How

64. The distance from the earth to the sun is about 95000006 miles; the distance to the moon is about 240000 miles. much farther to the sun than to the moon?

65. Methuselah died at the age of 969 years, and Washington

at 67; what was the difference of their ages?

66. Mr. Hale, owing a debt of $3762, paid $2486; how much remained unpaid?

EXAMPLES IN ADDITION AND SUBTRACTION.

1. From the sum of 76 and 92 take 14.

Ans. 154.

2. From the sum of the three numbers, 876, 493, and 916, take the sum of 842 and 397.

Ans. 1046.

3. I owe 3 notes, whose sum is $600. One of these notes is for $150, another for $200; for what is the third one?

4. My real estate is valued at $4500 and my personal property at $2596. I owe to A $600, to B $1358, and to C $318; what am I worth? Ans. $4820.

5. Bought a barrel of flour for $9, four yards of cloth for $2, and 8 pounds of sugar for $1. In payment I gave a ten and a five dollar bill; what change shall the merchant return to me?

6. Mr. Fox, owning 3762 acres of land, gave 563 acres to his oldest son, and 672 acres to his youngest son; how many acres had he remaining?

7. The area of Maine is 35000 square miles; N. H., 8030; Vt., 8000; Mass., 7250; R. I., 1200; Ct., 4750. Which is the greater, Maine or the rest of N. E.? How much?

8. Gave my note for $3465. Paid $1300 at one time, and $575 at another; how much do I still owe? Ans. $1590.

3. Mr. T., opening an account at the Andover Bank, deposited $187 on Monday, $362 on Tuesday, $580 on Thursday, and $675 on Friday. On Tuesday he withdrew $67, on Wednesday $213, on Friday $350, and on Saturday $125; how much remained on deposit at the close of the week? Ans. $1049.

10. A traveler who was 875 miles from home, traveled toward home 144 miles on Monday, 127 miles on Tuesday, 156 miles on Wednesday, and 157 miles on Thursday; how far fron home was he on Friday morning?

11. From the discovery of America by Columbus in 1492, to the settlement of Jamestown in 1607, was 115 years, from the settlement of Jamestown to the Declaration of Independence in 1776, was 169 years, and from the Declaration of Independence to the present time (1862) is 86 years. Methuselah died at the age of 969 years; how much longer did he live than from the discovery of America to the year 1862?

12. Four men, A, B, C, and D, commencing business together, furnished money as follows: A, $2475; B, $3475; C, $2850; and D, $4500. At the end of a year they closed business, having lost $3225; h w much money had they to divide between them?

MULTIPLICATION.

54. MULTIPLICATION is a short method of adding equal numbers; that is, multiplication is a short method of finding the sum of the repetitions of a number.

Or, MULTIPLICATION is a short method of finding how many units there are in any number of times a given number.

The MULTIPLICAND is the number to be repeated.

The MULTIPLIER is the number which shows how many times the multiplicand is to be taken.

The PRODUCT is the sum of the repetitions, or the result of the multiplication.

The Multiplicand and Multiplier are called FACTORS.

Ex. 1. There are 7 days in 1 week; how many days in 4 weeks?

This example may be solved by addition; thus, 7+7+74-7 =28; or more briefly, by multiplication; thus, 4 times 7 are 28, Ans.

54. What is Multiplication? Another definition? What is the Multiplicand? Multiplier? Product? What are the Multiplicand and Multiplier called?

55. The pupil, before advancing further, should learn the

Ex. 2. How many are 8 times 3? 3 times 8? 6 times 4?

4 times 6? 7 times 7? 5 times 9?

3. How many are 9 times 7? 9 times 11? 8 times 6? 6 times 12? 12 times 6? 9 times 8?

4. If I deposit $10 a month in a savings bank, how many dollars shall I deposit in 4 months? In 7 months? In 5 months? In 12 months?

5. When wood is worth $6 a cord, what shall I pay for 3 cords? 5 cords? 8 cords? 11 cords?

6. In one year there are 12 months, how many months in 2 years? 4 years? 7 years? 12 years?

7. If I study 11 hours in a day, how many hours shall I study in 3 days? 5 days? 8 days? 11 days?

56. To multiply by a single figure.

8. In one bushel are 32 quarts; how many quarts in 6 bushels?

BY ADDITION.

32

32

32

32

32

BY MULTIPLICATION.

32

6

Product, 192

In 6 bushels there are, evidently, 6 times as many quarts as in 1 bushel, and the number of quarts in 6 bushels may be obtained by adding, as in the margin; or, more briefly, by multiplying; thus, 6 times 2 units are 12 units ten and 2 units; write the 2 units in units' place, and then say 6 times 3 tens are 18 tens, which, increased by the 1 ten previously obtained, make 19 tens= 1 hundred and 9 tens, and these, written in the place of hundreds and tens respectively, give the true product. Hence,

32 Sum, 192

RULE.

Write the multiplier under the multiplicand, and draw a line beneath; multiply the units of the multiplicand, set the units of the product under the multiplier, and add the tens, if any, to the product of the tens, and so proceed.

56. Which figure of the Multiplicand is multiplied first? Where are the units of the product written? What is done with the tens? Repeat the rule.

57. To multiply by two or more figures. 19. How many quarts in 46 bushels?

OPERATION.

Multiplicand, 32 Multiplier, 46 192 128

First multiply by 6, as though 6 were the only figure in the multiplier; then multiply by 4, and set the first figure of this product in the place of tens; for multiplying by the 4 tens is the same as multiplying by 40, and 40 times 2 units are 80 units = 8 tens; i. e. the product of units by tens is tens. Having multiplied by each figure in the multiplier, the sum of the partial products will be the true product.

Product, 1472

NOTE. So much of the product as is obtained by multiplying the whole multiplicand by one figure of the multiplier is called a partial product; thus, in the 19th example, 192 is the first partial product and 128 tens is the second.

58. Similar reasoning applies however many figures there may be in the multiplier. Hence,

RULE. 1. Set the multiplier under the multiplicand and draw a line beneath.

2. Beginning at the right hand of the multiplicand, multiply the multiplicand by each figure in the multiplier, setting the first figure of each partial product directly under the figure of the multiplier which produces it.

3. The sum of these partial products will be the true product. 59. PROOF. Multiply the multiplier by the multiplicand, and, if correct, the result will be like the first product.

NOTE. This proof rests on the principle, that the order of the factors is immaterial; thus, 3X44X3; 5×2×7 = 2X7X5,

57. Which figure of the multiplier is first employed? Where is the first figure of each partial product written? What is a partial product? 58. Rule for multiplying by two or more figures? 59. Proof? Principle?