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87. The Multiplier is always an abstract number.

88. The Multiplicand may be an abstract, or concrete number.

89. The Product is always of the same kind as the true Multiplicand.

NOTE. In the explanation of the solution of problems which contain concrete numbers, the number used concretely is the true multiplicand. Thus, 4 chairs at $3 each, will cost 4 times 3 dollars" etc. In this explanation "3 dollars" is the number used concretely, and is, therefore, the true multiplicand.

90. The product is numerically the same in whatever order the factors are multiplied together.

+ Written Exercises+

CASE I.

91. When the Multiplier does not exceed 9. EXAMPLE.-How many are 564+564+564, or 3 × 564?

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EXPLANATION.-In the Solution by Addition

we find that the sum of 3 564's is 1692. But since 4 ones+4 ones +4 ones = 3 × 4 ones, and 6 tens + 6 tens + 6 tens = 3 x.6 tens, and 5 hundreds + 5 hundreds + 5 hundreds = 3 x 5 hundreds, the process can be very much shortened, as shown in the following Solution by Multiplication.

Multiplicand, 564
Multiplier, 3
Product, 1692

EXPLANATION. -In this Solution the Multiplicand 564 is written but once; and as it is to be repeated 3 times, we write the Multiplier 3 under it, and commence at the right hand to mul. tiply.

3 times 4 ones are 12 ones = 1 ten and 2 ones. Write the 2 ones in ones' order of the product and reserve the 1 ten to add to the product of tens. 3 times 6 tens are 18 tens, plus 1 ten reserved, are 19 tens = 1 hundred and 9 tens. Write the 9 tens in tens' order, and reserve the 1 hundred to add to the product of hundreds. 3 times 5 hundreds are fifteen hundreds, plus 1 hundred reserved, are 16 hundreds = 1 thousand and 6 hundreds, which write in the thousands' and hundreds' orders.

Hence the Product is 1692, equal to the Sum of the 1st Solution.

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19. What cost 456 barrels of flour, at $8 a barrel?

Although $8 is the true Multiplicand, for convenience we use 8 for the Multiplier and 456 for the Multiplicand (89, NOTE), but the Product is dollars. For if 1 barrel of flour cost $8, 456 barrels will cost 456 times $8 = $3648.

20. What cost 986 pounds of sugar, at 9 cents a pound? 21. What is the cost of 7 lots of ground, at $5648 each? 22. How much will 8 tons of hay cost, at $23 a ton?

Ans. $184.

23. How much will 4 horses cost, at $225 each?

Ans. $900.

24. How much will a year's board cost, at $6 a week, there Ans. $312.

being 52 weeks in a year?

25. There are 5280 feet in a mile. How many feet in

9 miles?

Ans. 47520 feet.

26. There are 1760 yards in a mile. How many yards in 7 miles? Ans. 12320 yards.

27. How far will a train of cars run in 8 hours, at the rate of 42 miles an hour? Ans. 336 miles.

28. An iron manufacturer pays his hands $4269 a month. How much do their wages amount to in 6 months? in 7 months? 1st Ans. $25614; 2d, $29883.

29. What will 8 farms cost at $12219 each?

30. How many square miles in 7 townships, if each contains 24375 square miles?

31. How much will 328 yards of cloth cost, at $5 per yard? Ans. $1640. 32. What will 375 head of sheep cost, at 4 dollars a head? Ans. $1500.

33. What will 2768 barrels of flour cost, at $7 a barrel ? Ans. $19376.

34. There are 1728 cubic inches in one cubic foot. How many cubic inches in 8 cubic feet?

Ans. 13824 cubic inches.

35. In one square foot there are 144 square inches. How many square inches in 6 square feet.

Ans. 864 square inches. 36. At $325 per acre what will a lot of 9 acres cost?

37. What cost 675 melons at 8 cents each?

Ans. $2925.

38. What will 7 buggies cost at $225 apiece?

39. A man travels 8 miles an hour. How far can he

travel in 4897 hours?

Ans. 39176 miles.

+ Written Exercises+

CASE II.

92. When the Multiplier exceeds 9.

EXAMPLE.-Multiply 583 by 47.

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But since 1 ten is 10 times as great as 1 one, 4 tens must be 10 times as great as 4 ones, and therefore each figure multiplied by 4 tens will produce a product ten times as great as the same figure multiplied by 4 ones. Therefore 583 multiplied by 4 tens, is ten times as great as when multiplied by 4 ones.

But when multiplied by 4 ones the product is written so that the right-hand figure stands in the ones' place. Hence, to make it express ten times as much, it must be written so that the right-hand figure shall stand in the tens' place.

Writing it thus and adding the two Partial Products, we have 27401, the required Product.

93. RULE.- Write the Multiplier under the Multiplicand so that units of the same order shall stand in the same column.

Multiply the entire Multiplicand by each significant figure in the Multiplier, and write the several Partial Products one under another so that the right-hand figure of

each Partial Product will represent the same order as the figure of the Multiplier producing it.

The sum of the Partial Products is the required Product.

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These 10 Problems may be increased in number to 100 by multi

plying each Multiplicand by each of the 10 given Multipliers.

11. If a vessel sails 231 miles in 1 day, how far will she sail in 35 days? in 72 days? 1st Ans. 8085 miles.

12. If a man spends $25 a day for store expenses, how much will he spend in 730 days?

Ans. $18250.

13. A man bought 16 lots at $750 apiece. How much did they all cost him ?

Ans. $12000.

14. A farmer raised 456 bushels of corn each year for 19 years. How much in all did he raise ?

Ans. 8664 bushels.

15. A gentleman's expenses for 1 year were $3450. At

the same rate what would they be for 25 years ?

Ans. $86250.

16. In a school there are 19 teachers and 43 pupils for each teacher. How many pupils are there ?

Ans. 817 pupils. 17. In a certain school district there are 351 pupils, and the cost of tuition per pupil is $14. What is the entire Ans. $4914. 18. What cost 450 tons of iron, at $45 per ton?

cost?

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