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When he came back, he found he had exactly $500. How much had he in the purse when he left home? Ans. $12.

7. Two men set out on a journey, travelling in the same direction, and at the end of a week one of them had travelled 200 miles, and the other 240 miles. How far were they then apart?

8. A merchant sold a ship for $8000, which was $1500 more than he paid for it. How much did he pay for it?

9. The value of the gold coined in the mint of the United States in 1831, was $714,270; in 1832, $798,435; in 1833, $978,550 ; in 1834, $3,954,270. How much more was coined in 1834 than in the other three years taken together ?

Ans. $1,463,015. 10. The following is a statement of the revenue of the government of the United States from the year 1837 to 1842, inclusive. The revenue is comprised in two classes, namely, receipts from customs, and from the sales of lands and miscella

The latter column is left blank, to be filled by the pupil by horizontal subtraction. If the work be correctly performed, the total amount of the receipts from customs and from the sales of lands, &c., will agree with that of the aggregate of receipts.

neous sources.

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11. In 1790, the first census under the constitution of the United States was taken by act of Congress, and it has been followed by similar enumerations every ten years. The following table shows the total population at these several periods. The column of free colored persons is left blank, to be filled by the pupil. The necessary addition and subtraction should be performed at one operation, by aid of the complements of the several numbers. Prove by horizontal addition of the totals.

Free Colored.

Slaves.

Total.

Years.

Whites. 1790 3,172,464

697,897 3,929,827 1800 4,304,489

893,041 5,305,925 1810 5,862,004

1,191,364 7,239,814 1820 7,861,937

1,538,038 9,638,181 1830 10,537,378

2,009,043 12,866.021 1840 14,195,695

2,487,455 17,069,458 1850 19,553,068

3,204,313 23,191,874 Total

12. The following is a statement of the commerce of the United States, from the year 1831 to 1842, inclusive. Complete the table by finding the difference in value between the exports and imports of each year, and balance the statement by finding the total difference, which may be considered the cost of the freight, and the amount of the profits of the commerce. The computation will be correct, if the balance of the second and third columns is the same as that of the fourth and fifth. Why?

[All the subtractions in the following table should be performed horizontally, as a suitable exercise to prepare the pupil for balancing books and accounts.]

Value of Exports. Value of Imports. Excess of Exports. Excess of Imports. 1831 $ 81,310,583 $103,191,124 1832 87,176,943 101,029,266 1833 90,140,433 108,118,311 1834 104,336,973 126,521,332 1835 121,693,577 149,895,742 1836 128,663,040 189,980,035 1837 117,419,376 140,989,217 1838 108,486,616 113,717,404 1839 121,028,416 162,092,132 1840 132,085,946 107,141,519 1841 121,851,803 127,946,177 1842 104,691,534 100,162,087 Total Bal. Aggr.

13. The following is a statement of the population of some of the largest cities and towns of the United States, by the census of 1840 and that of 1850. The column of increase in

Years.

ten years

Cities.

1840.

1850.

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is left blank, to be filled by the student by horizontal subtraction. When that is done, add the three columns, and if the difference between the sums of the first two columns agrees with the sum of the third, the work is correct.

Increase in

10 years. Portland, Me..

15,218 20,815 Boston, Mass..

93,383 136,881 Providence, R. I. 23,171 41,513 New Haven, Conn. 12,960 20,345 New York, N. Y. 312,710 515,547 Brooklyn, N. Y. . 36,233 96,838 Albany, N. Y.

33,721 50,763 Buffalo, N. Y.

18,213 42,261 Rochester, N. Y. 20,191 36,403 Troy, N. Y.

19,334 28,785 Newark, N. J.

17,290 38,894 Philadelphia, Pa. 220,423 340,045 Pittsburgh, Pa.

21,115 46,601 Baltimore, Md. 102,313 169,054 Washington, D. C. 23,364 40,001 Richmond, Va.

20,153 27,570 Charleston, S. C. 29,261 42,985 Savannah, Ga.

11,214 15,312 Mobile, Ala.

12,672 20,515 Louisville, Ky.

21,210 43,194 Cincinnati, Ohio . 46,338 115,436 St. Louis, Mo.

16,469 77,860 New Orleans, La. 102,193 116,375 Milwaukee, Wis.. 1,700

29,061 Cambridge, Mass. 8,409 15,215 Roxbury, Mass.

9,089 18,364 Worcester, Mass. 7,497 17,049 Poughkeepsie, N. Y. 10,006 13,944 Syracuse, N. Y.. 6,500

22,271 Newburg, N. Y.

6,000

11,415 Reading, Pa..

8,410 15,743 Wilmington, Del. 8,367 13,979 Cleveland, Ohio. 6,071 17,034 Columbus, Ohio

6,048 17,882 Detroit, Mich.

9,102 21,019

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SECTION III. – Multiplication. [For an explanation of the signs and terms used in Multiplication, see p. 56, 2; and 57, 7.]

[It is usual, in treatises on arithmetic, to present the pupil with a multiplication-table, and require him to commit it to memory. Fortunately this drudgery is wholly unnecessary where Oral Arithmetic has been properly attended to. Should the teacher, however, still think it requisite, the table should be a mere skeleton, as below, to be filled up by the pupil from his own mind. In mathematics, nothing should rest on authority. Neither book nor teacher should furnish ought but definitions, graduated exercises, and suggestive questions.]

1 2 3 4 5 6 7 8 9 10 11 12

3

4

5

6

9

10

11

12

Exercises for the Slate and Black-board. 1. Name the product of each of the following pairs of factors, without naming the factors, to be repeated as a daily exercise till it can be done correctly, as rapidly as the words can be spoken.

4 7
5 3

5 9 2 9 8 5 1 9
8 8 4 3 2 4 7

5 4 6 7 2 3
8 7 2 7 2 2

4 6 3 9 4 4 3 3

2
7

6 8 3 6 9 6 3 4 2 4 8
4 8 5 7 9 3 4 2 9 8 6

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2. Multiply 42579638 by 2, 3, 4, 5, 6, 7, 8, and 9, severally, and prove by addition.

Exemplification for the Black-board. Where the multiplier consists of one figure only.

42579638 Multiplicand, or 1st factor.

6 Multiplier, or 2d factor.

255477828 Product.

85159276 1st factor taken 2 times. Proof by addition, 127738914 1st factor taken 3 times.

255477828 1st factor taken 6 times.

how many

Solution. Suggestive Questions. How many are 6 times 8 ? 40 of first rank make how many of 2d rank? How many are 6 times 3 ? How many are 18+4 from the first rank? 20 of second rank how

many

of third ? How many are 6 times 6+2 from second rank ? 30 of third rank: of fourth ? How many are 6 times 9+3 from third rank? and so on till all the figures are taken 6 times; that is, multiplied by 6.

[The student can hardly be cautioned too frequently to avoid unnecessary words. All that are requisite in the above example are (if any are necessary at all) forty-eight, twenty-two, thirty-eight, fifty-seven, forty-seven, thirty-four, fifteen, twentyfive. The student should also be engaged in writing one figure while multiplying the adjoining one; for multiplication, with one figure as factor, should proceed as fast as the figures in the product can be written.

Proof by Addition. How was the first line of proof found by addition ? Ans. By adding the first factor to itself. How was the second found? The third ? By these three lines the product of the multiplicand by any significant figure could be found. How could four times the first factor be found ? (3+1.) How could five times be found ? (2+3.) How seven times ? Eight times? Nine times ?

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