 acre.

f of a = b $ 59.791 = cost of $ of an A. $59.793

7 x b= $418.563 = cost of } of an A. 7

$418.561 = Ans. NOTE. — The second form is better than the first, when a full statement and explanation of the successive steps are required; but in other cases the first is preferable.

2. How much will .9 of an acre of land cost at $394.26

per acre ?

* Since 8 = + 8,

Solution. - If i acre costs $394.26, 9 of an acre will cost .9 of $394.26, which, found as before explained, is $354.834. The following exhibits the written work. Full Form.

Abbreviated Form. a = $394.26 = cost of 1 acre.

$394.26 .1 of a=b=$ 39.426 = cost of .1 of an A. 9 Xb= $354.834 = cost of .9 of an A.

$354.834 = Ans. 3. How much is of a vessel worth, if the whole vessel is worth $29951.88 ?

4. I bought is of an acre of land at $397.36 per acre. How much ought I to pay for it?

5. If a bushel of tomatoes weighs 48.75 lb., what will of a bushel weigh?

6. What will .4 of a ton of iron cost at $47.93 per ton ? 7. What will .8 of a bag of coffee cost at $27.94 per bag ?

8. Mr. Foster bought 4377 bushels of corn, and sold .37 of it to Mr. Gardiner, .43 of it to Mr. Calder, and the rest to Mr. Godfrey. How many bushels did he sell to each ?

9. What will 9 acres of land cost at $126 per acre ?

10. What will 32% acres of land cost at $139.50 per acre ?

11. The ship Surprise is valued at $25427. Joseph Ward owns 4 of her, and Samuel Lowe owns the remainder. What is the value of Mr. Ward's share ? Of Mr. Lowe's ?

12. The brig Maria sailed from Philadelphia to Boston, with 207 T. 13 cwt. 2 qr. 19 lb. of coal, of which was landed at one wharf, and the remainder at another. What was the weight of that landed at each wharf?

13. If John walks as fast as George, how far will John walk while George is walking 97 m. 6 fur. 37 rd. 2 yd. ?

14. A company of 9 California gold diggers found in 1 month 37 lb. 4 oz. 16 dwt. 17 gr. of gold, which they divided equally. What was the share of each ?

15. An English ship, valued at £8743, carries a cargo which is worth { as much as the ship. How many £, s., and d. is the cargo worth ? In a storm the sailors were obliged to throw of the cargo overboard. What was the value of what they threw overboard ?

16. A merchant bought 17 T. 14 cwt. 2 qr. 23 lb. 10 oz. of sugar. He sold of it to one man, of the remainder to another, and what there was left to another. What was the weight of that which he sold to the first man ? To the second ? To the third ?

Suggestion. - After selling of the sugar, he must have had of it left, and of this, or what he sold to the second man, must be of or of the original quantity; and the remaining of , or of the original quantity, must have been sold to the third man.

Or, the share of the third man may be determined thus: Since the first man took of the sugar, and the second Ý, both took of it, which would leave for the third man.

17. A father, dying, left to his son an estate valued at $87646.44. The first year after the son came into possession of the property, he spent of it in dissipation, the second year he spent of the remainder, and at the end of the third year he was penniless. How much did he spend in the first year? How much in the second year? How much in the third ?

Suggestion. — Compare this example with the last.

18. If Mr. Smith uses 9.678 tons of coal per year, and Mr. Parkhurst uses .7 as much, how many tons does Mr. Parkhurst use?

Note. — The term per cent is often used in arithmetic, and in business transactions, instead of one-hundredths. Thus, 6 per cent has the same meaning as 6 one-hundredths. 8 per cent=.08 = 18.

5 per cent of 340 acres =.05 of 340 acres.

6 per cent of $86.38 = .06 of $86.38. Every number is 100 per cent of itself.

19. A teamster carted 7486 bushels of potatoes to a railroad depot, receiving in payment 9 per cent of them. How many bushels did he receive ?

20. A man who owned 378.37 acres of land gave 8 per cent of it to his son. How many acres did he give to his son?

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