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Art. 33. Ex. 1. What will three lemons cost, at 2 cents apiece?
Analysis.-Since 1 lemon costs 2 cents, 3 lemons will cost 3 times 2 cents; and 3 times 2 cents are 6 cents. Therefore, 3 lemons, at 2 cents apiece, will cost 6 cents.
Obs. The preceding operation is a short method of finding how much 2 cents will amount to, when repeated or taken 3 times, and is called Multiplication. Thus, 2 cents + 2 cents + 2 cents are 6 cents. Hence,
34. MUTIPLICATION is the process of finding the amount of a number repeated or added to itself, a given number of times.
The number to be repeated or multiplied, is called the multiplicand.
The number by which we multiply, is cailed the multiplier, and shows how many times the multiplicand is to be repeated or taken.
The answer, or number produced by multiplication, is called the product.
Thus, when we say 5 times 7 are 35, 7 is the multipli. cand, 5 the multiplier, and 35 the product.
OBs. When the multiplicand denotes things of one kind, or de. nomination only, the operation is called Simple Multiplication.
QUEST.-34. What is multiplication ? What is the numbrr to be repeated or multiplied called ? What the number by which we multiply? What does the multiplier show? What is the answer called ? When wo say 5 times 7 aro 35, which is the multiplicand ? Which the multiplier ? Which the product? Obs. When the inultiplicand donotes thinge of onu denomination only, what is the operation called 3
38. The product of any two numbers will be the same, whichever factor is taken for the multiplier. Thus,
If a garden contains 3 rows of trees as represented by the number of horizontal rows of stars in the margin, and each row has 5 trees as represented by the number of stars in a row, it is evident, that the whole number of trees in the garden is equal either to the number of stars in a horizontal row, taken three times, or to the number of stars in a perpendicular row taken five times; that is, equal to 5 X3, or 3 X 5.
39. When the multiplier contains but one figure. Ex. 1. What will 3 horses cost, at 123 dollars apiece ?
Analysis. -- Since 1 horse costs 123 dollars, 3 horses will cost 3 times 123 dollars.
Directions.-Write the multi Operation. plier under the multiplicand; 123 multiplicand. then, beginning at the right
3 multiplier. hand, multiply each figure of the
Dolls. 369 product. multiplicand by the multiplier. Thus, 3 times 3 units are 9 units, or we may simply say 3 times 3 are 9; set the 9 in units' place under the figure multiplied. 3 times 2 are 6; set the 6 in tens' place. 3 times 1 are 3; set the 3 in hundreds' place.
Note.--The pupil should be required to analyze every example, and to give the reasoning in full; otherwise the operation is liable to become mere guess-work, and a habit is formed, which is alike destructive to mental discipline and all substantial improvement. Solve the following examples in a similar manner :
(2.) (3.) (4.) (5.) Multiplicand 34 312 2021 1110 Multiplier 2
(6.) (7.) (8.) (9.) Multiplicand, 4022 6102 7110 8101 Multiplier, 3
Suggestion. In this example the product of the different figures of the multiplicand into the multiplier, exceeds 9; we must therefore write the units' figure under the figure multiplied, and carry the tens to the next product on the left, as in addition. Thus, begin. ning at the right hand as before, 6 times Operation. 3 units are 18 units, or we may simply 23 dolls. say 6 times 3 are 18. Now it requires 6 two figures to express 18; we there. Ans. 138 dollars. fore set the 8 under the figure multiplied, and reserving the 1, carry it to the product of the next figure, as in addition. (Art. 23.) Next, 6 times 2 are 12, and 1 (to carry) makes 13. Since there are no more figures to be multiplied, we set down the 13 in full. The product is 138 dollars. Hence,
40. When the multiplier contains but one figure.
Write the multiplier under the multiplicand, units un.. der units, and draw a line beneath them.
Begin with the units, and multiply each figure of the multiplicand by the multiplier, setting down the result arut carrying as in addition. (Art. 23.) Multiply the following numbers together. 11. 78X4.
18. 524 x 6. 12. 96 x 5.
19. 360 X7. 13. 83 X3.
20. 475 X 4. 14. 120x7.
21. 792 X 5. 15. 138 X 6.
22. 820 X 8. 16. 163 X 5.
23. 804 X 7. 17. 281 X 8.
24. 968 X 9.
25. Wiat will 115 barrels of flour cost, at 6 dollars
per barrel ?
26. A man bought 460 pair of boots, at 5 dollars a pair: wow much did he pay for the whole ?
27. What cost 196 acres of land, at 7 dollars per acre? 28. What cost 3 10 ploughs, at 8 dollars apiece ? 29. What cost 691 hats, at 7 dollars apiece ? 30. What cost 865 heifers, at 9 dollars per head ? 31. What cost 968 cheeses, at 8 dollars apiece? 32. What cost 1260 sheep, at 7 dollars per head? 33. What cost 9 farms, at 2365 dollars apiece ? 41. When the multiplier contains more than one figure.
34. A man sold 23 sleighs, at 54 dollars apiece: how much did he receive for them all ?
Suggestion.--Reasoning as before, if i sleigh costs 64 dollars, 23 sleighs will cost 23 times as much. Directions. As it is not
Operation. convenient to multiply by 23
54 Multiplicand. at once, we first multiply by
23 Multiplier. the 3 units, then by the 2
162 cost of 3 s. tens, and add the two results
108 together. Thus, 3 times 4 Dolls. 1242 are 12, set the 2 under the figure 3, by which we are multiplying, and carry the 1 as above. 3 times 5 are 15, and 1 (to carry) makes 16. Next, we multiply by the 2 tens thus: 20 times 4 units are 80 units or 8 tens; or we may simply say 2 times 4 are 8.
Set the 8 under the figure 2 by which we are multiplying, that is, in tens' place, because it is tens. 2 times 5 are 10. Finally, adding these two products together as they stand, units to units, tens to tens, &c., we have 1242 dollars, which is the whole product required.
6 20 8. « 23 s.
Note.- When she multiplier contains more than one figure, the several products of the roultiplicand into the separate figures of the multiplier, are called partial products. 35. Multiply 45 by 36, and prove the operation,
Operation. Beginning at the right hand, we 45 Multiplicand. proceed thus: 6 times 5 are 30; 36 Multiplier. set the 0 under the figure by which 270 we are multiplying ; 6 times 4 are 135 24 and s (to carry) are 27, &c. 1620 Prod.
Proof. PROOF.—We multiply the mul 36 tiplier by the multiplicand, and 45 since the result thus obtained is 180 the same as the product above, 144 the work is right.
1620 Prod. 36. What is the product of 234 multiplied by 165 ?
Operation. Suggestion.—Proceed in the same man 234 ner as when the multiplier contains but 165 two figures, remembering to place the 1170 right hand figure of each partial product 1404 directly under the figure by which you 234 inultiply.
38610 Ans 37. What is the product of 326 multiplied by 205 ?
Suggestion.—Since multiplying by a Operation. cipher produces nothing, in the operation 326 we omit the 0 in the multiplier. Thus, 205 having multiplied by the 5 units, we next 1630 multiply by the 2 hundreds, and place the 652 first figure of this partial product under · 66830 Ans, the figure by which we are multiplying.