MULTIPLICATION.

41. 1. What will 4 oranges cost, at 2 cents apiece? ANALYSIS.-1 orange costs, 2 cents;

For the cost of 1 orange, 2 is taken once; for the cost of 2 oranges, it is taken twice; for the cost of 3, it is taken three times; and for the cost of 4, it is taken four times: Hence, one time 2 is 2; two times 2 are 4; three times 2 are 6; and four times 2 are 8.

2. What is the cost of 6 yards of ribbon, at 7 cents a yard?

ANALYSIS.-Six yards of ribbon will cost 6 times as much as 1 yard; since 1 yard costs 7 cents, 6 yards will cost 6 times 7 cents, which are 42 cents: Therefore, 6 yards of ribbon, at 7 cents a yard, will cost 42 cents.

42. MULTIPLICATION is the operation of taking one number as many times as there are units in another.

43. The MULTIPLICAND is the number to be taken.

44. The MULTIPLIER is the number denoting how many times the multiplicand is to be taken.

45. The PRODUCT is the result of the operation.

46. The FACTORS of the product are the multiplicand and multiplier.

47. The sign X, is called the sign of Multiplication. When placed between two numbers, it denotes that they are to be multiplied together; thus,

7X5 35; and is read, 5 times 7 are 35.

41. What will 4 oranges cost, at 2 cents apiece?-42. What is Multiplication?-43. What is the multiplicand?-44. What is the multiplier?-45. What is the product ?-46. What are the factors? -47. What is the sign of Multiplication? What is it called? When placed between two numbers, what does it denote ?

9X10 90 10x10=100 11x10=110 | 12×10=120 9X11 99 10x11=110 | 11x11=121 12×11=132 9X12=108 10x12=120 | 11x12=132 12×12=144

77 12X 7= 84

CASE I.

48. When the multiplier does not exceed 9.

1. What is the product of 236 multiplied by 4?

ANALYSIS. Since the entire number 236 is to be taken 4 times, each order of units must be taken 4 times: hence, the product must contain 24 units, 12 tens, and 8 hundreds:

Therefore, the product is.

8

In practice, the operation is performed thus: Say, 4 times 6 are 24; set down the 4, and then say, 4 times 3 are 12, and 2 to carry are 14; set down the 4, and then say, 4 times 2 are 8, and 1 to carry are 9; set down the 9, and the product is 944, as before.

Hence, we have the following

Rule.

OPERATION.

236

4

944

Multiply each figure of the multiplicand by the multiplier, carrying and setting down as in Addition.

49. Since the multiplier denotes times, it is always an abstract number; and since the repetition of a number does not change its unit, the unit of the product will be the same as that of the multiplicand.

Reading.

50. The operations of Multiplication may be much shortened, by pronouncing only the final results.

Thus, in the last example, instead of saying, 4 times 6 are 24; 4 times 3 are 12, and 2 are 14; 4 times 2 are 8, and 1 are 9: we pronounce only the final results, 24, 14, 9,-performing the operations mentally.

12. Multiply 30416 by 7. 13. Multiply 204127 by 0. 14. Multiply 30497 by 6. 15. Multiply 69507 by 5. sheeting are sufficient to sup

16. If 104 yards of cotton ply 1 family for a year, how many yards would supply 9 families?

17. A farmer had 309 sheep in each of 4 fields: how many sheep had he altogether?

18. Mrs. Simpkins purchased 2 rolls of table linen, each containing 149 yards: how many yards did she buy?

19. Each of 9 grocers bought 2974 pineapples: how many did they all buy?

20. If each of 7 children receive 4073 dollars, how much do they all receive?

21. How many sheep are there in 9 droves, if in each drove there are 598 ?

22. What will be the cost of 9 horses, at 165 dollars apiece?

48. What is the rule, when the multiplier contains but one figure?

49. What kind of a number is the multiplier? What is the unit of the product?

50. How may the operations of Multiplication be abridged? Give an example.

51. The product of two numbers is the same, whichever be taken as the multiplier.

Multiply any two numbers together; as, 8 by 6.

ANALYSIS.-Place as many

1's in a horizontal row as there are units in the multiplicand, and make as many rows as there are units in the multiplier: the product is equal to the number of 1's in one row taken as many times as there are rows; that is, to 8 × 6 = 48.

8

1

1

1

1 1

1

1 1

1

1

1

1

1

1

1

1

If we consider the number of 1's in a vertical row to be the multiplicand, and the number of rows the multiplier, the product will be equal to the number of 1's in a vertical row taken as many times as there are vertical rows; that is, 6 x 8 = 48. Hence,

The product of two numbers is the same, whichever be taken as the multiplier.

CASE II.

52. When the multiplier contains two or more figures.

1. Multiply 8204 by 603.

ANALYSIS.-The multiplicand is to be taken 603 times. Taking it 3 times, we obtain 24612.

When we come to take it 6 hundred times, the lowest order of units will be hundreds: hence, 4, the first figure of the product, must be written in the third place.

OPERATION.

8204

603

24612

49224

4947012

NOTE. The product obtained by multiplying by a single figure of the multiplier, is called, a partial product. The sum of the partial products, is the required product.

51. Is the product of two numbers altered by interchanging the factors?