is evident that the product of 8×6=48, the number of units in all the lines. But let us first connect the lines in sets of 2 each, as on the right; there will then be in each set 8x2=16; or 16 units in each set. But there are 3 sets; hence, 16x3=48, the number of units in all the sets. If we divide the lines into sets of 3 each, as on the left, the number of units in each set will be equal to 8x3=24, and there being 2 sets, 24×2=48, the whole number of units. As the same may be shown for all numbers we have the following RULE. When the multiplier is a composite number, multiply by each of the factors in succession, and the last product will be the entire product sought. EXAMPLES. 1. Multiply 327 by 12. The factors of 12, are 2 and 6, or they are 3 and 4, or they are 3, 2 and 2: for, 2×6=12, 3×4=12, and 3×2×2=12. 2. Multiply 5709 by 48; the factors being 8 and 6, or 16 and 3. 3. Multiply 342516 by 56. 4. Multiply 209402 by 72. 5. Multiply 937387 by 54. 6. Multiply 91738 by 81. 7. Multiply 3842 by 144. APPLICATIONS. Ans. Ans. 19180896. Ans. 50618898. Ans. 553248. 1. There are ten bags of coffee, each containing 48 pounds how much coffee is there in all the bags ? Ans. lbs. 2. There are 20 pieces of cloth each containing 37 yards, and 49 other pieces, each containing 75 yards: how many yards of cloth are there in all the pieces? Ans. 4415 yards. Ans. hours. 3. There are 24 hours in a day, and 7 days in a week: how many hours in a week? 4. A merchant buys a piece of cloth containing 97 yards, at 3 dollars a yard: what does the piece cost him? 5. A farmer bought a farm containing 10 fields; three of the fields contained 9 acres each; three other of the fields 12 acres each; and the remaining 4 fields, each 15 acres: how many acres were there in the farm, and how much did the whole cost at 18 dollars an acre? The farm contained 123 acres. Ans. {It cost 2214 dollars. 6. A merchant bought 49 hogsheads of molasses, each containing 63 gallons: how many gallons of molasses were there in the parcel ? Ans. gallons. 7. Suppose a man were to travel 32 miles a day: how far would he travel in 365 days? Ans. 11680 miles. 8. In a certain city, there are 3751 houses. If each house on an average contains 5 persons, how many inhabitants are there in the town? Ans. inhabitants. 9. When a person sells goods he generally gives with them a bill, showing the amount charged for them, and acknowledging the receipt of the money paid; such bills are usually called Bills of Parcels. BILLS OF PARCELS. James Johnson New-York, Oct. 1, 1838. 4 Chests of tea, of 45 pounds each, at 1 doll. a pound. 3 Firkins of butter at 17 dolls. per firkin 4 Boxes of raisins at 3 dolls. per box 36 Bags of coffee at 16 dolls. each 14 Hogsheads of molasses at 28 dolls. each. Received the amount in full, Amount 1211 dollars. W. Smith. Hartford, Nov. 1, 1837. James Hughes 27 Bags of coffee at 14 dolls. per bag 18 Chests of tea at 25 dolls. per chest 75 Barrels of shad at 9 dolls. per barrel 87 Barrels of mackerel at 8 dolls. per barrel 67 Cheeses at 2 dolls. each 59 Hogsheads of molasses at 29 dolls. per hogshead, Amount 4044 dollars. per James Cross. Received the amount in full, for W. Jones, DIVISION OF SIMPLE NUMBERS. § 28. Charles has 12 apples, and wishes to divide them equally between his four brothers. OPERATION. 12 4 8 1st remain. 4 4 2d remain. He gives one to each, which takes 4. Subtracting 4 from 12, 8 remains. He then gives another to each, which takes 4 more. Subtracting this 4 from 8 leaves 4. He then gives one more to each, which takes all his apples, and leaves nothing. He has then divided them equally, and found that 12 contains 4, three times, for he has three times subtracted 4 from 12. Suppose he had 28 apples and wished to divide them equally among 8 boys. Giving each one, would take 8 and leave 20. Giving each one, a second time, would take 8 and leave 12. Giving each one, a third time, would take 8 and leave 4. Hence, 8 is contained three times in 28, and there are 4 over. 4 0 3d remain. OPERATION. 28 8 20 1st remain. 8 12 2d remain. 8 By continued subtraction we can always find how many times one number is contained in another, and also, what is left when it is not contained an exact number of times. 4 3d remain. We can arrive at the same result by a shorter method, called Division. DIVISION teaches the manner of finding how many times a less number is contained in a greater. It is a short method of subtraction. The less number is called the divisor. The greater number is called the dividend. The number expressing how many times the dividend contains the divisor, is called the quotient. If there is a number left, it is called the remainder, which is always less than the divisor. There are three signs used to denote division. They are the following. 18÷4 expresses that 18 is to be divided by 4. 18 4 expresses that 18 is to be divided by 4. 4) 18 expresses that 18 is to be divided by 4. When the last sign is used, a curved line is also drawn on the right of the dividend to separate it from the quotient, which is generally set down on the right. Q. When Charles divides 12 apples equally, among his four brothers, how many does he give to each? How many times does 12 contain 4? In dividing 28 apples equally among 8 boys, how many does each receive? How many remain? Which number is the dividend? Which the divisor? Which the quotient? Which the remainder? What does division teach? What is the less number called? What is the greater called? What is the answer called? What is the number called which is left? Is this number greater or less than the divisor ? How many signs are there in division? Make them? § 29. Let the following table be committed to memory. It is read 2 in 2, 1 time; 2 in 4, 2 times, &c. 5 in 5 1 time 5 in 10 2 times 5 in 15 3 times 5 in 20 4 times 5 in 25 5 times 5 in 30 6 times 5 in 35 7 times 5 in 40 8 times 5 in 45 9 times 6 in 6 1 time 6 in 12 2 times 6 in 18 3 times 6 in 24 4 times 6 in 30 5 times 6 in 36 6 times 6 in 42 7 times 6 in 48 8 times 6 in 54 9 times 7 in 7 1 time 7 in 14 2 times 7 in 21 3 times 7 in 28 4 times 7 in 35 5 times 7 in 42 6 times 7 in 49 7 times 7 in 56 8 times 7 in 63 9 times 11 in 11 1 time 11 in 22 2 times 8 in 8 1 time 88 8 times 9 in 45 5 times 9 in 54 6 times 9 in 63 7 times 9 in 72 8 times 9 in 81 9 times 10 in 10 1 time 10 in 20 2 times 10 in 30 3 times 10 in 40 4 times 10 in 50 5 times 10 in 60 6 times 10 in 70 7 times 10 in 80 8 times 10 in 90 9 times QUESTIONS. 11 in 36÷ 9 or 36 72÷ 8 or 12 54÷ 9 or 54 60÷ 5 or 60 96÷12 or 99 108÷12 or 108 12 = = = 1. If 12 apples be equally divided between 4 boys, how many will each have? How many times is 4 contained in 12? 2. If 24. peaches be equally divided between 6 boys, how many will each have? How many times is 6 contained in 24? 3. A man has 32 miles to walk, and can travel 4 miles an hour, how many hours will it take him? 4. A farmer receives 28 dollars for 7 sheep: how much was that apiece? |