3. How many times is contained in 12 ? or 12-3= how many ? Ans. 18. 4. How many times ģ can I have in 27 ? Ans. 483. 5. How many times is 7 contained in 34 ? Ans. 40. 6. How many men can I divide 75 dollars among, so as to give each of a dollar ? Ans. 100 men. Note. It will be seen by the 6 preceding examples, that the quotient is greater than the dividend. The reason of this is as follows. 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 j, the quotient will be increased, for 1 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. 3 times į is š Ans. 9 1. Whạt part of 5 yards is 3 yards ? Thus, lyd. is } of 5yds., and 3 yards are 3 times as much; the answer. 2. What part of 71b. is 41b. ? Ans. 4. 3. What part of 17 cents is 9 cents ? 17 4. What part of 18 dollars is 4 dollars ? Ans. 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. Ans. to 12 Ans. 15 16 Ans. š. 8. What part of 19 acres is 5 acres ? Operation. £1=20s. 12 153 240 4 4 Numerator 615qrs. Denominator 960qrs.=%26= 11. What part of a shilling is 41d.? Ans. 12. What part of a pound Troy is 7 oz. 4pwt. Ans. 13. What part of lcwt. is 2qr. 16lb. ? Ans. 14 14. What part of a yard is 3qr. 3na. ? 15. What part of a hogshead is 35gal. 2qts. ? Ans. 126 16. What part of a furlong is 6rd. 3yds. 2 feet? 17. What part of an acre is 3 roods 21rds. ? Ans. 160 18. Reduce 54 gallons to the fraction of a hogshead. Ans. 19. Reduce 6fur: 26rds. 11ft. to the fraction of a mile. Ans. Á What part of 1 year is 7 weeks 1 day? Ans. 79 21. What part of a yard is 2nr. 2 na. ? Ans. syd. (Reduce 2qr. 2na. to nails, then reduce the nails to thirds, adding in the 2 thirds, for the numerator ; then reduce i yard to thirds of nails, for the denominator.) 22. What part of a day is 11 hours 54 minutes ? 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 26th day of October, 1836, including that day? Ans. bi 25. What part of a dollar at 8 shillings, is 2 shillings 8 pence? Ans. $ 20. Ans. 4661 Àns: $1. PROBLEM VIII. To find the Value of a Fraction in Whole Numbers of less Denominations. RULE. 1. Multiply the numerator by the parts in the next lower dénomination, and divide the product by the denominator: 2. Multiply the remainder, if any, by the next lower de. nomination, and divide by the denominator, as before ; and the several quotients will be the answer. EXAMPLES. S. How much is klb. avoirdupoise? How much is llb. ? How much is klb.? How much is klb. ? How much is şib.? How much is lb..? How much is of a shilling? How much is į of a shilling? How much is of a shilling? How much is of a shilling ? How much is s? How much is į of a shilling? 1. What is the value of į of a pound sterling ? Operation. £1=20s. and į of £l is Numerator=7 same as į of 20s.; and to Shillings in £l=x20 gets of 20s. ye multiply the Denominator=8)140(17 numerator of the fraction, 7, and 20, together; and the 8 product divided by the de60 nominator, g, gives 17$. and 56 of another shilling remain 4 ing; then of a shilling is Pence in Ishil. =12 of 12 pence, and to get of 12d. we multiply the numer8)48(6 ator, 4, and 12, together, and 48 divide by the denominator, gives 6 pence. Ans. 17s. 6ų, 2. What is the value of 11 of a pound sterling ? Ans. 18s. 4d. 3. What is the value of g of a shilling ? Ans. 41. 4. What is the value of of a shilling ? Ans. 10 pence 1{qrs. 5. What is the value of 34 of a pound Troy? Ans. 9oz. 6. What is the value of of a pound avoirdupoise ? Ans. 12oz. 12dr. 7. Reducc of a hundred weight to its proper quantity ? Ans. 3qr. 3lb. loz. 12 dr. 8. What is the value of g of an Ell English ? Ans. 2qr. 3 na. 9. What is the value of of a yard ? Ans. 3qr. 1&na. -d. 10. How much is of a hogshead of wine ? Ans. 35gals. 11. How much is of a mile ? Ans. 6fur. 26rd. 3yds. 2ft. 12. How much is of a day? Ans. 12h. 55min. 23 13 sec. 13. How much is of an acre ? Ans. 3 roods. 25 rods. Questions I. What are Fractions? From what terms of a fraction ! do all fractions arise ? Of how many II. How do you change a whole or kinds are fractions ? mixed number to an improper fraction ? II. How is a Vulgar fraction repre III. Ilow do you change an improper sented ? fraction to a whole or mixed number? III. What is the number below the IV. How do you multiply a whole line called ? and why? number by a fraction ? IV. What is the number above the line V. How do you multiply a fraction by called ? and why? a whole number? In Division, what is the denominator? VI. How do you divide a whole numand what is the numerator ? ber by a fraction ? V. What is a simple or proper frac VII. How do you reduce a given quantion? What is an improper fraction ? tity to the fraction of a greater denomiWhat is a mixed nnmber? How may a nation of the same kind ? whole number be expressed as a frac VIII. How do you find the value of a tion? fraction in whole numbers of less de. Prob. I. How do you reduce fractions nominations? to their lowest terms ? What are the DECIMAL FRACTIONS. 1. A Decimal* Fraction is that whose denominator is al. ways 1 with a cipher, or a number of ciphers annexed to it. Thus, fo, TÖb To 00 &c. &c. 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, Po is written, ,5; 1,,25; 1040, ,645. 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, Tog is written ,06 and 70% 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 signihcant 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 TABLE. 5 10 · Hundreds. Hundredths. Tenths. Units. 0 100 ,05 6 5 1000 1234 10000 450.12 100000 1000000 65 =6 tenths. 5 hundredths. ,065 66 65 thousandths. ,1 2 3 4 “1234 ten thousandths. "45012 hun. thousandths 8,000005.“ 8 and 5 millionths. .45,6-5 “ 45 and 65 hundredths. 365,1 2 3 4 5 6" 365 & 123456 millionths. ,45012 45100 365 123456 1000000 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 |