17. A merchant bought 472 yards of cloth at $1.25 a yard; 147 were damaged and had to be sold at $.67 a yard. He sold the remainder at $1.58 a yard; did he gain or lose on the transaction, and how much? Ans. $21.99 gain.

18. A mechanic employed on a building 78 days received $2.75 a day. His family expenses during the same time were $1.86 a day; how much did he save? Ans. $69.42.

19. Bought 167 bushels of wheat at $1.65 a bushel, and 287 bushels of oats at $.37 a bushel. I sold the wheat at a loss of 4 cents on a bushel, and 34 bushels of oats at a gain of 18 cents a bushel, the remainder at a gain of 13 cents. did I gain on the transaction?

What

20. A merchant purchased 16 pieces of cloth, each containing 48 yards, at $2.75 a yard. He sold the entire lot at an advance of $.45 per yard. How much did he pay for the cloth, and what was his entire gain?

DEFINITIONS.

100. Multiplication is the process of taking one number as many times as there are units in another.

101. The Multiplicand is the number taken, or multiplied.

102. The Multiplier is the number which denotes how many times the multiplicand is taken.

103. The Product is the result obtained by multiplication.

104. A Partial Product is the result obtained by multiplying by one order of units in the multiplier, or by any part of the multiplier.

105. The Total or Whole Product is the sum of all the partial products.

106. The Process of Multiplication consists, first,

in finding partial products by using the memorized results of the Multiplication Table; second, in uniting these partial products by addition into a total product.

10%. A Factor is one of the equal parts of a number. Thus, 12 is composed of six 2's, four 3's, three 4's, or two 6's; hence, 2, 3, 4, and 6 are factors of 12.

The multiplicand and multiplier are factors of the product. Thus, 36 × 25925. The product 925 is composed of twenty-five 37's, or thirty-seven 25's. Hence, both 37 and 25 are equal parts or factors of 925.

108. The Sign of Multiplication is ×, and is read times, or multiplied by.

When placed between two numbers, it denotes that either is to be multiplied by the other. Thus, 8 x 6 shows that 8 is to be taken 6 times, or that 6 is to be taken 8 times; hence it may be read either 8 times 6 or 6 times 8.

109. PRINCIPLES.—I. The multiplicand may be either an abstract or concrete number.

II. The multiplier is always an abstract number. III. The product is of the same denomination as the multiplicand.

REVIEW AND TEST QUESTIONS.

110. 1. Define Multiplication, Multiplicand, Multiplier, and Product.

2. What is meant by Partial Product? Illustrate by an example.

3. Define Factor, and illustrate by examples.

4. What are the factors of 6? 14? 15? 9? 20? 24? 25? 27? 32? 10? 30? 50? and 70?

5. Show that the multiplicand and multiplier are factors of the product.

6. What must the denomination of the product always be, and why?

7. Explain the process in each of the following cases and illustrate by examples:

I. To multiply by numbers less than 10.

II. To multiply by 10, 100, 1000, and so on.
III. To multiply by one order of units.

IV. To multiply by two or more orders of units.

V. To multiply by the factors of a number (92-2).

8. Give a rule for the third, fourth, and fifth cases.

9. Give a rule for the shortest method of working examples where both the multiplicand and multiplier have one or more ciphers on the right?

10. Show how multiplication may be performed by addition. 11. Explain the construction of the Multiplication Table, and illustrate its use in multiplying.

12. Why may the ciphers be omitted at the right of partial products?

13. Why commence multiplying the units' order in the multiplicand first, then the tens', and so on? Illustrate your answer by an example.

14. Multiply 8795 by 629, multiplying first by the tens, then by the hundreds, and last by the units.

15. Multiply 3572 by 483, commencing with the thousands of the multiplicand and hundreds of the multiplier.

16. Show that hundreds multiplied by hundreds will give ten thousands in the product.

17. Multiplying thousands by thousands, what order will the product be?

18. Name at sight the lowest order which each of the following examples will give in the product:

(1.) 8000 × 3000; 2000000 × 3000; 5000000000 × 7000. (2.) 40000 × 20000; 7000000 × 4000000.

19. What orders in 3928 can be multiplied by each order in 473, and not have any order in the product less than thousands?

111. STEP I.-To find how many times a number expressed by one figure is contained in any number not greater than 9 times the given number.

1. This is done by our knowledge of the Multiplication Table. Thus, if asked how many 3's in 15, we can answer at once five 3's.

Answer the following questions:

1. How many 2's in 4? In 8? In 12?

5. How many 6's in 8, and what remaining? In 13? In 26? In 37?

In 45? In 32?

6. How many 3's in 4, and what remaining? In 7? In 11? In 16 ? In 14? In 25?

2. The method of finding how many times one number is contained in another by using the Multiplication Table is called Division.

3. The number we divide is called the Dividend, and the number by which we divide is called the Divisor.

4. The number which tells how many times the divisor is contained in the dividend is called the Quotient, and what is left of the dividend after the division is performed is called the Remainder.

112. The sign

stands for the words, how many

in,

or divided by. Thus, 93 is read, How many 3's in 9, or

9 divided by 3.

Express on your slate with the sign ÷ each of the following questions, and then give the answer:

Read and give the answer for each of the following: 5. 248. 40 8. 16 ÷ 8. 568. 168. 728. 98. 278. 198.

6. 186. 24 ÷ 6.

488.

366. 48 6. 126. 306.

42 6. 54÷ 6. 14 ÷ 6.

7. 279. 45 9. 99. 369. 549. 819. 63 9. 189. 119. 30 ÷ 9.

113. STEP II.-To apply the Multiplication Table in finding at sight how many times a number expressed by one figure is contained in any number not greater than 9 times the given number.

Pursue the following course:

1. Write on your slate in irregular order the products of the Multiplication Table, commencing with the products of 2. Write immediately before, the number whose products you have taken; thus,

2)10 2)4 2)14 2)6 2)12 2)16 2)8

2. Write under the line from memory the number of 2's in 10, in 4, in 14, etc. When this is done, erase each of these results, and rewrite and erase again and again, until you can give the quotients at sight of the other two numbers.