MULTIPLICATION. 10. A farm-house is worth $3246, the farm is worth 3 times as much plus $1200, and the stock is worth twice as much as the house, less $1875. What is the value of the whole, and of the farm and stock? 11. What is the difference in the cost of 48 horses at $184.50 each, and of 130 sheep at $4.80 a head? 12. Bought 150 barrels of flour for $1150, and finding 25 barrels of it worthless, sold the remainder at $9 a barrel. Did I gain or lose, and how much? Complete the following equations: 13. (142405) x (1000850) - 5000 = ? 14. (97 × 1000) (75 x 500420) + 1500? 115. 1. How many 4's are 12? Are 16? Are 24? 2. How many lots, of 5 acres each, in 20 acres? 3. How many 5's in 15? In 30? In 35? In 50? 4. How many barrels, each holding 3 bushels, will be required for 18 bushels of apples? 21 bushels? 5. How many times can 6 yards of cloth be taken from a piece containing 30 yards. 6. How many times can 6 cents be taken from 23 cents, so as to have 5 cents remaining? 7. Distribute $28 equally among 7 men. dollars will each receive? How many Do you find how many times 7 men are contained in $28, or do you find one of 7 equal parts of $28? 8. How do you find one of 8 equal parts of a number? Of 9 equal parts? Of 6 equal parts? Of 48? 9. What is one of 4 equal parts of 40? Of 36? Of 48? 10. What is one of 6 equal parts of 30? Of 42? 11. What is one of 7 equal parts of 56 pounds? 12. How many times 8 cents are 48 cents? Is the result a concrete or an abstract number? 13. What is one of 8 equal parts of 48 cents? Is the result a concrete or an abstract number? DEFINITIONS. 116. Division is the process of finding how many times one number is contained in another of the same kind, or of finding one of the equal parts of a number. 117. The Dividend is the number to be divided. 118. The Divisor is the number by which to divide. 119. The Quotient is the result of the division, and shows how many times the dividend contains the divisor. The division is said to be exact when there is no remainder. The part of the dividend remaining when the division is not exact is called the Remainder, and must always be less than the divisor. 120. The Sign of Division is. It is read divided by. It shows that the number before it is to be divided by the one after it; thus 54 ÷ 9 is read 54 divided by 9. 121. Division is also indicated by placing the dividend above the divisor with a line between them; thus, 2 is read 72 divided by 8. 122. PRINCIPLES.-In finding how many times one number is contained in another: 1. The divisor and dividend are like numbers, and the quotient an abstract number. In finding one of the equal parts of a number: 2. The dividend and quotient are like numbers, and the divisor an abstract number. 3. The dividend is equal to the product of the divisor by the quotient, plus the remainder. 1. Divide by 2, from 2 in 2 to 2 in 24. OPERATION.-2 in 2, once; 2 in 4, twice; 2 in 6, 3 times; 2 in 8, 4 times; 2 in 10, 5 times, and so on to 2 in 24, 12 times. In the same manner, divide 2. By 3, from 3 in 3, to 3 in 36. 3. By 4, from 4 in 4, to 4 in 48. 4. By 5, from 5 in 5, to 5 in 60. 5. By 6, from 6 in 6, to 6 in 72. 6. By 7, from 7 in 7, to 7 in 84. 7. By 8, from 8 in 8, to 8 in 96. 8. By 9, from 9 in 9, to 9 in 108. 9. By 10, from 10 in 10, to 10 in 120. The pupil may reverse the above; thus, 2 in 24, 12 times; 2 in 22, 11 times; 2 in 20, 10 times, and so on. Also combine the two; thus, 3 in 3, once; 3 in 6, twice, 2 in 6, 3 times; 3 in 12, 4 times, 4 in 12, 3 times; and so on to 3 in 36, 12 times, 12 in 36, 3 times. 124. Division may also be regarded as a short method of performing several subtractions of a number. Thus, 24-618; 186 = 12; 12 6 6; 6-60. We have performed four subtractions of 6, hence there are four 6's in 24, or 6 is contained in 24, 4 times. 125. Since one number is contained in another as many times as it is a factor of the other, division may be regarded as the reverse of multiplication. In Multiplication, both factors are given to find the product; in Division, one factor and the product (answering to the dividend) are given to find the other factor, which answers to the quotient. Thus, 6×424, the factor 6 being taken 4 times; hence there are four 6's in 24, or 6 is contained in 24, 4 times. 126. The Object of Division is twofold. First. To find how many times one number is contained in another of the same kind. Ex. At 5 cents each, how many pencils can be bought for 20 cents. Since 5 cents taken 4 times equals 20 cents (5 × 4=20), it follows that 5 cents is contained in 20 cents 4 times. ANALYSIS. As many pencils can be bought for 20 cents, as 5 cents are contained times in 20 cents, which are 4 times. Hence, etc. 127. Second. To separate a given number into as many equal parts as there are units in another. Ex. If 4 pencils cost 20 cents, what is the cost of 1 pencil? Since 5 cents taken 4 times equals 20 cents, it follows that 5 cents is one of the four equal parts of 20 cents (5+5+5+5=20), and we say one-fourth of 20 cents is 5 cents. ANALYSIS. Since 4 pencils cost 20 cents, 1 pencil costs one-fourth of 20 cents, which are 5 cents. 128. The equal parts into which a unit or whole thing is divided are called fractions. 129. The names of these equal parts of a unit vary according to the number of these parts; thus, one-half is one of two equal parts, one-third is one of three equal parts into which the whole thing or number is divided. |