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22. If the interest of one dollar be 6 cents a year, what will be the interest of 2 dollars, for the same time? of 3? of 4 ? of 5 ? of 6? of 7 ? of 8? of 9? of 10 ? of 11? of 12?
23. If the interest of a dollar be 6 cents for 1 year, what will it be for 2 years ? for 3? 4? 5? 6? 7? 8? 9? 10? 11 ? 12 ?
24. At 6 dollars a week, what cost 3 weeks board ? 5 weeks? 7 weeks? 9 ? 11 ? 13 ?
25. Two men start from Hartford towards New-Haven at the same moment. One goes 7 miles an hour, the other, 3 miles an hour. How far apart are they at the end of one hour ? 2 hours ? 3 hours ? 6? 9? 8? 7? 11? 12?
26. Two men leave Hartford at the same moment. One
goes north, and the other south. One travels 4 miles an hour, and the other 2 miles an hour. How far apart are they at an hour's end? How far in 2 hours ? 3? 4? 5? 6 ? 7 ? 8? 9 ? 12 ? 11 ?
27. If 3 men can dig a ditch in 1 day, how many days would one man be doing it?
28. If 3 men can now a field in 2 days, how many days would 1 man be doing it ?
29. If 7 men can build a wall in one day, how many days would it take one man to do it?
30. If 4 men can make a fence in 3 days, how long would 1 man be making it?
31. If 5 men can do a piece of work in 7 days, in how many days could 1 man do it ?
32. If a barrel of cider will last 10 men 4 weeks, how many weeks will it last 1 man ?
33. If 8 men can do a piece of work in 5 days, how many men can do it in 1 day?
34. If 7 men can do a piece of work in 3 days, how many men can do it in 1 day?
35. If 9 men can live on a quantity of provision 6 weeks, how many men could live on the same provision 1 week ?
36. A man purchased 8 sheep at 4 dollars apiece; 5 barrels of cider, at 2 dollars a barrel ; 4.yards of cloth, at 1 dollar a yard ; and 3 bushels of apples, at 3 dollars a bushel. How much did they all come to ?
37. A lady bought 3 yards of ribbon for 6 cents a yard ; 1 yard of silk for 50 cents; 6 needles at 1 cent apiece ; 1
paper of pins at 12 cents; and 1 pair of scissors at 30 cents. How much did they all come to ?
§ XI. Let the pupil write the following exercises on his slate.
i. A man bought 5 pounds of beef at 6 cents a pound. How much did he give for the whole ? PROCESS.
down the 6 once, with the 5 under it, we may write down
the answer immediately. Thus, 30
6 are 30.
performed by addition, and likewise by a shorter process.
Let the pupil perform, by both these methods, the following. 2. At the rate of 7 miles an hour, how far can I travel in 8 hours ? 3. At 9 dollars a barrel, what will 7 barrels of flour cost ? 4. At 3 dollars a ream, what will 6 reams of paper come at ? 5. At 5 cents a pound, what cost 4 pounds of pork ?
Thus far, we have been repeatedly adding numbers, consisting of one figure only. Let us now try an example in which larger num. bers are concerned.
6. At 75 cents a bushel, what will 4 bushels of salt cost ?
Here we have a number, consisting of 7 tens and 5 units, to be repeated 4 times. By addition, the process would be as follows: 75
75 That is, 75 added 4 times to itself, makes 300. 75
300 The same may be performed as follows:
Here we say, 4 times 5 units are 20 units, and set down the 20. We then say, 4 times 7 tens are 28 tens, and set dow nthe 28, with the 8, of course, in the tens' place. We then add the two together, and the answer is 300, as before.
4 20 28 300
7. At 58. cents a yard, what will 6 yards of cloth cost ? A. 348 cts. The pupil will see that he may carry, as in Addition. Thus,
8. At 18 cents a pound, what cost 8 pounds of raisins ? 18 8 times 8 units are 64 units, equal to 6 tens and 4 units. 8 The 4 belongs to the units' place. Reserve the 6 tens, in the mind. 8 times 1 ten are 8 tens. To the 8 tens add the 6 144 tens reserved, and the amount is 14 tens, or 1 hundred, and 4 tens. The answer, then, is 1 hundred, 4 tens, and 4 units, or 144.
[The pupil should frequently be required to explain his processes, in the man ner of the above illustrations.)
9. In one hogshead are 63 gallons. How many gallons in 7 hogsheads ?
441 Ans. 10. A man has 7 lots, each containing 54 acres of ground. How many acres in all ?
Ans. 328. 11. A man employed 27 laborers, and gave them 4 dollars a week. llow much did he give the whole a week?
Ans. 108. 12. At 5 dollars a firkin, what cost 39 firkins of butter? A. 195.
You see that if we were not to employ this short process in performing these examples, it would be necessary to use Addition. We have, then, found a means of performing Addition more rapidly than by the ordinary mode. This is called MULTIPLICATION.
You see, also, that Multiplication consists in repeating the same number ; that is, in adding the SAME NUMBER TO ITSELF. Then,
MULTIPLICATION IS A CONCISE MODE OF PERFORMING ADDITION, WHEN THE NUMBERS TO BE ADDED ARE EQUAL. Or,
MULTIPLICATION IS PINDING A NUMBER, EQUAL TO A GIVEN NUMBER OF REPETITIONS OF SOME OTRER NUMBER.
