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10. If 4 pecks make a bushel, how many pecks in 2 bushels ? In 3 bushels ? In 4 bushels ? In 6 bushels ? In 7 bushels ? In 9 bushels ?
11. If 12 inches make 1 foot, how many inches in 3 feet? In 4 feet? In 5 feet? In 7 feet? In 8 feet? In 9 feet ? In 10 feet? In 12 feet?
12. If ere be feet in a square yard, how many feet in 4 yards? In 5 yards? In 6 yards ? În 8 yards? In 9 yards? In 12 yards ?
13. What cost 3 yards of cloth at 5 dollars per yard? 4 yards ? 5 yards ? 6 yards ? 7 yards ? 8 yards? 9 yards ? 10 yards ? 11 yards ? 12 yards?
14. If 1 pound of iron cost 7 cents, what cost 2 pounds ? 3 pounds ? 5 pounds : 6 pounds ? 7 pounds ? 8 pounds ? 9 pounds ? 12 pounds ?
15. If 1 pound of raisins cost 6 cents, what cost 4 pounds ? 5 pounds ? 6 pounds 7 pounds ? 8 pounds ? 9 pounds? 10 pounds? 12 pounds ?
16. In 1 acre there are four roods; how many roods in 2 acres? In 3 acres ? In 4 acres ? In 5 acres ? In 6 acres? In 9 acres ?
17. A good pair of boots is worth 5 dollars; what must I give for 5 pairs ? For 6 pairs ? For 7 pairs ? For 8 pairs ?
18. A cord of good walnut wood may be obtained for 8 dollars; what must I give for 4 cords ? For 6 cords ? For 9 cords ?
19. What cost 4 quarts of milk at 5 cents a quart, and 8 gallons of vinegar at 10 cents a gallon ?
20. If a man earn 7 dollars a week, how much will he earn in 3 weeks ? In 4 weeks? In 5 weeks? In 6 weeks ? In 7 weeks? In 9 weeks ?
21. If 1 thousand feet of boards cost 12 dollars, what cost 4 thousand : 5 thousand ? 6 thousand ? 7 thousand ? 9 thousand ? 12 thousand ?
22. If 3 pairs of shoes buy 1 pair of boots, how many pairs of shoes will it take to buy 7 pairs of boots ?
23. If 5 bushels of apples buy 1 barrel of flour, how many bushels of apples are equal in value to 12 barrels of flour ?
24. If 1 yard of canvas cost 25 cents, what will 12 yards cost ?
ILLUSTRATION. The number 25 is composed of 2 tens and 5 units; 12 times 2 tens are 24 tens; and 12 times 5 units are
60 units, or 6 tens; 24 tens added to 6 tens make 30 tens, or 300. Therefore, 12 yards will cost 300 cents, or 3 dollars.
25. In 1 pound there are 20 shillings ; how many shillings in 3 pounds ? In 4 pounds ? In 6 pounds ?
26. A gallon of molasses is worth 25 cents; what is the value of 2 gallons ? Of 3 gallons ? Of 4 gallons ? Of 5 gallons? Of 6 gallons ? Of 9 gallons ?
27. If 12 men can do a piece of work in 16 days, how long will it take 1 man to do it ?
28. If a steam-engine runs 28 miles in 1 hour, how far will it run in 4 hours ? In 6 hours ? In 9 hours ?
29. If the earth turns on its axis 15 degrees in 1 hour, how far will it turn in 7 hours ? In 11 hours ? In 12 hours ?
30. In a certain regiment there are & companies, in each company 6 platoons, and in each platoon 12 soldiers; how many
soldiers are there in the regiment ? Art. 34. The learner, having performed the foregoing questions, will perceive that
MULTIPLICATION is the repetition of a number any proposed number of times, and is therefore a compendious method of addition.
In multiplication, three terms are employed, called the Multiplicand, the Multiplier, and the Product.
The multiplicand is the number to be multiplied or repeated.
The multiplier is the number by which we multiply, and denotes the number of repetitions to be made.
The product is the answer, or number produced by the multiplication.
The multiplicand and multiplier are often called factors.
Art. 35. SIGNS. — The sign of multiplication is formed by two short lines crossing each other obliquely ; thus, X. It shows that the numbers between which it is placed are to be multiplied together; thus, the expression 7 x 5 = 35 is read, 7 multiplied by 5 is equal to 35.
QUESTIONS. — Art. 34. What is multiplication ? What three terms are employed ? What is the multiplicand ? What is the multiplier ? What is the product? What are the multiplicand and multiplier often called ? - Art. 35. 'What is the sign of multiplication ? What does it show ?
EXERCISES FOR THE SLATE.
Art. 36. Method of operation, when the multiplier does not exceed 12. Ex. 1. Let it be required to multiply 175 by 7.
