ART. 28. 1. If 1 orange cost 2 cents, what will 3 cost? ANALYSIS.-Three oranges will cost 3 times as much as one. That is, 2 cents taken 3 times: 2+2+2=6. Ans. 2. If 1 lemon cost 3 cents, what will 4 lemons cost? 3+3+3+3=12. Ans. 3. In an orchard there are 4 rows of trees, in each row 21 trees: how many trees in the orchard? SOLUTION.-1ST. By writing 21 four times, as in the margin, and adding, the whole number of trees is 84. 1st row, 21 trees. 2d row, 21 trees. 3d row, 21 trees. 4th row, 21 trees. 84 2D. Instead of writing 21 four times, write it once, place the number 4 under, it being the number of times 21 is to be taken, and say, 4 times 1 (unit) are 4 (units), which put in units' place: 21 then, 4 times 2 (tens) are 8 (tens), to be put in tens' place; the result is 84, the same as found by addition. The latter method is termed Multiplication. DEFINITIONS. 4 84 Multiplication is a short method of Addition, when the numbers to be added are equal. Multiplication is also taking one number as many times. as there are units in another. The number to be taken is the multiplicand. The number denoting how many times the multiplicand is taken, is the multiplier. The result is the product. Thus, 4 times 5 are 20; 5 is the multiplicand, 4 the multiplier, and 20 the product. The multiplicand and multiplier are together called factors, because they make or produce the product. REVIEW.-28. What is multiplication? What is the multiplicand? What does the multiplier denote? What is the product? What are the multiplicand and multiplier together called? Why? REM. When is multiplication termed simple? 3 3d Bk. 33 REMARK. When the multiplicand is of one denomination, the operation is called Simple Multiplication. ART. 29. The Sign X, read multiplied by, or times, denotes that the numbers between which it is placed are to be multiplied together. Thus, 4×3=12, shows that 4 multiplied by 3, or 4 times 3, are 12, or equal 12. In the table, the sign X, may be read times: thus, 2 times 2 are 4; 2 times 3 are 6; and so on. 9X 0= 0 10X 0= 0 11X 0 0 12X 0= 0 9X11- 99 10X11-110 9X12 108 11x11-121 12X11-132 10x12=120 11x12=132 | 12×12=144 ART. 30. Here are several rows of stars. Counting upward, there are 5 rows of 4 stars each; 5 rows contain 5 times as many as 1 row; 5 times 4 stars are 20 stars. ***** Counting across, there are 4 rows of 5 stars each; 4 rows contain 4 times as many as one row; 4 times 5 stars are 20 stars, the same as before. ** Hence, the product of two numbers is not altered by changing the order of the factors. What is the difference between 6×5 and 5×6? 9X8 and 6×12? 10×12 and 12X10? REMARK 1.—The product is always of the same kind or denomi nation as the multiplicand. PRINCIPLES AND EXAMPLES. 1. What is the product of 5 multiplied by 3; that is, what is the amount of 5 taken 3 times? Ans. 15. 3 times 5 are equal to 5+5+5=15. Here, the multiplicand, 5, is an Abstract number; that is, it denotes no particular thing, and the product, 15, is also abstract. REVIEW.-29. What does the sign ( X ) of multiplication denote? Give an example. 30. Does it alter the product to make either of the factors the multiplier? Illustrate this. 2. What will 3 yards of muslin cost, at 5 cents a yard? ANALYSIS.-Three yards will cost three times the price of one yard, that is, 5 cents taken 3 times, which is 5 cts.+5 cts.+5 cts.-15 cts. Ans. Here, the multiplicand is a Concrete number; that is, it denotes some particular thing, as Cents; the product, 15 cents, denoting the same thing, is also concrete. Hence, in general, if the multiplicand is money, the product will be money of the same name; if it is pounds, the product will be pounds, &c. REM. 2. The multiplier shows the number of times the multiplicand is to be taken; hence, it must always be considered an abstract number. In Example 2, 5 cents are not multiplied by 3 yards, but are taken three times; as three yards will cost three times the price of 1 yard. To speak of multiplying 3 dollars by 2 yards, or 25 cents by 25 cents, is as absurd as to propose to multiply 3 apples by 2 potatoes. ART. 31. When the Multiplier does not exceed 12. EXAMPLES. 1. How many yards of cloth are there in 3 pieces, each containing 123 yards? OPERATION. 123 multiplicand. 3 multiplier. SOLUTION. Having placed the multiplier under the multiplicand, as in the margin, say, 3 times 3 units are 9 units, which write in units' place; 3 times 2 tens are 6 tens, which write in tens' place; 3 times 1 hundred are 3 hundreds, which write in hundreds' place. 369 product. 2. What will 2 houses cost at $231 each? 3. What will 3 horses cost at $132 each? Ans. $462. Ans. $396. 4. What is the product of 201×4? Ans. 804. Ans. 6903. 5. What is the product of 2301X3? REVIEW.-30. REM. 1. What is the denomination of the product? Show that when the multiplicand is an abstract number, the product is also abstract. Show that when the multiplicand is a concrete number, the product must be concrete. REM. 2. What does the multiplier show? What must it be considered? OPERATION. $43 6. At $43 an acre, what will 5 acres of land cost? SOLUTION.-Say, 5 times 3 (units) are 15 (units); write the 5 (units) in units' place, and carry the 1 (ten); then, 5 times 4 (tens) are 20 (tens), and 1 (ten) carried make 21 (tens), which write as in the margin. 7. What is the product of 347 X12? The product of each of the figures by 12 is known from the Mul. Table; multiply then by 12, as by a single figure. PROOF.-Since 11 times 347 and 1 time 347 are equal to 12 times 347, therefore, take 1 from the multiplier, and use the remainder instead of the former multiplier; to the product, add the multiplicand; the result must be the same as the first product. RULE 5 $215 347 4164 347 11 3817 347 4164 For Multiplication.-1. Write the multiplicand, with the multiplier under it, and draw a line beneath. 2. Begin with units; multiply each figure of the multiplicand by the multiplier, and carry as in addition. PROOF.—Take 1 from the multiplier, and use the remainder for a multiplier; to the product thus found, add the multiplicand; if the sum equals the first product, the work is correct. REM.-Begin at the right hand to multiply, so that the excess of tens a any lower order may be carried to the order next higher. REVIEW.-31. How are the numbers written, Rule? begin to multiply? How do you multiply? How do you prove the operation? REM. Why begin at the right hand to multiply? Where do you |