54. Two Kinds of Division. Since 2 x $4 $8 ÷ $4 = 2, and $8 ÷ 2 = $4. That is, If the dividend and the divisor are concrete, they must be like numbers, and the quotient is abstract. For example, $8 $42. This is sometimes called measuring because the $8 is measured by the $4. If the dividend is concrete and the divisor abstract, the quotient is like the dividend. For example, $8÷2=$4. This is sometimes called partition, because the $8 is separated into 2 equal parts. 55. Check. If there is no remainder, the dividend is the product of the quotient and the divisor. To check the work in division, therefore, we multiply the quotient by the divisor; this product added to the remainder, if any, should equal the dividend. For example, 45 ÷ 5 = 9; then 5 × 945. Again 48 ÷ 5 = 9 with remainder 3; then 5 × 9 = 45, and 45 + 3 = 48. 56. Short Division. When the divisor is so small that the work may be done mentally the process is called short division. (1) Required to divide 342 by 3. 3)300+30 +12 100+10 + 4 = 114 If 342 is separated as shown, we can divide each part by 3. This is what we do mentally below. We write the numbers as shown; then 3 is contained in 3 once (that is, 100 times in 300); 3 is contained in 4 once with remainder 1 (that is, 10 times in 40 with remainder 10); this 1 is 10 units, and with the 2 makes 12. 3)342 Then 3 is contained in 12 four times. Therefore the quotient is 114. We think: "3 in 3, 1; in 4, 1; in 12, 4." 114 (2) Required to divide 42,364 by 7. 7)42364 Here 7 is not contained in 4, but 7 is contained in 42 six times. Then 7 is not contained in 3, so we write a 0 6052 beneath the 3. Then 7 is contained in 36 five times with remainder 1, and in 14 twice. Therefore the quotient is 6052. (3) Required to divide 35,282 by 9. Here 9 is contained in 35 three times with remainder 9)35282 8; in 82 nine times with remainder 1; in 18 twice; in 2 3920 no times with remainder 2. Therefore the quotient is 3920 with remainder 2, and this is usually written 39203. (4) Required to divide $27.53 by 4. 4)$27.53 $6.88 Dividing as before, we have $27÷ 4 = $6, with $3 remaining; $3.50 ÷ 4 = $0.8 with $0.30 remaining; $0.334 $0.08, with $0.01 remaining. Therefore the quotient is $6.881. = 54. Two Kinds of Division. Since 2 × $4 = $8, therefore $8 ÷ $4 2, and $8÷ 2 = $4. That is, = If the dividend and the divisor are concrete, they must be like numbers, and the quotient is abstract. For example, $8 $42. This is sometimes called measuring because the $8 is measured by the $4. If the dividend is concrete and the divisor abstract, the quotient is like the dividend. For example, $8÷2=$4. This is sometimes called partition, because the $8 is separated into 2 equal parts. 55. Check. If there is no remainder, the dividend is the product of the quotient and the divisor. To check the work in division, therefore, we multiply the quotient by the divisor; this product added to the remainder, if any, should equal the dividend. For example, 45 ÷ 5 = 9; then 5 × 9 = 45. Again 48 ÷ 5 = 9 with remainder 3; then 5 × 9 = 45, and 45 + 3 = 48. 56. Short Division. When the divisor is so small that the work may be done mentally the process is called short division. (1) Required to divide 342 by 3. 3)300 +30 +12 If 342 is separated as shown, we can 100 + 10 + 4 = 114 divide each part by 3. This is what we do mentally below. We write the num bers as shown; then 3 is contained in 3 once (that is, 100 times in 300); 3 is contained in 4 once with remainder 1 (that is, 10 times in 40 with remainder 10); this 1 is 10 units, and with the 2 makes 12. 3)342 Then 3 is contained in 12 four times. Therefore the quotient is 114. We think: "3 in 3, 1; in 4, 1; in 12, 4." 114 (2) Required to divide 42,364 by 7. Here 7 is not contained in 4, but 7 is contained in 42 7)42364 six times. Then 7 is not contained in 3, so we write a 0 6052 beneath the 3. Then 7 is contained in 36 five times with remainder 1, and in 14 twice. Therefore the quotient is 6052. (3) Required to divide 35,282 by 9. Here 9 is contained in 35 three times with remainder 9)35282 8; in 82 nine times with remainder 1; in 18 twice; in 2 3920 no times with remainder 2. Therefore the quotient is 3920 with remainder 2, and this is usually written 39203. (4) Required to divide $27.53 by 4. 4)$27.53 $6.88 $6, with $3 Dividing as before, we have $274 remaining; $3.50 ÷ 4 = $0.8 with $0.30 remaining; $0.33 ÷ 4 = $0.08, with $0.01 remaining. Therefore the EXERCISE 18 1. If 3 head of cattle cost $117, how much will 1 head cost? 2. It is 108 feet around a square. What is the length of each side? 3. There being 7 days in a week, how many weeks are there in 364 days? 4. There being 3 feet in a yard, how many yards are there in 432 feet? 5. There being 4 quarts in a gallon, how many gallons are there in 236 quarts? 6. It is 132 feet around a triangle whose sides are all equal. What is the length of each side? 7. At $9 a dozen, how many dozen cups and saucers can be bought for $1116? 8. At $4 a dozen, how many dozen handkerchiefs can be bought for $1152? 9. A dealer pays $1224 for 9 wagons. What is the average price? 10. A wholesale grocer pays $1728 for a shipment of chocolate at $3 per dozen cans. How many dozen cans does he buy? 11. A wholesale grocer pays $4320 for some cocoa at $6 per dozen cans. How many dozen cans does he buy? 12. A hotel pays $13.20 for 8 dozen packages of breakfast food. How much does it pay per dozen? 13. If a grocer pays $12.15 for 9 dozen packages of breakfast cereal, how much does he pay per dozen? 14. If a grocer pays $19.84 for 8 dozen tins of biscuit, how much does he pay per dozen? |