5. What difference will it make to the man who pays interest on $100 for 1 year, whether it be computed by days, or according to true rule in page 165? DISCOUNT. Discount is an abatement of a certain part of a debt, when the debt is paid before it becomes due. For instance; suppose that A is bound to pay B $106, in one year from the present time; but B, wanting the money now, agrees to receive $100 for the debt, on condition of present payment: in this case, $100 is the present worth of the debt, and $6 is the discount. The present worth of any debt due at a future period, is that sum of money, which, if put at interest, would amount to the debt by the time it becomes due. Therefore, when the interest is 5 per cent., that is, 15 of the principal, then the discount is 15 of the principal RULE FOR COMPUTING DISCOUNT. Multiply the principal by the number of cents found to be the interest of one dollar for the time, and divide the product by the number which results from adding 100 to the multiplier. The quotient will be the discount. For example, at the rate of 6 per cent., the discount on $124.25, due in 1 year and 10 months, is found thus 124.25 111)1366.75(13.21 44 76. What is the discount on $48.51, due in 3 years; the rate of interest being 5 per cent.? 77. What is the discount on $247, due in 1 year, the rate of interest being 6 per cent.? 78. What is the present worth of $150, due in 1 year, the rate of interest being 6 per cent.? Find the discount, and subtract it from the debt. 79. What is the present worth of $1640, due in 2 years, the rate of interest being 5 per cent.? 80. Find the difference between the discount and the interest of $100 for 1 year, the rate being 6 per cent. 81. Find the present worth of $75, due in 2 years and 9 months, [2 years], interest being 6 per cent. SECTION 24. 1. Suppose of a piece of broad-cloth to be worth $118.87; what is of the piece worth? What is the whole piece worth? 2. 11887 is of what number? 3. If the interest of $100 be $3.50 for 7 of a year, what is the interest of $100 for of a year? Then what would be the interest for 1 year? 4. If of an acre of land produce 133 bushels of potatoes, how many bushels does of an acre produce? How many bushels would 1 acre produce? 5. 9071 is of what number? 6. If a man earn $190 a year by working of the time, how much could he earn by working constantly? 7. $14 is 8 per cent. or 18 of what sum of money? SECTION 25. CHANGE OF THE TERMS OF FRACTIONS. The numerator and denominator of a fraction, are called the two terms of a fraction. These terms may be changed, and the fraction may still express the same quantity. For instance, the terms 2 and 3, in the frac tion, may be changed to 4 and 6, and the fraction will become, which is still equal to 3. 1. § is equal to how many twenty-fourths? Direction. 8-eighths are equal to 24-twenty-fourths; therefore, find & of 24, and this number will be the required numerator of 24⚫ 2. is equal to how many fourteenths? 3. Change to eighteenths and add to it. 4. is equal to how many forty-fifths? 5. Chauge to fortieths, and then take from it. SECTION 26. REDUCTION OF FRACTIONS TO LOWER TERMS. When a number can be found, that will divide both terms of a fraction, without a remainder, the two quotients arising from the division, will express the fractio reduced to lower terms. For example, both term P the fraction can be divided by 3, and the reduced fraction will be 2. Again, both terms of can be divided by 2, and the reduced fraction will be. Thus any fraction may be reduced to its lowest terms, by repeatedly dividing the terms, until no number will divide them both without a remainder. 6 600 6 18 12 1. Reduce each of the following fractions to its low est terms. . . 12. 12. 13. 18. 17. 14 2. Reduce each of the following fractions to its lowest terms. 10. 20. 300. 100. 15. 