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33. Add together E. D. and 1 cts. Ans. 7D. 53c. 29 m 34. Take 3}cis. from } of 2] dollars.

Ans. 43 cts. Note.-The reason of the rules, both for Addition and Sub. traction of Fractions, is manifest. If the given fractions have the same denominator and are of the same denomination, the sum of the numerators written over the given denominator, will be the sum of the fractions. A fraction is subtracted from a whole number, by taking the numerator of the fraction from its denominator, and placing the remainder over the denominator, then taking one from the whole number, thus :-From 123

12

12 Take 74

5} 44 113 Note.—To multiply by a fraction, whether the multiplicand be a whole number or a fraction, as before observed, you can divide hy the denominator of the multiplying fraction, and multiply the quotient by the numerator, thus : 20 multiplied by , the product is 15. To divide by a fraction, whether the dividend be a whole number or a fraction, we multiply by the denominator of the dividing fraction, and divide the product by the numerator, thus : 20 divided by is 263, and 12 divided by a is 16, &c. You will observe in multiplication, the multiplier being less than unity, or 1, will require the product to be less than the multiplicand; and in division, the divisor being less than unity or 1, it will be contained a greater number of times, and consequently will require the quotient to be greater than the dividend, to which it will be equal when the divisor is 1, and less when the divisor is more than 1.

By Vulgar and Decimal Fractions. 35. If 2 bushels sow an acre, how many acres will 22 bushels sow?

Thus, 23 X4+3=1 improper fraction; 22 X411=8 acres Ans. Or, 22-2.75=8 Ans. 36. At 43 dollars per yard, how many yards of cloth may

be bought for 37 dolls.? Thus, 43 x 5+2=22; 37 x 5-22=8293 yards Ans.

D37-4.40=8 yards, 1 quarter, 1 nail + Ans. 37. If 143 yards cost 75 dollars, how much per yard ? Thus, 143 X 8+3=115; 75 X 8=600--115=515 - Ans. Or, D75- 14.375 yards=D5.21.7 Ans.

38. How many times is a contained in 746 ? Thus, 746 X8-3=19891; 746_375=198947= Ans.

1

39. Divide by =l; divide by 1 =3; divide by=2. 40. If g of a yard cost 5 dollars, how much per yard ?

41. If z of a bushel cost 3 dollars, how much for 3 bushcls? how much for į a bushel ?

42. Add ž, s, = ; add la 1, 16, 18=:; add } ?, ?, 13, 2, together.

43. There are 3 pieces of cloth, one containing 73 yards, another 13} yards, and the other 15% yards, how many yarus in all ? First reduce the fractional parts to their least common denominator, thus :--3 x 2 x8=48

68 +33 +59

+39=138=28=3
1 X4 X8=32 Then,
7X4 X2=56

13

15 4X2 X 8--64

374= }

Ans. 37

48
6

1

64

Adding together all the 61ths, namely; 48, 32, and 56, you have 136, that is 1.3.6 =261= write down the fractions 8 under the other fractions, and carry the two integers to the other integer, and you have 37}. Ans.

Or, thus : 7.75+13.5+15.875=37.125=5. The last method is the most convenient.

44. From of an ounce take į of a pwt. Ans. 11 pwt. 3 grs. 45. From 4 days, 7 hours, take 1 day 9 35 hours.

Ans. 2 days, 22 hours, 20 min. 46. A man purchased is of 7 cwt. of sugar, how much did he purchase ?

47. 14 hours, 48 minutes, 549 seconds, is what part of a day?

48. Add { lb. troy to iż of an oz. Ans. 6 oz. 11 pwt. 16 grs 49. At 3 dollars per cwt., what will 6 lbs. cost ?

50. If of a yard costs of a dollar, what will 15 of a yard cost?

51. Add together D., * cents, į mills. 52. Add gof a yard, 3 of a foot, and 19 of a mile, together. 53. Add į of a week, of a day, 1 of an hour, and s of a minute, together.

54. Froin 5 weeks, 21 days, 9 hours, take 7 days, 14; hours.

55. Multiply of 5, by of} of 113.
56. What is the continual product of 7, 1, of s, and 3?

Ans. 641

Ans. 211

57. In a certain school zo of the pupils study Greek, to Latin, ş arithmetic, read and write, and 20 attend to other studies ; required the number of pupils.

Ans. 100. 58. In 3423 gallons, how many of a gallon?

Ans. 1371 of a gallon, or 1371 quarts. Note.—Any whole number may be made an improper frac- . tion by drawing a line under, and putting a unit or 1 for a denominator, as 5 may be expressed thus, s, and 10, I, &c. Reduce .275 to a vulgar fraction.

Thus : = 25 com. m.=11. Ans.

275 1000

SECOND REVIEW OF VULGAR FRACTION.

What do you mean by common divisor ? by the greatest common divisor? How can you multiply a fraction by a whole number? How will you multiply a mixed number? When the multiplier is less than a unit, what is the product compared with the multiplicand? How do you multiply a whole number by a fraction ? How can you multiply one fraction by another, and can you do it in more than one way? How do you multiply a mixed number by a mixed number? How can you divide a fraction by a whole number ? What is understood by common denominator ? the least common denominator ? Do cases frequently occur in vulgar fractions, where the terms may be reduced, and solved by decimals? What can you say of the importance of a correct knowledge of all kinds of fractions ?

