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Pounds of
Butter.

6. Give the amounts of the productions of the United States and Territories for the year 1850, as expressed in the following columns:

Alabama...... 4,008,811

Arkansas...... 1,854,239

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150

.....

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SUBTRACTION.

69. Subtraction is the process of determining the difference, between two numbers of the same unit value.

70. The Minuend is the number to be subtracted from. 71. The Subtrahend is the number to be subtracted.

72. The Difference or Remainder is the result obtained by the process of subtraction.

73. When the given numbers contain more than one figure each, the method of subtraction depends upon the following principles:

I. If the units of each order in the subtrahend be taken separately from the units of like order in the minuend, the sum of the differences must be equal to the entire difference of the given numbers. (Ax. 10.)

II. If both minuend and subtrahend be equally increased, the remainder will not be changed.

1. From 928 take 275.

OPERATION.

Minuend,

Subtrahend,

Remainder,

928
275

653

ANALYSIS. We first subtract 5 units from 8 units, and obtain 3 units for a partial remainder. As we cannot take 7 tens from 2 tens, we add 10 tens to the 2 tens, making 12 tens; then 7 tens from 12 tens leave 5 tens, the second partial remainder. Now, since we added 10 tens, or 1 hundred, to the minuend, we will add 1 hundred to the subtrahend, and the true remainder will not be changed (II); thus, 1 hundred added to 2 hundreds makes 3 hundreds, and this sum subtracted from 9 hundreds leaves 6 hundreds; and we have for the total remainder, 653.

NOTE. The process of adding 10 to the minuend is sometimes called borrowing 10, and that of adding 1 to the next figure of the subtrahend, carrying 1.

74. From these principles and illustrations we deduce the following

RULE. I. Write the less number under the greater, placing units of the same order under each other.

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.

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5. From 180037561 take 5703746.

6. From 2460371219 take 98720342.

7. 89037426175 — 2435036749: 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 did 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 in quantity 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? Ans. $35,260.

TWO OR MORE SUBTRAHENDS.

76. Two or more numbers may be taken from another at a single operation, as shown by the following example:

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?

OPERATION.

Minuend,

1278

236 Subtrahends, 362

Remainder,

387
293

ANALYSIS. Since the remainder sought, added to the subtrahends, must be equal to the minuend, we add the columns of the subtrahends, and supply such figures in the remainder as, combined with these sums, will produce the minuend. Thus, 7 and 2 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.

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