1300. Only once dividing the terms of a fraction, will reduce it to its lowest terms, if we use the greatest common divisor, that is, the greatest number that will divide both terms without a remainder. TO FIND THE GREATEST COMMON DIVISOR of two numbers,-Divide the greater number by the smaller, then divide the divisor by the remainder; and thus continue dividing the last divisor by the last remainder, till nothing remains. The divisor used last of all, will be he greatest common divisor. 3. Find the greatest common divisor of 91 and 117. 91)117(1 26)91(3 13)26/2 72 This operation is performed according to the direction above, and 13 is found to be the greatest common divisor; or the greatest number by which 91 and 117 can be divided without a remainder. 4. Find the greatest common divisor of 15 and 235. 5. Reduce to its lowest terms, by using the 189 est common divisor of the two terms. 92 great 6. Reduce to their lowest terms, 122, 138, and 223 SECTION 27. COMPOUND FRACTIONS. 425 A compound fraction arises from dividing a unit to a certain number of equal parts, and then dividing one of these parts into other equal parts. TO REDUCE A COMPOUND FRACTION TO A SIMPLE FRACTION,-Multiply all the numerators together for a new numerator, and all the denominators for a new denominator: then reduce the new fraction to its lowest terms. 1. Reduce of to a simple fraction. 2. of a water melon was divided equally among 6 boys. What fraction of the melon did 1 boy receive? 3. Reduce of to a simple fraction. 4. of an acre of land was divided into 4 equal lots. What fraction of an acre did 2 lots contain? 5. Reduce of to a simple fraction. Suggestion. 7 pence is of 1 shilling, and 1 shilling is 20 of £1. Therefore, 7 pence is 12 of 20 of £1. 10. Reduce 10 grains to the fraction of an ouncì; that is, reduce of to a simple fraction. 11. Reduce 3 nails to the fraction of a yard. 6 16. Reduce of 33 of to a simple fraction. When the lower denominations of a compound number are to be reduced to the fraction of a higher denomination,- First, reduce the given quantity to the lowest de nomination mentioned, and this number will be the numerator: then reduce a unit of the higher denomination, to the same denomination with the numerator, and this number will be the denominator. 17. Reduce 14s. 10d. 2qr. to the fraction of £1. 14s. 10d. 2qr. 12 17 8d. Num. 714 qr. 18. Reduce 2s. 7d. 1qr. to the fraction of £1. 19. Reduce 11d. 3qr. to the fraction of a pound. 20. Reduce 15s. Od. 3qr. to the fraction of a pound 21. Reduce 10d. 1qr. to the fraction of a shilling. 22. Reduce 2s. 9d. 3 qr. to the fraction of a pound. Direction. Find the number of fifths of a farthing in 2s. 9d. 3 qr., for a numerator; then find the number of fifths of a farthing in £1, for a denominator. 23. Reduce 8 pence to the fraction of a pound. 24. Reduce 5qt. Ipt. to the fraction of a bushel. 25. Reduce 9gal. 3qt. 1pt. to the fraction of 1hhd. 26. Reduce 6 rods 3yd. 2ft. to the fraction of a mile. 27. Reduce 35 seconds to the fraction of a day. When the fraction of a higher denomination is to be reduced to its value in whole numbers of lower denomination, Multiply the numerator by that number of the next lower denomination which is required to make a unit of the higher, and divide the product by the denominator, the quotient will be a whole number of the lower denomination, and the remainder will be the numerator of a fraction. Proceed with this fraction as before, and so on. 28. Reduce of £1 to its value in shillings &c. 2 20 7)40 5 5 760 8 4 4 7)16 22 Since of £ is the same as of 20 shillings, we find 2 of 20 shillings, in shillings and the fraction of a shilling; it is 5 shillings. Then, since of 1 shilling is the same as of 12 pence, we find of 12 pence;- it is 1⁄2 8 pence. Then, since of 1 penis the same as of 4 farthings, we find of 4 farthings;-it is 24 farthings. Thus by finding one de nomination at a time, we finally ob tain, 5s. 8d. 23 qr. ny 29. Reduce of £1 to its value in shillings &c. &c.? 32. Change £15 to pounds, shillings, pence, &c 33. Reduce of Icwt. to quarters, pounds, &c. |