II. If you have a remainder, how do you proceed? A. Find how many of the next lower denomination this remainder is equal to, which add to the next denomination; after which divide as in whole numbers. PROOF. What is the proof? A. The same as in Simple Division. More Exercises for the Slate. 2. If 8 tons of hay cost 40£ 14 s. 8 d., what will 1 ton cost? A. 5£ 1 s. 10 d. 3. If 11 gals. of brandy cost 5£ 16s. 5 d., what will 1 gallon cost? A. 10 s. 7 d. 4. If a man spend 60£ 13 s. 4 d. a week, how much is that a day? A. 8£ 13 s. 4 d. 5. If 1 cwt. of rice cost 2£ 6 s. 8 d., what will 1 lb. cost? A. 0£ 0 s. 5 d. 6. You have 31£ 9s. 6 d. to be divided equally among 2 men; how much would it be apiece? (1514-9) How much would it be apiece to be divided among 3? (10.9-10) Among 6? (5-4-11). A. 31£ 9 s. 6d. 7. Divide 2 gals. 2 qts. by 4; (0-2-1) by 5; (0-2) by 10; (0-1) by 2; (1-1). A. 2 gals. 2 qts. 1 pt. 8. Divide 96 acres 2 roods, 16 rods, by 7; (13-3-8) by 8; (12-0-12) by 12; (8-0-8). A. 33 acres, 3 roods, 28 rods. Questions to exercise the foregoing Rules. 1. What is the sum of the following numbers, viz. one, two thousand, thirty thousand, four millions, twenty thousand, nineteen, four hundred millions? A. 404052020. 2. Bought a coat for 15 dollars, a vest for 1 dollar 371⁄2 cents, a pair of boots for 6 dollars 12; what did the whole cost me? A. $22,50. 3. Bought a horse for $75, and sold him for 37 cents less than he cost me; what did I get for him? A. $74,62,5. 4. What will 3200 yards of tape come to at 64 cents, or of a dollar, a yard? (200) At 124 cents, or of a dollar? (400) At 25 cents, or of a dollar? (800). A. $1400. 5. How many yards in 31557600 rods? A. 173566800. 6. How many years in 31557600 seconds, allowing the year to contain 365 days? A. 1 year. 7. At 4 cents a gill, what will 1 tun of wine cost? A. $322,56. 8. How much wine can be bought for $322,56, at 4 cents a gill? A. 1 tun. 9. How many rods in 1100 yds.? In 3300 yds.? A. 800 rods. 10. How many dollars in 300£? In 900£? A. $4000. 14. What will 1 ton of clover-seed cost, at 5 mills an ounce ? A. $179,20. 15. At 2 cents an inch, what will 1 yard of cloth cost? A. 72 cents. 16. Reduce 1 tun to gills. A. 8064 gills. 17. Reduce 20 bushels to pints. A. 1280 pints. 13. Reduce 4 tons to drams. A. 2293760. 19. How many barley-corns will reach across the Atlantic Ocean, allowing it to be 3000 miles? A. 570240000. 20. How many times will a watch click in 20 years, if it click at the usual rate of 60 times in a minute? A. 631152000. 21. A father left legacies to his children as follows: to Thomas, 75£ 14 s. 6 d., to William 3 times as much as Thomas, to his daughter Mary as much as Thomas, and to Susan, his youngest child, as much as all the rest, lacking 20£ 13 s. 8 d.; how much did each receive? A. William 227£ 3 s. 6 d., Mary 12€ 12 s. 5 d., Susan 294£ 16s. 9 d. ↑ XXXIV. 1. If one third (3) of an apple cost 2 cents, what will a whole apple cost? 2. If one third cost 3 cents, what will a whole one cost? If one third cost 4 cents, what will one whole apple cost? If one third cost 6 cents? 8 cents? 9 cents? 20 cents? 50 cents? 100 cents? 3. If you pay 3 cents for one fifth (†) of an orange, what will a whole orange cost? 4. If you pay 2 dollars for one eighth (§) of a ticket, what will a whole ticket cost? Q. How many halves to an apple, or any thing? Q. How many thirds? Fifths? Eighths? Sixteenths? Q. When an apple, or any thing, is divided into two equal parts, would you call one of these parts a half or a third? Into 3 equal parts, what is one part called? Q. Into 4 parts, what is one part called? Q. When an apple, or any thing, is divided into two equal parts, how would you express one part, on the slate, in figures? A. I set the 1 down, and draw a line under it; then write the 2 under the line. Let me see you write down, in this manner, on the slate, one half. One third. One fourth. One fifth. One sixth. Two sixths. Three sixths. Three eighths. Eight twelfths. Q. What are such expressions as these called? A. Fractions. Q. When, then, any whole thing, as an apple, a unit, &c. is broken or divided into equal parts, what are these parts called? A. Fractions. Q. Why called fractions? broken. A. Because fraction signifies Q. You have seen, that, when any whole thing is divided into 3 parts, these parts are called thirds; into 4 parts, called fourths: