PROBLEM III. To change an Improper Fraction to a Whole or Mixed Number. RULE. Divide the numerator by the denominator, and the quotient will be the value of the fraction. EXAMPLES. 1. In 45 of a dollar, how many dollars? of a dollar are equal to 1 dollar, and 6 is contained in 45, 7 times and of another time; therefore the answer is 7 dollars 7 dollars. I. Divide the whole number by the denominator of the fraction, (when it can be done without a remainder,) and multiply the quotient by the numerator; or, II. Multiply the whole number by the numerator of the fraction and divide the product by the denominator. EXAMPLES. 1. What is the product of 48 multiplied by ?? By this example we see, there are two ways of multiplying a whole number by a fraction, and that both methods produce the same result. Thus, by the first method, we get of 48, and this repeated 3 times is evidently equal to 2, for 3 times of any number is equal to of that number. By the second method, we repeat 48, 3 times, and then take of that product, which is the same as 3 times of 48. 2. At 25 dollars per acre, what is the cost of 15 of an acre of land? Ans. 23 dolls. 3. If a ship sail 246 miles a day, how far will she sail in 7 of a day? 4. How much is of $1845,56 ? 5. Multiply 400 by 6. Multiply 750 by Ans. 191 miles. Ans. 150. Ans. 450. 7. The interest of $750 for 1 year, is $45; what is the interest on the same sum for 5 months, or of a year? Ans. $18,75. Note. If the multiplier of any sum be greater than a unit or 1, the multiplicand will be increased as many times as the multiplier is greater than a unit; that is, the multiplicand will be taken as many times as the multiplier contains units. But when the multiplier is a fraction or part of a unit, the product will be only a part of the multiplicand. Hence in multiplying by a proper fraction, the product is always less than the multiplicand, as will be seen by the preceding examples. PROBLEM V. To Multiply a Fraction by a Whole Number. RULE. Multiply the whole number and the numerator of the fraction together, and write the product over the denominator; and if it produce an improper fraction, change it to a whole or mixed number, by Prob. 3. EXAMPLES. 1. If a man spend & of a dollar a day, how much will he spend in 11 days? If he spend in 1 day, he will spend 11 times 5=55 in 11 days, and 55 of a dollar =91 dollars, the answer. 2. If 1 yard of cloth cost of a dollar, what will 15 yards cost? Ans. $9, 3. If a bushel of oats cost of a dollar, what will 23 bushels cost? Ans. $972. 4. A certain lot contains of an acre of land; how much land would 37 such lots contain? Ans. 273 acres. 5. If a bushel of potatoes cost of a dollar, what will 56 bushels cost? Ans. $16. Note. The process of multiplying a fraction by a whole number, may be shortened, thus: Divide the denominator of the fraction by the whole number, (when it can be done without a remainder,) and over the quotient write the numerator. 6. If a pound of sugar cost of a dollar, what will 20lb. cost. 6 59 Divide the denominator, 100, by 20, and the quotient is a new denominator; then write the numerator over it, and it becomes of a dollar=2} dollars, the answer. 7. If a pound of nails cost of a dollar, what will 11lb. cost? Ans. $1. 8. If a pound of butter cost of a dollar, what will 5lb. cost? Ans. $3. 9. At of a dollar per pound, what will 11lb. raisins come to ? PROBLEM VI. Ans. $3. To divide a Whole Number by a Fraction. RULE. Multiply the whole number by the denominator of the fraction, and divide the product by the numerator. EXAMPLES. 1. How many times is of a dollar contained in $9 ? 1 dollar is, and 9 dollars is 9 times as many; 9x4=38; and is contained in 36 as many times as 3 is contained in 35. 2. How many times is & contained in 16? Thus, 16 Ans. 12. 3. How many times is contained in 12? how many ? or 12÷3= Ans. 18. Ans. 48. Ans. 40. 4. How many times can I have in 27 ? 5. How many times is 17 contained in 34 ? 6. How many men can I divide 75 dollars among, so as to give each of a dollar? Ans. 100 men. examples, that The reason of number, 12 for Note. It will be seen by the 6 preceding the quotient is greater than the dividend. this is as follows. If we divide a whole example by 2, the quotient will be 6, which is equal to half the dividend; and if we divide it by 1, the quotient will be 12, for 1 is contained in any number twice as often as 2. Again, if we divide by 2, the quotient will be increased, for is contained in any number twice as often as 1; thus, 12 is 24 halves, and is contained in 24, 24 times. Hence when the divisor is less than a unit, it will be contained in the dividend a greater number of times. Therefore dividing a whole number by any proper fraction, the quotient will always exceed the dividend. PROBLEM VII. To Reduce any given Quantity to a Fraction of a higher Denomination of the same kind. RULE. 1. Reduce the given quantity to the lowest denomination mentioned, for a numerator. 2. Reduce 1 of the higher denomination to the same name, for a denominator. EXAMPLES. 1. What part of 5 yards is 3 yards? Thus, lyd. is of 5yds., and 3 yards are 3 times as much; 3 times is, the answer. 2. What part of 7lb. is 4lb.? Ans. 4. 3. What part of 17 cents is 9 cents? Ans. 17 4. What part of 18 dollars is 4 dollars? * Ans. 3. Note. Reduce all the fractions to their lowest terms. 5. What part of £15 is £6? Ans. 6. What part of 25 rods is 15 rods? Ans. 7. What part of 63 gallons is 9 gallons? Ans. 8. What part of 19 acres is 5 acres? 9. 18 inches is what part of 56 inches? 10. What part of £1 is 12s. 9d. 3qrs. ? Operation. 12s. 9d. 3qrs. 12 153 4 Numerator 615qrs. £1=20s. 12 240 Denominator 960qrs.=815=44. 11. What part of a shilling is 44d.? Ans. Ans. Ans. 9 12. What part of a pound Troy is 7 oz. 4pwt. Ans. 15 16. Ans. 126 1 Ans. Ans. 1. Ans. 9. 19. Reduce 6fur. 26rds. 11ft. to the fraction of a mile. 20. What part of 1 year is 7 weeks 1 day? 21. What part of a yard is 2qr. 2na.? Ans. . Ans.. Ans. yd. (Reduce 2qr. 2na. to nails, then reduce the nails to thirds, adding in the 2 thirds, for the numerator; then reduce 1 yard to thirds of nails, for the denominator.) 22. What part of a day is 11 hours 59 minutes? Ans. 4661 10080' 23. Reduce 4 shillings 6 pence to the fraction of a dollar or 6 shillings. 24. What part of the year had transpired the October, 1836, including that day? 25. What part of a dollar at 8 shillings, is 2 pence ? PROBLEM VIII. Ans. $3. 26th day of Ans.. shillings 8 Ans. $. To find the Value of a Fraction in Whole Numbers of less Denominations. RULE. 1. Multiply the numerator by the parts in the next lower denomination, and divide the product by the denominator. |