1. In 35 pieces of cloth, each measuring 277 yds. how many yards? Ans. 971 yds. I qr. 2. In 9 fields, each containing 14 acres, 1 rood, and 25 poles, how many acres ? Ans. 129 a. 2 roods, 25 rods. 3. In 6 parcels of wood, each containing 5 cords and 96 feet, how many cords? Ans. 34 cords, 64 feet. 4. A gentleman is possessed of 1} dozen of silver spoons, cach weighing 2 oz. 15 pwt, 11 grs., 2 dozen of tea-spoons, each weighing 10 pwt. 14 grs., and 2 silver tankards, each 21 oz. 15 pwt. Pray what is the weight of the whole ? Ans. 8 lb. 10 oz. 2 pwt. 6 grs. MULTIPLICATION OF VULGAR FRACTIONS. MULTIPLICATION WHEN ONLY THE MULTIPLICAND IS A FRACTION A man gave one child three quarters of a dollar, and another four times as much, how much did he give the last? A man has 12 barrels of wine, and takes a half pint from each 3 times, how many half pints does he take ? If a man has an ounce of silver, and takes 2 sixteenths from it 6 times, how many sixteenths does he take? How much is 4 times two sixths ? From the above examples it appears, that we can mul. tiply a fraction by a whole number, by multiplying its numerator. Let the pupil perform the following sums, first mentally, and then on the slate. 27 25 1. What is 9 times 2 How can a fraction be multiplied by a whole number? 47 100 In performing these sums on the slate, let the pupil use the signs, thus: Two twentieths multiplied by nine, equals eighteen twentieths; and is expressed by signs as follows: 22 × 9=18. There is another method, by which the value of a fraction is multiplied, by increasing the size of the parts expressed by the denominator. For example, when we wish to multiply by 2, the most common way is to multiply the numerator by 2, thus: X2= 1/2 But the same effect is produced, if we divide the denominator by 2, thus: ×2=4. It will easily be seen, that and are the same quantity. The only difference is, that in one case the unit is divided into 12 parts and 8 are expressed, and in the other case, the unit is divided into 6 parts, and 4 are expressed. In one case, we make twice as many pieces, and in the other we make them twice as large. When we multiply the numerator, the number of parts is What is the second method of increasing the value of a fraction? What is the difference between the two methods? multiplied, and when we divide the denominator the size of the parts is multiplied. If we multiply by 3, in what 'two ways can it be done? If we multiply the numerator, what is it that is multiplied? If we divide the denominator, what is it that is multiplied? Multiply by 3 in both ways, and tell what each method multiplies. RULE FOR MULTIPLYING WHEN ONLY THE MULTIPLICAND IS A FRACTION. Multiply the numerator, or divide the denominator by the multiplier. Let the following sums be performed, and explained as above. MULTIPLICATION WHERE ONLY THE MULTIPLIER IS A FRACTION. 1. If you have twelve cents, and give away a sixth of them to each of four children, how many cents do you give away? Ans. A sixth of twelve cents is two cents. Two cents What is the rule for multiplying when the multiplicand only is a frac given to each of four children would be eight cents given away. 2. If a man has fifteen cents, and gives a fifth of them to each of three children, how many does he give away ? Ans. One fifth of fifteen is three. Three times three is nine. He gives away nine cents. From the above examples it appears that when we mul. tiply by a fraction, we take a part of the multiplicand, and repeat it à certain nuniber of times. In the last case the man had fifteen cents, which is the multiplicand. We take a fifth of it and repeat it three times. 3. If a man had eighteen cents, and gave a ninth of them to six different boys, how many cents did he give away? In the above question, what is the multiplicand ? What part are you to take from it, and how often are you to re. peat it ? 4. If a man has twelve dollars, and gave a fourth of them to three different workmen, how many did he give away? What is the multiplicand? What part are you to take from it, and how often are you to repeat it? 5. How do you multiply twelve by three fourths ? Ans. We take a fourth of twelve and repeat it three times. One fourth of twelve is three. Three fourths are three times as much. Three times three is nine. 6. How do you multiply eight by three fourths ? 7. How do you multiply eighteen by three ninths ? 8. If you multiply twelve by three, do you make it larger or smaller ? If you multiply it by three fourths, do you make it larger or smaller ? Why is the multiplicand made smaller when you multi- . ply by three fourths ? Ans. Because we do not repeat the whole number, but only a fourth part of it; and this is repeated only three times, which does not make it as large a number as the multiplicand. 9. If you multiply eight by three, do you make it larger or smaller? If you multiply it by three fourths, do you make it larger or smaller ? Why? 10. Multiply fifteen by two thirds.. 11. Multiply twenty-four by five sixths. 12. Multiply thirty-two by three eighths. Multiplication has been defined, as repeating a number, as often as there are units in another number. In multiplying by a fraction, we take such a part of a number, as is expressed by the denominator, and repeat it as often as there are units in the numerator. Thus in multiplying 12 by we take a sixth part of 12, and repeat it 4 times, and the answer is 8. NOTE. The propriety of calling the number in the numerator units, is explained on page 40, where the distinction is shown between units that are whole numbers, and units that are fractions. It is shown also on page 57, where it appears that the numerator may be considered as whole numbers, divided by the denominator. In multiplying let the pupil use the signs thus: 12+6= 2. 2 X 36. Answer. In doing the above sum what part of 12 is taken? How often is it repeated? Is the product larger or smaller than the multiplicand? Multiply 12 by .. Is a proper or improper fraction? Is there a whole unit in ? Is the product larger or smaller than the multiplicand, when 12 is multiplied by ?? Why is it larger when multiplied by and smaller when multiplied by 3? Let the following sums be stated thus; 16X. One fourth of 16 is 4, and two fourths, is twice as much, or 8. Multiply 16 by by 2 by 2 18 How do we multiply by a fraction? |