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II. Begin at the right hand, and take each figure of the subtrahend from the figure above it, and write the result underneath.
III. If any figure in the subtrahend be greater than the corresponding figure above it, add 10 to that upper figure before subtracting, and then add 1 to the next left hand figure of the subtrahend.
75. Proof. It is evident that the subtrahend and remainder must together contain as many units as the minuend; hence, to prove subtraction, we have three methods :
1st. Add the remainder to the subtrahend; the sum will be equal to the minuend. Or,
2d. Subtract the remainder from the minuend; the difference will be equal to the subtrahend. Or,
3d. Find the excess of 9's in the remainder and subtrahend together, and it will be equal to the excess of 9's in the minuend.
5. From 180037561 take 5703746.
how many? 8. 10000033421 - 999044110 :
how many? 9. A certain city contains 146758 inhabitants, which is 3976 more than it contained last year; how many did it contain last year?
Ans. 142,782. 10. The first newspaper published in America was issued at Boston in 1704; how long was that before the death of Benjamin Franklin, which occurred in 1790 ?
11. A merchant sold a quantity of goods for $42017, which was $1675 more than they cost him; how much aid they cost him ?
Ans. $40,342. 12. In 1858 the exports of the United States amounted to $324644421, and the imports to $282613150; how much did the exports exceed the imports ?
Ans. $42,031,271. 13. In 1858 the gold coinage of the United States amounted to $52889800, and the silver to $8233287; how much did the gold exceed the silver coinage ?
14. The South in 1850 produced 978311690 pounds of cotton, valued at $101834616, and 237133000 pounds of sugar valued at $16599310; how much did the cotton exceed the sugar
quan tity and in value ? Ans. 741,178,690 pounds; $85,235,306.
15. The area of the Chinese Empire is 5110000 square miles, and that of the United States 2988892 square miles; the estimated population of the former is 340000000, and that of the latter in 1850 was 23363714. What is the difference in area and in population ?
16. The population of London in 1850 was 2362000, and that of New York city 515547; how many more inhabitants had London than New York ?
Ans. 1,846,453. 17. The total length of railroads in operation in the United States, January 1, 1859, was 27857 miles, and the total length of the canals 5131 miles; how many miles more of railroad than of canal ?
Ans. 22,726. 18. The entire deposit of domestic gold at the United States Mint and its branches, to June, 1859, was $470341478, of which $451310840 was from California; how much was received from other sources ?
Ans. $19,030,638. 19. During the year ending September 30, 1858, the number of letters exchanged between the United States and Great Britain were 1765015 received, and 1603609 sent; between the United States and France, 624795 received, and 639906 sent. How many letters did the exchange with Great Britain exceed those with France ?
Ans. 2,103,923. 20. The Southern States in 1850 had a population of 6696061, the Middle States 6624988, and the Eastern States 2728116; how many more inhabitants had the Middle and Eastern States than the Southern States ?
21. Having $20000, I wish to know how much more I must accumulate to be able to purchase a piece of property worth $23470, and have $5400 left?
Ans. $8,870. 22. A has $3540°more than B, and $1200 less than C, who has $20600; D has as much as A and B together. How much has D?
TWO OR MORE SUBTRAHENDS.
76. Two or more numbers may be taken from another at a
24., single operation, as shown by the following example:
33411 1. A man having 1278 barrels of flour, sold 236 barrels to' A," }, 362 to B, and 387 to C; how many had he left ?
/ 46 ANALYSIS. Since the remainder sought, id Minuend, 1278 added to the subtrahends, must be equal to
the minuend, we add the columns of the 236
subtrahends, and supply such figures in the Subtrabends, 362
remainder as, combined with these sums, 387
will produce the minuend. Thus, 7 and 2 Remainder, 293
are 9, and 6 are 15, and 3 (supplied in the
remainder sought) are 18; then, carrying the tens' figure of the 18, 1 and 8 are 9, and 6 are 15, and 3 are 18, and 9 (supplied in the remainder) are 27; lastly, 2 to carry to 3 are 5, and 3 are 8, and 2 are 10, and 2 (supplied in the remainder) are 12; and the whole remainder is 293. Hence, the following
RULE. I. Having written the several subtrahends under the minuend, add the first column of the subtrahends, and supply such a figure in the remainder sought, as, added to this partial sum, will give an amount having for its unit figure the figure above in the minuend.
