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II. When the multiplier is 10, 100, 1000, &c.
¶ 12. It will be recollected, (T3.) that any figure, on being removed one place towards the left hand, has its value increased tenfold; hence, to multiply any number by 10, it is only necessary to write a cipher on the right hand of it. Thus, 10 times 25 are 250; for the 5, which was units before, is now made tens, and the 2, which was tens before, is now made hundreds. So, also, if any figure be removed two places towards the left hand, its value is increased 100 times, &c. Hence,
When the multiplier is 10, 100, 1000, or 1 with any number of ciphers annexed, annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the multiplicand, so increased, will be the product required. Thus, Multiply 46 by 10, the product is 460.
EXAMPLES FOR PRACTICE.
1. What will 76 barrels of flour cost, at 10 dollars a barrel? 2. If 100 men receive 126 dollars each, how many dollars will they all receive?
3. What will 1000 pieces of broadcloth cost, estimating each piece at 312 dollars?
4. Multiply 5682 by 10000.
13. On the principle suggested in the last ¶, it follows, When there are ciphers on the right hand of the multiplicand, multiplier, either or both, we may, at first, neglect these ciphers, multiplying by the significant figures only; after which we must annex as many ciphers to the product as there are ciphers on the right hand of the multiplicand and multiplier, counted together.
EXAMPLES FOR PRACTICE.
1. If 1300 men receive 460 dollars apiece, how many dollars will they all receive?
Ans, 598000 dollars.
The ciphers in the multiplicand and multiplier, counted together, are three. Disregarding these, we write the significant figures of the multiplier under the significant figures of the multiplicand, and multiply; after which we annex three ciphers to the right hand of the
product, which gives the true answer.
2. The number of distinct buildings in New England appropriated to the spinning, weaving, and printing of cotton goods, was estimated, in 1826, at 400, running, on an average, 700 spindles each; what was the whole number of spindles ?
3. Multiply 357 by 6300.
In the operation it will be seen, that multiplying by ciphers produces nothing. Therefore,
III. When there are ciphers between the significant figures of e multiplier, we may omit the ciphers, multiplying by the significant figures only, placing the first figure of each product directly under the figure by which we multiply.
EXAMPLES FOR PRACTICE.
8. Multiply 154326 by 3007.
2. What is the number to be
1. What is multiplication? multiplied called? 3. to multiply by called? 4. What is the result or answer called? 5. Taken collectively, what are the multiplicand and multiplier called? 6. What is the sign of multiplication? 7. What does it show? 8. In what order must the given number be placed for multiplication? 9. How do you proceed when the multiplier is less than 12? 10. When it exceeds 12, what is the method of procedure? 11. What is a composite number? 12. What is to be understood by the component parts, or factors, of any number? 13. How may you proceed when the multiplier is a composite number? 14. To multiply by 10, 100, 1000, &c., what suffices? 15. Why? 16. When there are ciphers on the right hand of the multiplicand, multiplier, either or both, how may we proceed? 17. When there are ciphers between the significant figures of the multiplier, how are they to be treated?
1. An army of 10700 men, having plundered a city, took so much money, that, when it was shared among them, each man received 46 dollars; what was the sum of money taken?
2. Supposing the number of houses in a certain town to be 145, each house, on an average, containing two families, and each family 6 members, what would be the number of inhabitants in that town? Ans. 1740.
3. If 46 men can do a piece of work in 60 days, how many men will it take to do it in one day?
4. Two men depart from the same place, and travel in opposite directions, one at the rate of 27 miles a day, the other 31 miles a day; how far apart will they be at the end of 6 days? Ans. 348 miles. 5. What number is that, the factors of which are 4, 7, 6, and 20? Ans. 3360. 6. If 18 men can do a piece of work in 90 days, how long will it take one man to do the same?
7. What sum of money must be divided between 27 men, so that each man may receive 115 dollars?
8. There is a certain number, the factors of which are 89 and 265; what is that number?
9. What is that number, of which 9, 12, and 14 are factors?
10. If a carriage wheel turn round 346 times in running 1 mile, how many times will it turn round in the distance from New York to Philadelphia, it being 95 miles.
Ans. 32870. 11. In one minute are 60 seconds; how many seconds in 4 minutes? in 5 minutes? in 20 minutes?
in 40 minutes?
12. In one hour are 60 minutes; in two hours?
how many seconds in how many seconds from
13. In one dollar are 6 shillings; how many shillings in
in 300 dollars?
in 467 dollars?
14. Two men, A and B, start from the same place at the same time, and travel the same way; A travels 52 miles a day, and B 44 miles a day; how far apart will they be at the end of 10 days?
15. If the interest of 100 cents, for one year, be 6 cents, how many cents will be the interest for 2 years?
for 10 years?
for 35 years?
16. If the interest of one dollar, for one year, be six cents, what is the interest for 2 dollars the same time?
17. A farmer sold 468 pounds of pork, at 6 cents a pound, and 48 pounds of cheese, at 7 cents a pound; how many cents must he receive in pay?
18. A boy bought 10 oranges; he kept 7 of them, and sold the others for 5 cents apiece; how many cents did he receive? 19. The component parts of a certain number are 4, 5, 7, 6, 9, 8, and 3; what is the number?
20. In 1 hogshead are 63 gallons; how many gallons in 8 hogsheads? In 1 gallon are 4 quarts; how many quarts in 8 hogsheads? In 1 quart are 2 pints; how many pints in 8 hogsheads?
OF SIMPLE NUMBERS.
¶ 14. 1. James divided 12 apples among 4 boys; how many did he give each boy?
2. James would divide 12 apples among 3 boys; how many must he give each boy?
3. John had 15 apples, and gave them to his playmates, who received 3 apples each; how many boys did he give them to? 4. If you had 20 cents, how many cakes could you buy at 4 cents apiece?
5. How many yards of cloth could you buy for 30 dollars, at 5 dollars a yard?
6. If you pay 40 dollars for 10 yards of cloth, what is one yard worth?
7. A man works 6 days for 42 shillings; how many shil úngs is that for one day?
8. How many quarts in 4 pints?
in 10 pints?
9. How many times is 8 contained in 88 ?
in 6 pints?
10. If a man can travel 4 miles an hour, how many hours would it take him to travel 24 miles?
11. In an orchard there are 28 trees standing in rows, and there are 3 trees in a row; how many rows are there? Remark. When any one thing is divided into two equal parts, one of those parts is called half; if in:0 3 equal parts, one of those parts is called a Aird; if into four equal parts, one part is called a quarter or a fourth; if into tive, one part is called a fifth, and so on.