what, then, does the fraction take its name or denomination from? A. From the number of parts into which any thing is divided. Q. When an apple is divided into 6 parts, and you áre desirous of giving away 5 parts, how would you express these parts? A. §. Q. What is the 6 (in §) called ? A. The denominator. Q. Why so called? A. Because it gives the name or denomination to the parts. Q. What is the 5 (in ) called? A. Numerator. Q. Why so called? A. Because it numerates or numbers the parts. Q. Which is the numerator, then? A. The number above the line. Q. Which is the denominator? A. The number below the line. Q. What, then, does the denominator show? A. The number of parts a unit, or any thing, is divided into. Q. What does the numerator show? A. How many parts are taken, or used. Q. In the expressions, 16, 12, 30, which are the numerators, and which are the denominators ? Q. If you own & of a vessel, how many parts is the vessel supposed to be divided into? and how many parts do you own? A. 40 parts, and I own 28 parts. Q. Is of an apple more than of it? Q. What fraction, then, is greater than †? Than? Than Than? Than? What fraction is less than? Than Than? Than? Q. From these remarks, what appears to be a correct definition of fractions? A. They are broken parts of a whole num ber. Q. How are they represented? A. By one number placed above another, with a line drawn between them. Q. In Simple Division, you recollect, that the remainder was represented in like manner; what, then, may justly be consid-· ered the origin of fractions? A. Division. Q. What may the numerator be considered? A. The dividend. Q. What may the denominator be considered? A. The divisor. Q. What, then, is the value of a fraction? A. The quotient of the numerator divided by the denominator. Q. What is the quotient of 1 dollar divided among 2 men? A.. Q. What is the quotient of 7 divided by 8? Q. How, then, are fractions represented? of division. Q. What does express? A. The quotient, of which{ A. Z. A. By the sign 2 is the dividend. 3 is the divisor. 1. If 3 apples be divided equally among 8 boys, what part of one apple will each boy receive? 1 apple among 8 boys would be of an apple apiece, and 3 apples would be 3 times as much; that is, of an apple apiece. Ans. §. = 2. If 4 oranges be divided equally among 8 boys, what part of an orange is each boy's part? 1 orange among 8 boys §, and 4 oranges are 4 times as much; that is, §, Ans. If 2 oranges among 7 boys? A. . 9 oranges among 13 boys? 20 oranges among 37 boys? 3. One orange among 2 boys is of an orange apiece; how much is 1 divided by 2, then? Ans. . How much is 1 divided by 3? A. The quotient of 5 divided by 6? A. §. Of 3 by 5? Of 7 by 9? Of 8 by 13? Of 11 by 15? 4. What part of one apple is a third part of 2 apples? A third part of one apple is, and a third part of 2 apples must be twice as much; that is, of 1 apple. A. 3. 5. What part of 1 apple is one fourth (4) part of 3 apples? of 3 apples is 3 times as much as of 1 apple; that is, of 1 apple. A. 1. 6. What part of one apple is of 3 apples? A. 3. What part of 1 apple is of 4 apples? A. of 4 apples is what part of 1 apple? Ans. . A PROPER FRACTION. Q. We have seen that the denominator shows how many parts it takes to make a whole or unit; when, then, the numerator is less than the denominator, is the fraction greater, or less, than a whole thing or unit? A. It must be less. Q. What is such a fraction called? A. A Proper Fraction. Q. How may it always be known? A. The numerator is less than the denominator. Q. What kind of fractions are 1, 2, 3, &c.? AN IMPROPER FRACTION. Q. When the numerator is as large, or larger than the denominator, as, §, 1f, †, it is plain, that the fraction expresses 1 whole, or more than 1 whole; what is such a fraction called? 4. An Improper Fraction.' Q. How may it be known? A. The numerator is greater than the denominator. Q. What kind of fractions are,,, &c. A MIXED NUMBER. Q. What is a mixed number? A. A fraction joined with a whole number. Q. What kind of fractions are 15, 167, &c. Q. What kind of fractions are each of the following expressions, viz. 15, 8, 21, 84, 18, 71, 5o ? ¶ XXXV. To change an Improper Fraction to a Whole or Mixed Number. 1. How many whole apples are there in 6 thirds (§) of an apple? In 8 quarters ()? In? In? In 2? In f? In 488 ? 2. How many weeks in 56? In 84? of a week? In 28? In ? In |