II. Carry the tens' figure of this amount to the next column of the subtrahends, and proceed as before till the entire remainder is obtained.
EXAMPLES FOR PRACTICE. (1.)
(4.) From 47962
127368 903486 2503734 21435 56304 430164
89763 Take 15672 4782 132875
9156 67321 237564 Rem. 10399 57126 273126 2082200
5. From 4568 take 1320 + 275 + 320. 6. Subtract 1200 + 750 + 96 from 4756 +575 + 110 + 84.
7. A man bought four city lots, for which he paid $15760. For the first he paid $2175, for the second $3794, and for the third $1587; how much did he pay for the fourth ? Ans. $5,201.
8. John Wise owns property to the amount of $75860, of which he has $45640 invested in real estate, $25175 in personal property, and the remainder he has in bank; how much has he in bank ?
9. Lake liuron contains 20000 square miles; by how much does it exceed the area of Lake Erie and Lake Ontario, the former containing 11000 square miles, and the latter 7000 ?
Ans. 2000 square miles. 10. In the year 1852, there arrived in the United States 398170 immigrants, of whom 157548 were born in Ireland, and 143429 were born in Germany; how many were born in other countries ?
Ans. 97,493. 11. The entire amount of coinage in the United States for the year ending June, 1858, was $61357088, of which $52889800 was of gold, $234000 of copper, and the remainder of silver; how much was of silver ?
12 A speculator gained $5760, and afterward lost $2746; at another time he gained $3575, and then lost $4632. How much did his gains exceed his losses ?
Ans. $1,957. 13. The Eastern States have an area of 65038 square miles, the Middle States 114624 square miles, and the Southern States 613166 square miles; how many more square miles have the Southern than the Middle and Eastern States?
14. The entire revenue of the United States Post Office Department for the year ending Sept. 30, 1858, was $8186793, of which sum $5700314 was received for stamps and stamped letters, and $904299 for ietter-postage in money; how much was received from all other sources ?
Ans. $1,582,180. 15. The total expenditures of the Department for the same year were $12722470, of which sum $7821556 was paid for the transportation of inland mails, $124497 for the transportation of foreign mails and $2355016 as compensation to postmasters; how much was expended for all other purposes?
MULTIPLICATION. 77. Multiplication is the process of taking one of two given numbers as many times as there are units in the other
78. The Multiplicand is the number to be taken.
79. The Multiplier is the number which shows how many times the multiplicand is to be taken.
80. The Product is the result obtained by the process of multiplication.
81. The Factors are the multiplicand and multiplier.
Notes. — 1. Factors are producers, and the multiplicand and multiplier are .cailed factors because they produce the product.
2. Multiplication is a short method of performing addition when the numbers to be added are equal.
82. The method of multiplying when either factor contains more than one figure, depends upon the following principles :
It is evident that 5 units taken 3 times is the same as 3 units taken 5 times; and the same is true of any two factors Hence,
I. The product of any two factors is the same, whichever is used as the multiplier. If units be multiplied by units, the product will be units; if tens be multiplied by units, or units by tens, the product will be tens; and so on. That is,
II. If either factor be units of the first order, the product will be units of the same order as the other factor.
III. If the units of each order in the multiplicand be taken separately as many times as there are units in the multiplier, the sum of the products must be equal to the entire product of the given numbers, (Ax. 10). 1. Multiply 346 by 8.
ANALYSIS. In this example it is reMultiplicand, 316 quired to take 346 eight times. If we take Multiplier,
8 the units of each order 8 times, we shall
take the entire number 8 times, (III). Product, 2768
Therefore, commencing at the right hand, we say; 8 times 6 units are 48 units, or 4 tens and 8 units; writing the 8 units in the product in units' place, we reserve the 4 tens to add to the next product; 8 times 4 tens are 32 tens, and the 4 tens