3. How many times is contained in 12? how many? or 12÷}= Ans. 18. Ans. 483. 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? Note. It will be seen by the 6 preceding the quotient is greater than the dividend. this is as follows. Ans. 100 men. examples, that The reason of If we divide a whole number, 12 for 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, 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.? 3. What part of 17 cents is 9 cents? 4. What part of 18 dollars is 4 dollars? 5. What part of £15 is £6 ? Ans. 4. Ans.. Ans.. Note. Reduce all the fractions to their lowest terms. 6. What part of 25 rods is 15 rods? 7. What part of 63 gallons is 9 gallons? 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.=815-4 11. What part of a shilling is 44d.? Ans. 12. What part of a pound Troy is 7 oz. 4pwt. 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. 23na. ? (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? 10080 Ans. 4661 23. Reduce 4 shillings 6 pence to the fraction of a dollar or 6 shillings. Ans: $3. 24. What part of the year had transpired the 26th day of October, 1836, including that day? Ans.. 25. What part of a dollar at 8 shillings, is 2 shillings 8 pence ? Ans. $. PROBLEM VIII. 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 dénomination, and divide the product by the denominator: 2. Multiply the remainder, if any, by the next lower denomination, and divide by the denominator, as before; and the several quotients will be the answer. EXAMPLES. How much is lb. avoirdupoise? How much is lb. ? How much is How much is of a shilling? of a shil How much is 3lb.? How much is lb.? ling? How much is of a shilling? How much is? How much is of a shilling? 1. What is the value of of a pound sterling? 2. What is the value of of a pound sterling? 3. What is the value of 3 of a shilling? 4. What is the value of of a shilling? 5. What is the value of 3 6. What is the value of Ans. 18s. 4d. Ans. 10 pence 14qrs. of a pound Troy? Ans. 9oz. of a pound avoirdupoise? Ans. 12oz. 12 dr. 7. Reduccof a hundred weight to its proper quantity? Ans. 3qr. 3lb. 1oz. 12 dr. 8. What is the value of § of an Ell English? 9. What is the value of of a yard? Ans. 2qr. 3 na. 10. How much is 15 of a hogshead of wine? Ans. 35gals. Ans. 6fur. 26rd. 3yds. 2ft. of a day? Ans. 12h. 55min. 23 sec. 13. How much is 32 of an acre? Ans. 3 roods. 25 rods. Questions II. How is a Vulgar fraction represented? III. What is the number below the line called? and why? IV. What is the number above the line called? and why? In Division, what is the denominator? and what is the numerator? V. What is a simple or proper fraction? What is an improper fraction? What is a mixed nnmber? How may a whole number be expressed as a fraction? Prob. I. How do you reduce fractions to their lowest terms? What are the terms of a fraction? II. How do you change a whole or mixed number to an improper fraction? III. How do you change an improper fraction to a whole or mixed number? IV. How do you multiply a whole number by a fraction? V. How do you multiply a fraction by a whole number? VI. How do you divide a whole number by a fraction? VII. How do you reduce a given quantity to the fraction of a greater denomination of the same kind? VIII. How do you find the value of a fraction in whole numbers of less denominations? DECIMAL FRACTIONS. 1. A Decimal* Fraction is that whose denominator is always 1 with a cipher, or a number of ciphers annexed to it. Thus, fo, 180, 1800, &c. &c. 5 6 56 2. The integer is always divided into 10, 100, 1000, &c. equal parts. Therefore the denominator is always 10, 100, 1000, &c. 3. The true value of a decimal fraction is expressed by writing the numerator only with a point before it. Thus, is written,,5; 256,25; 6456,645. 1000' 4. If the numerator has not so many places of figures as the denominator has ciphers, we must put as many ciphers on the left hand as will make up the defect. Thus, Too is written,06 and Too is written ,006, &c. 5. The point prefixed is called a separatrix. * So called from the Latin word decem, which signifies ten. 6. Each figure takes its value by its distance from the unit's place; the first figure on the right hand of units, or the separatrix, signifies so many tenths; the second so many hundredths; the third so many thousandths, &c., thus decreasing in a tenfold proportion from the left towards the right hand. 7. Ciphers placed at the right hand of a decimal fraction do not alter its value, since every significant figure continues to possess the same place. Thus, ,5,50,500, &c. are all of the same value, and each equal to or 8. Every cipher placed at the left hand of a decimal fraction decreases its value tenfold, by removing each significant figure farther from the place of units. Thus, ,5 in the first place is 5 tenths; ,05 in the second place is 5 hundredths; ,005 in the third place is 5 thousandths, &c. See the following 9. The magnitude of a decimal fraction depends mostly on the first, or left hand figure, which, if it be less than ,9, we may extend to an infinite number of figures, and it will not be equal to ,9. Thus,,899999 is not equal to ,9. 10. Decimals are read in the same manner as whole num |