PRACTICE.

PRACTICE is a contraction of simple proportion, when the first term happens to be a unit, or one, and has its name from its daily use among merchants and tradesmen; being an easy and concise method of working many questions which occur in trade and business; also, many questions in simple interest may be readily and easily solved by aliquot parts, after the pupil has made himself thoroughly acquainted with the tables, which he should commit to memory.

One number is an aliquot, or even part of another when it forms an exact part of it, as 25c. is an aliquot part of a D=1; 75c.=; 10c.=id; 5c.=zo, &c. 56 lb. is an aliquot part of a cwt.={; 28 lbs. = 1; 14 lbs. = 1; 7 lbs = 1, &c

Many questions that occur in practice may be easily solved by decimals; as it is mure immediately connected with sterling money, and when pounds, shillings, and pence, occur, no method of calculation is better than the various rules of practice. .

RULE.

1. When the price is in dollars and cents, multiply the given quantity by the dollars, and take aliquot parts for the cents, and add the products together, and you will have the answer in D., C., &c.

2. But when the quantities are of various denominations, write down the given price of one of the highest denomination, and multiply it by the whole of the highest denomination ; then take aliquot parts of the next lowest denomination continually, then add the products together for the answer.

Proof, by Simple Proportion.

A TABLE OF ALIQUOT, OR EVEN PARTS.

6

12

10

10 days

To of a D.

20

25

Time.

Dolls.

Avoir. Weight. 6 mo. is 1 a year 50c. is 1 of a D. 84 lb. is of a cwt. 4 mo, is

a year 33 c. is of a D. 56 lb. is of a cwt. 3 mo. is

a year 25c. is of a D. 28 lb, is of a cwt. 2 mo. is a year 20с. is

of a D. 14 lb. is of a cwt. 1 mo. is 1

a year 12.5c.is į of a D. 7 lb. is of a cwt. equal to of 3 mo. 10 is of a D. 3.5 lb.is of a cwt. 15 days of a mo. 8. is 1 of a D. 1 lb. is bĀ of a cwt. of a mo. 61 is

Cloth Measure. 71 days of a mo.5

is of a D. 6 days

of a mo.4 is of a D. 27 in. or 3 qrs. * yd. 5 days of a mo. 2

is of a D. 18 in. or 2 qrs.
of a mo. 1
is too of a D. 9 in.

or 1 qr. 2 days 15 of a mo.

4.5 in. or 2 na. i yd. Land Measure.

2.25 or 1 na. Cubic measure.

80 po.or 2 R. } an A. 1} in. 3 yd. 96 ft. is of a cord 40 po.or1 R. Parts of a cwt. 64 ft. is į of a cord 32 po. 32 ft. is i of a cord 20 po.

an A. 14 lb. is 16 ft. is of a cord 16 po.

an A.17 ib. is 8 ft. is of a cord 8 po.

A. 4 lb. is 4 ft. is of a cord 4 po. do an A. 2 lb. is a of a gr. 2 ft. is of a cord 2 po.

san A. 1 lb.

is zo of a gr.

50

3 days to

yd. yd.

1

16 yd.

an A. an A.

8

of a qr. of .a qr. of a qr.

8

17

1 1 20

an

EXAMPLES.

1. What is the value of 5 cwt. 1 qr. 14 lbs., at D2.50 per cwt. ? Thus: D2.50

First multiply D2.50 by 5 cwt., 5 X

and this will give the value of 5

cwt.; then of D2.50 is 62c., 12.50=5 cwt. the price or value of the 1 qr. ;

621 cwt. then the 1 of 62} is 31į, the value =3114 lbs.

of 14 lbs., and their products give

the whole value, &c. D13.433 2. What cost 2 cwt. 3 qrs. 7 lbs. 8 oz. of iron, at 5 dollars

per cwt. ?

2)5.00D.

2 cwt.

Or, 1792 oz. : 5048 oz. :; 5.000.

5.00

D. c. m.

10.00 =2 cwt. 1792)2524000(14.08.4+ Ans proof. 3.75 = 3 cwt. .311=7 lbs. .02 = 8 oz. (nearly.)

yard ?

D14.084 Ans. 3. What is the value of 498 yards of tape, at 6 mills per

Ans. D2.98.8. 4. What cost 1724 yards of flannel, at 37} cents per yard ?

Ans. D646.50. 5. What cost 190 pounds of cotton, at 20 cents per pound ?

Ans. D38.00. 6. What cost 16 cwt. 2 qrs. of sugar, at D5.18 per cwt. ?

Ans. D85.47. 7. What cost 560 yards of sheeting, at c.20.5 per yard ?

Ans. D114.80. 8. What cost 270.5 yards, at 15 cents per yard ?

Ans. D40.57.5. 9. What cost 45 gallons, at 183 cents per gallon ?

Ans. 18.43.75. 10. What is the cost of 56.5 pounds of butter, at c.12.5 per

Ans. 17.06.25. 11. What is the cost of 15 A. 1 R. at D10.25 per acre ?

Ans. D156.31.25 12. What cost 5.30 cords of wood, at D2.50 per cord ?

Ans. Di3.25. 13. What cost 6 yards, 3 quarters of cloth, at D1.50 per yard.?

Ans. D10.13)

pound?

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