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9. What is the value of 9 acres of land, at 125 dollars an acre?
10. There are 1,760 yards in one mile; how many yards are there in 10 miles?
11. In one year there are 365 days; how many days are there in 11 years?
12. There are 2,240 pounds in one ton; how many pounds are there in 12 tons?
Illustration of the process.
What will be the product of 7,654 multiplied by 543? 7,654 multiplicand. 543 multiplier.
3 times the multiplicand. 40 times the multiplicand. = 500 times the multiplicand.
22962= 30616 38270
Product, 4,156,122 543 times the multiplicand. When the multiplier consists of several figures, we first multiply the multiplicand by the units of the multiplier, as directed in the preceding rule. We next multiply the multiplicand by the tens of the multiplier, and write the first figure of the product in the place of tens, because units multiplied by tens produce tens. Then we multiply the multiplicand by the hundreds of the multiplier, and write the first figure of the product in the place of hundreds, because units multiplied by hundreds produce hundreds. Finally, we add the several products, and the total product thus obtained is 4,156,122.
From the preceding illustration we deduce the following rule for multiplication, when the multiplier consists of several figures:
RULE. Write down the multiplicand; then write the multiplier under the multiplicand, placing units under units, tens under tens, and hundreds under hundreds.
Multiply the multiplicand by each significant figure
of the multiplier, in succession, beginning with units, and write the first figure of each product directly under the figure by which you are multiplying.
Find the sum of the several products; their sum will be the total product required.
1. What will be the product of 325 multiplied by 23? 2. What will be the product of 436 multiplied by 45? 3. What will be the product of 578 multiplied by 67? 4. What will be the product of 908 multiplied by 98? 5. What will be the product of 315 multiplied by 234? 6. What will be the product of 435 multiplied by 506? 7. What will be the product of 508 multiplied by 805? 8. What will be the product of 645 multiplied by 744? 9. What will be the product of 719 multiplied by 912? 10. What will be the product of 915 multiplied by 814? 11. If a ship sail uniformly 175 miles each day, what number of miles will she sail in 25 days?
12. A man purchased a wood-lot, containing 45 acres, at 35 dollars an acre; what did it cost him?
13. If an acre of land produce 32 bushels of wheat, how many bushels will 64 acres produce?
14. If a bale of sheeting contain 36 pieces, and each piece measures 32 yards, what number of yards does the bale contain?
15. If an acre of land produce 225 bushels of potatoes, what number of bushels will 25 acres produce?
16. If an acre of land is worth 225 dollars, what is the value of a farm containing 175 acres?
17. A merchant imported 350 boxes of oranges. What number of oranges did he import, supposing each box to contain 180 oranges ?
18. In an orchard there are 120 apple-trees. Supposing each tree to bear 10 bushels of apples, and each bushel to contain 240 apples, and each apple to contain 12 seeds, what number of seeds are there in the whole number of apples?
DIVISION is the method of finding how many times, or what part of a time, one of two given numbers is contained in the other; or, it is the process of finding any required part of any given number. Division is also a short method of performing several subtractions of the same number. One of the two given numbers is called the dividend, and is the number to be divided.
The other is called the divisor, and is the number to divide by, and indicates what part of the dividend is required.
The number found by the operation is called the quotient, and it shows the number of times, or part of a time, that the divisor is contained in the dividend. It expresses also the number of units, or part of a unit, in the required part of the dividend.
The sign of division is a short line between two points, thus,, and is read, divided by. When placed between two numbers, it shows that the number before it is to be divided by the number after it; thus, 20÷5—4, which is read, 20 divided by 5 equals 4.
Division is also denoted by writing the divisor under the dividend, with a short line between them; thus, 12,, which may be read, 12 divided by 6, 3 divided by 4. Illustration First. - Suppose we wish to find how many yards of cloth can be purchased with 36 cents, at 12 cents a yard. It is plain that as many times as 12 cents are contained in 36 cents, so many yards can be purchased. In the first method, we write 36, the number of cents, for the dividend, and then write 12, the
1st method. The divisor 12) 36 the dividend.
3 the quotient.
number of cents in the price of a yard, at the left of the dividend, for the divisor; we find that 12 is contained in 36, 3 times; hence 3 is the number of yards that can be
In the second method, we write down 36 cents, and subtract 12 cents from 36 cents, and 24 cents remain; we next subtract 12 cents from 24 cents, and 12 cents remain; finally, we subtract 12 cents from 12 cents, and nothing remains. We here see that 12 cents has been subtracted from 36 cents 3 times; hence 3 is the number of yards that can be purchased with 36 cents, at 12 cents a yard. We have thus found that the same number of yards can be purchased with 36 cents, by each of the methods.
2d method. 36 cents. 12 cents.
12 cents. 12 cents.
Illustration Second. - How many times is 9 contained in 675? We write down the dividend and draw a curve line on each side of it, and then write the divisor at the left of the dividend.
We next find the number of times that the divisor, 9, is contained in 67 tens, which is 7 (tens) times; we write the 7 (tens) at the right of the dividend for the first figure in the quotient, and multiply the divisor by the 7 (tens); the product is 63 tens, which we write under the 67 tens: we then subtract 63 tens from 67 tens; the remainder is 4 tens. We then place the 5 units of the dividend at the right of the remainder, and we have the number 45. The divisor, 9, is contained 5 times in 45; we write the 5 units at the right of the 7 tens in the quotient, and multiply the divisor by the 5 units; the product is 45, which we write under the 45. There being no remainder, the operation is completed, and we have found that 9 is contained 75 times in 675. Illust. 3d. If 10,624 dollars be equally divided among 15 men, what number of dollars will each man receive? We first write down the dividend, and then write the divisor at its left, as before. We perceive
that the divisor, 15, is contained in 106 (hundred) 7 (hun
9) 675 (75 63
15) 10624 (708 Ans.
dred) times; we write 7 (hundred) in the quotient, and multiply the divisor by it; the product is 105 hundred, which we write under 106 hundred; we then subtract, and the remainder is 1 hundred: we then place the 2 (tens) of the dividend at the right of the remainder, and the number is 12 tens, which being less than the divisor, we write a cipher in the place of tens in the quotient, and then place the 4 units of the dividend at the right of the 12 tens, and the number is 124: we find that 15 is contained in 124,8 times; we write 8 in the quotient, and multiply the divisor by it; the product is 120, which we subtract from 124, and the remainder is 4. This remainder, 4, will contain 15 4 fifteenths of one time, which we write in a fractional form at the right of the quotient figures before found, and we have the complete quotient, 708, which is the number of dollars that each man will receive.
From the preceding questions and illustrations we derive the following general rule for division:
RULE.-Write down the dividend; draw a curve line on each side of it, and write the divisor at its left.
Take so many of the highest orders of figures of the dividend as will contain the divisor; find the number of times the divisor is contained in them; write a figure expressing the number at the right of the dividend, for the first figure of the quotient; then multiply the divisor by this quotient figure, and write the product under those figures of the dividend taken.
Subtract this product from those figures, and place the next undivided figure of the dividend at the right of the remainder; then divide these orders of figures as before, and thus proceed until all the figures of the dividend are divided.
If there be a final remainder, write it over a short line at the right of the quotient figures already found, and place the divisor under it, which will express what part of a time the remainder contains the divisor, and completes the quotient.
Whenever a figure of the dividend has been annexed to the remainder, if this partial dividend is less than the