After having written the multiplier unMultiplicand 175
der the unit figure of the multiplicand, and Multiplier 7 drawn a line below it, we multiply the 5 Product 1225
in the multiplicand by 7, saying, 7 times
5 are 35, and set down the 5 units directly under the 7, and reserve the 3 tens in the mind. We then multiply the 7 in the multiplicand, saying 7 times 7 are 49, and, adding the 3 tens which were reserved, we have 52 tens, or 5 hundreds and 2 tens. Setting down the 2 tens, and reserving the 5 hundreds, we multiply 1 by 7, and, adding the reserved 5 hundreds, we have 12 hundreds, which, as it completes the multiplication, we set down in full, and the product is 1225.
EXAMPLES FOR PRACTICE.
5 Ans. 35 0 24 13701
8. 89 7 6 5
284035 2873 58 431599 807885 9. Multiply 767853 by 9.
Ans. 6910677. 10. Multiply 876538765 by 8.
Ans. 7012310120. 11. Multiply 7654328 by 7.
Ans. 53580296. 12. Multiply 4976387 by 5.
Ans. 24881935. 13. Multiply 8765448 by 12.
Ans. 105185376. 14. Multiply 4567839 by 11.
Ans. 50246229. 15. What cost 8675 barrels of flour at 7 dollars
barrel ? Ans. 60725 dollars.
QUESTIONS. - Art. 36. How must numbers be written for multiplication ? At which hand do you begin to multiply? Why? Where do you write the first or right-hand figure of the product of each figure in the multiplicand ? Why? What is done with the tens or left-hand figure of each product ? How, then, do you proceed when the multiplier does not exceed 12 1
16. What cost 25384 tons of hay at 9 dollars per
Ans. 228456 dollars. 17. If on 1 page in this book there are 2538 letters, how many are there on 11 pages ?
Ans. 27918 letters.
Art. 37. Method of operation, when the multiplier exceeds 12. Ex. l. Let it be required to multiply 763 by 24.
Here we write the multiplicand and Multiplicand 763
multiplier as before, and proceed to mul
tiply the multiplicand by 4, the unit figMultiplier 24
ure of the multiplier, precisely as in Art. 3052
36. We then, in like manner, multiply 1526
the multiplicand by the 2 tens in the mul
tiplier, taking care to set the first figure Product 18312
obtained by this multiplication directly
under the 2 of the multiplier, and, adding together the products obtained by the two multiplications and placed as in the operation, we have the full product of 763 multiplied by 24, which is 18312.
Art. 38. The preceding examples sufficiently illustrate the principle and method of multiplication ; and the learner is now prepared to understand and apply the following general
RULE. 1. Place the larger number uppermost for the multiplicand, and the smaller number under it for a multiplier, arranging units under units, tens under tens, fc.
2. Then multiply each figure of the multiplicand by each figure of the multiplier, beginning with the right-hand figure, and carrying for every ten as in addition.
3. If the multiplier consists of more than one figure, the right-hand figure of each product must be placed directly under the figure of the multiplier that produces it. The sum of the several products will be the whole product required.
NotE.— When there are ciphers between the significant figures of the multiplier, pass over them in the operation, and multiply by the significant figures only, remembering to set the first figure of the product directly under the figure of the multiplier that produces it.
QUESTIONS. - Art. 37. How do you proceed when the multiplier exceeds 12? Where do you set the first figure of each partial product? Why? How is the true product found? - Art. 38. What is the general rule for multiplication? When there are ciphers between the significant figures of the multiplier, how do you proceed?
Art. 39. First Method of Proof. – Multiply the multiplier by the multiplicand, and if the result is like the first product, the work is supposed to be right.
The reason of this proof depends on the principle, That, when two or more numbers are multiplied together, the product is the same, whatever the order of multiplying them. Ex. 2. Multiply 7895 by 56.
Ans. 442120. OPERATION. Multiplicand 7895
56 Multiplier 56
448 Product 442120
Product 442 120 NotE. — The common mode of proof in business is to divide the product by the multiplier, and, if the work is right, the quotient will be like the multiplicand. This mode of proof anticipates the principles of division, and therefore cannot be employed without a previous knowledge of that rule.
Art. 40. Second Method of Proof. Begin at the left hand of the multiplicand, and add together its successive figures toward the right, till the sum obtained equals or exceeds the number nine. If it equals it, drop the nine, and begin to add again at this point, and proceed till you obtain a sum equal to, or greater than, nine. If it exceeds nine, drop the nine as before, and carry the excess to the next figure, and then continue the addition as before. Proceed in this way till you have added all the figures in the multiplicand and rejected all the nines contained in it, and write the final excess at the right hand of the multiplicand.
Proceed in the same manner with the multiplier, and write the final excess under that of the multiplicand. Multiply these excesses together, and place the excess of nines in their product at the right.
Then proceed to find the excess of nines in the product obtained by the original operation; and, if the work is right,
QUESTIONS. — Art. 39. How is multiplication proved by the first method ? What is the reason for this method of proof? What is the common mode of proof in business ? — Art. 40. What is the second method of proving multiplication ?