You see, further, that two numbers are always given you ; one to be repeated, and another, which shows how often it is to be repeated.
TAE NUMBER TO BE MULTIPLIED, OB REPEATED IS CALLED THE MUL, TIPLICAND.
The NUMBER WHICU SHOWS HOW MANY REPETITIONS ARE TO BE BIADE, IS CALLED THE MULTIPLIER.
THE RESULT OBTAINED FROM MULTIPLICATION, IS CALLND TIN PRO. DUCT.
Whenever two or more numbers are multiplied together, each of them is called a factor.
Then, in Multiplication, BOTH MULTIPLIER AND MULTITLICAND ARE CALLED FACTORS.
X between two numbers signifies that they are to be multiplied together. Thus, 6 X5=30, signifies that 6 multiplied by 5, equals 30.
Here is a window. A person wishes to discover how many panes of glass are contained in it. He counts the upper row of panes, and finds that there are 4 panes. He then counts the number of rows, and finds 6 rows. Then he says, 6 rows of panes, with 4 panes in each
make up all the panes in the window. Therefore, there are 24, because 6 times 4 are 24. Another per. son counts the panes in one row up and down, and finds 6 panes. He counts the rows, and finds 4 rows. Then
he says, 4 rows with 6 panes in a row, make up all the panes in the window. Therefore, there are 24 panes, because 4 times 6 are 24.
Four times 6, then, is equal to 6 times 4; because either is equal to all the panes in the window, that is, to 24.
Hence it appears that, IN PERFORMING MULTIPLICATION, EITHER FACTOR MAY BE MADE THE MULTIPLICAND, WITHOUT ALTERING THE
For convenience, however, we usually place the larger number upper most.
EXAMPLES FOR PRACTICE. 1. What will 2 wagons cost at 23 dollars apiece ?
Ans. 46 dolls. 2. What cost 3 head of cattle at 17 dollars a head ?
Ans. 51 dolls. 3. What will 18 pounds of pork cost at 4 cents a pound?
Ans. 72 cts. 4. What cost 21 barrels of cider at 3 dollars a barrel ?
Ans. 63 dolls. 5. At 5 dollars a yard, what cost 27 yards of cloth ?
Ans. 135 dolls. 6. At 6 cents per pound, what cost 32 pounds of beef ?
Ans. 192 cts. 7. At 7 dollars a barrel, what cost 35 barrels of flour ?
Ans. 245 dolls. 8. At 8 dollars a hundred weight, what cost 43 hundred weight of
Ans. 344 dolls. 9. At 9 dollars a barrel, what cost 653 barrels of flour ?
Ans. 5,877 dolls. 10. A man bought 23 kegs of brandy, each containing 9 gallons. How many gallons did he buy?
Ans. 207 galls. 11. He paid for it at the rate of 8 cents a quart. How much did it come to, allowing 4 quarts to the gallon ? Ans. 6,624 cts.
Nors. As 100 cents make a dollar, the pupil will readily tell how niany dollars are contained in any number of cents. For all the
even hundreds will be dollars; and hence, if we cut off the two low. er orders, viz. tens and units, they will be cents, and all to the left will be dollars. According to this principle, the answer in the 5th example is 1 dollar 92 cents; and in the last, 66 dollars 24 cents.
12. A man bought 17 barrels of ale at 9 dollars a barrel; and 25 barrels of flour at 7 dollars a barrel ; and 9 pieces of broadcloth, each piece containing 47 yards, at 6 dollars a yard. What did it all
Ans. 2,866 dolls. 13. A man bought 15 pounds of beef at 7 cents a pound ; 8 pounds of sugar at 18 cents a pound ; 9 yards of cotton cloth at 22 cents a yard; and 3 pails at 62 cents apiece. What did the whole amount to ?
Ans. 6 dolls. 33 cts. 14. 365X3
Ans. 1,095 | 19. 74,635X3 Ans. 223,905 15. 273x8 2,184 20. 53,894X4
215,576 16. 786X3 2,358 21. 83,528X5
417,640 17. 1,338X5 6,690 ) 22. 123,456X8
987,648 18. 2,487X9
22,383 23. 13,052,068X5 65,260,340 24. 12,345,006,789X7 Ans. 86,415,047,523 25. 236,120,013X5
1,180,600,065 Thus far, none of our multipliers have exceeded 12; and from observation, we derive the following rule.
I. PLACE THE MULTIPLIER UNDER THE MULTIPLICAND. II. BEGINNING WITH UNITS, MULTIPLY EACH FIGURE OF THE MULTIPLICAND BY THE MULTIPLIER, AND CARRY AS IN ADDITION.
[When the pupil employs technical language, the teacher should frequently require him to explain its meaning. Thus, he should inquire the meaning of multiplier, multiplicand, frc.]
1. At 255 cents per pound, what cost 24 pounds of Indigo ?
Here our multiplier is more than 12. Therefore, we cannot proceed by the last rule.
But we know that 24 is twice as much as 12. There. fore, 24 times any number will be twice as much as 12 times the same number.
By the last rule, 12 times 255=3,060.
2. At 16 dollars a hogshead, what cost 472 hogsheads of molasses ! 16 is 4 times 4. Therefore, 16 times aný number will be 4 times as much as 4 times the same number.