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14. Divide 7360416 by
82. When the divisor is 10, 100, 1000, etc.
1. Divide 374 acres of land equally among 10 men. How many acres will each have?
(2 × 6 × 8).
(2 × 5 × 7).
796950 by 175, (7 × 5 × 5).
SOLUTION. - As has been shown, to remove a figure one place toward the left by annexing a cipher increases its value tenfold, or multiplies it by 10 (69). On the contrary, by cutting off or taking away the right hand figure of a number, each of the other figures is removed one place toward the right, and, consequently, the value of each is diminished tenfold, or divided by 10 (33).
Quotient. 37... 4 Rem. or, 37 acres.
83. For similar reasons, if we cut off two figures, we divide by 100, if three, we divide by 1000, and so on.
RULE. - From the right hand of the dividend cut off as many figures as there are ciphers in the divisor. Under the figures so cut off, place the divisor, and the whole will form the quotient.
2. Divide 47600 by 10. By 100.
3. Divide 1306321 by 1000.
4. Divide 9760347 by 10000.
5. Divide 2037160310 by 100000.
84. When there are ciphers on the right hand
of the divisor.
1. Divide 437661 by 800.
547... 61 Rem.
SOLUTION. - In this example we resolve 800 into the factors 8 and 100, and divide first by 100, by cutting off two right hand figures of the dividend (82), and we have a quotient of 4376, and a remainder of 61. We next divide by 8, and obtain 547 for a quotient; and the entire quotient is 5478.
2. Divide 34716 by 900.
5 2d Rem.
5 x 100+16=516 True Rem. 38518. Ans.
SOLUTION. Dividing as in the last example, we have a quotient of 38, and two remainders, 16 and 5; but the 5 being in hundreds' place represents 500, to which we add 16, and we have 516 for the true remainder.
85. When there is a remainder after dividing by the significant figures, it must be prefixed to the figures cut off from the dividend to give the true remainder; if there is no other remainder, the figures cut off from the dividend will be the true remainder.
3. Divide 34716 by 900.
4. Divide 1047634 by 2400.
5. Divide 47321046 by 45000.
7. Divide 976031425 by 92000.
8. Divide 80013176321 by 700000.
9. Divide 19070367428 by 4160000. 4584. 10. Divide 379025644319 by 554000000. 11. Divide 897654321234 by 940000000. 12. Divide 394298765984 by 898000000. 13. Divide 647281909085 by 102030400.
14. The circumference of the earth at the equator is 24898 miles. How many hours would a train of cars require to travel that distance, going at the rate of 50 miles an hour? Ans. 49748.
15. The sum of $350000 is paid to an army of 14000 What does each man receive? Ans. $25.
16. If 800 shares of railroad stock are valued at $840999, what is the value of each share?
GENERAL PRINCIPLES OF DIVISION.
86. The quotient in Division depends upon the relative values of the dividend and divisor. Hence any change in the value of either dividend or divisor must produce a change in the value of the quotient. But some changes may be produced upon both dividend and divisor, at the same time, that will not affect the quotient.
The laws which govern these changes are called General Principles of Division. They are as follows: 549 = 6.
I. Multiplying the dividend by 3, we have 54 x 3÷9-162÷9=18,
and 18 the quotient, 6, multiplied by 3. Hence,
Multiplying the dividend by any number, multiplies the quotient by the same number.
Using the same example, 54÷9=6.
II. Dividing the dividend by 3, we have
and 2=the quotient, 6, divided by 3.
Dividing the dividend by any number, divides the quotient by the same number.
III. Multiplying the divisor by 3, we have
and 2=the quotient, 6, divided by 3. Hence,
Multiplying the divisor by any number, divides the quotient by the same number.
IV. Dividing the divisor by 3, we have
and 18 the quotient, 6, multiplied by 3.
Dividing the divisor by any number, multiplies the quotient by the same number.
V. Multiplying both dividend and divisor by 3, we have 54 × 3÷9×3=162÷27=6. Hence,
Multiplying both dividend and divisor by the same number, does not alter the value of the quotient.
VI. Dividing both dividend and divisor by 3, we have 54+3=18÷3-6. Hence,
Dividing both dividend and divisor by the same number, does not alter the value of the quotient.
These examples illustrate all the changes we ever have occasion to make upon the dividend and divisor. The principles upon which these changes are based may be stated as follows:
PRINCIPLES.-I. Multiplying the dividend multiplies the quotient; dividing the dividend divides the quotient. (I. and II.)
II. Multiplying the divisor divides the quotient; dividing the divisor multiplies the quotient. (III. and IV.)
III. Multiplying or dividing both dividend and divisor by the same number, does not alter the quotient. (V. and VI.)
These three principles may be embraced in one law. GENERAL LAW. A change in the DIVIDEND produces a LIKE change in the quotient; but a change in the DIVISOR produces an OPPOSITE change in the quotient.
If a number is multiplied and the product divided by the same number, the quotient will be equal to the number multiplied. Thus, 15 x 4 = 60, and 60 ÷ 4 = 15.
EXAMPLES IN THE PRECEDING RULES.
1. George Washington was born in 1732, and lived 67 years. In what year did he die? Ans. In 1799.
2. How many dollars a day must a man spend, to use an income of $1095 a year? Ans. $3. 3. If I give $141 for a piece of cloth containing 47 yards, for what must I sell it in order to gain $1 a yard? Ans. $188. 4. A speculator who owned 500 acres, 17 acres, 98 acres, and 121 acres of land, sold 325 acres. How many acres had he left? Ans. 411 acres.
5. A dealer sold a cargo of salt for $2300, and gained $625. What did the cargo cost him? Ans. $1675.
6. If a man earns $60 a month, and spends $45 in the same time, how long will it take him to save $900 from his earnings?
7. If 9 persons use a barrel of flour in 87 days, how many days will a barrel last 1 person at the same rate? Ans. 783 days.
second is 8 the second. Ans. 324.
8. The first of three numbers is 4, the times the first, and the third is 9 times What is their sum?
9. If 2, 2, and 7 are three factors of 364, what is the other factor? Ans. 13. 10. A man has 3 farms; the first contains 78 acres, the second 104 acres, and the third as many acres as both the others. How many acres are there in the 3 farms?
11. If the expenses of a boy at school are $90 for board, $30 for clothes, $12 for tuition, $5 for books, and $7 for pocket money, what would be the expenses of 27 boys at the same rate? Ans. $3888.
12. Two men travel in opposite directions, one at the rate of 35 miles a day, and the other at the rate of 40