7. 41, 61, 81, and 9.

Ans. 287.

Ans. 1911.

8. 21, 33, 621, and 751.

9. 5, 6, 7, 85, and 94.

17

10. 3, 8, 11, 14, 13, 38, and 111.

11. 5, of, of §, 7 of 14, and 17 of 25.

12. 1, 3, 4, 4, 6, 9, 3, 8, 1, 19, and 11.

Ans. 3817.

Ans. 5188.

Ans. 3.

Ans. 824958

SUBTRACTION OF FRACTIONS.

77

235. Subtraction of Fractions is the process of finding the difference between two fractions.

PRINCIPLES.

1. To subtract one fraction from another they must express similar fractional units.

2. To subtract one fraction from another they must be reduced to a common denominator.

1. What is the difference between and &? SOLUTION. Reducing the fractions to a common denominator that they may express similar fractional units, we have & and 45; 64 seventyseconds minus 45 seventy-seconds equals 19 seventyseconds. Hence the following

OPERATION.

=

Rule. Reduce the fractions to a common denominator, take the difference of the numerators and write it over the common denominator.

NOTE.-Reduce compound fractions to simple ones, and reduce each fraction and the difference to lowest terms.

13. (+3+18) — (1 +1 +15).

14. Find the difference between 155 and 128.

SOLUTION. We cannot take & from 5, so we take 1 from 15, which equals, and this added to =14; 1—8—§; 14—12=2; hence 155-128=26 or 23.

(3+)− ( } +†).

Ans. 17.

Ans.

Ans. 3.

1. A merchant sold a customer 221 yd. of silk, 34 yd. of paper muslin, 11 yd. of silesia, 53 yd. of cambric, and 5 yd. of ruffling; how many yards were sold? Ans. 3723.

2. A lady in shopping makes a bill of $373 for silk, $124 for merino, $2 for calico and $13 for braid, sewing-silk, etc.; she gave the merchant a 100 dollar bill in payment; what change should she receive? Ans. $46.

3. A person engages 4 tons of coal (2000 lb.); but when delivered, he finds the first ton falls short 221⁄2 lb., the second 23 lb., the third 1972 lb., and the fourth 263 lb.; how much coal should he pay for? Ans. 790714 lb.

4. From a barrel of vinegar containing 564 gallons, there were drawn at one time 21 gallons; at another time 131 gallons, of which 31 gallons were returned to the barrel; at a third time 155 gallons were drawn; and at a fourth time 19 gallons were poured in; how many gallons were in the barrel finally? Ans. 283 gallons.

SPECIAL SOLUTIONS

IN ADDITION AND SUBTRACTION OF FRACTIONS.

CASE I.

236. To add or subtract two fractions whose numerators are a unit.

1. Find the sum and the difference of and .

SOLUTION.-Reducing to a common denomina

tor and adding, we see that we have the sum of

the denominators divided by their product.

If we subtract, we shall have the difference of the denominators divided by their product. Hence we have the following

OPERATION.

卄=옳十옳=

5+3

15

Ans.

Ans.

15

5-3

Rule. Take the sum or difference of the two denominators and write it over the product of the denominators.

237. To add or subtract two fractions having a common numerator.

1. Find the sum and difference of 3 and §. SOLUTION.-Reducing to a common denominator, and adding, we see we have 3 times the sum of the denominators divided by their product.

If we subtract we shall have three times the difference of the denominators divided by their product. Hence the

OPERATION.

3(5+4)

5X4

=

=27. Ans.

Rule. Take the sum or difference of the denominators, multiply it by the common numerator, and write the product over the product of the denominators.

238. To add or subtract two fractions by reducing to a common numerator.

Rule. Reduce to a common numerator and proceed as

NOTE. It will be noticed that this is sometimes less work than reducing

to a common denominator.

MULTIPLICATION OF FRACTIONS.

239. Multiplication of Fractions is the process of finding a product when one or both factors are fractions.

240. There are Three Cases: 1st. A fraction by an integer; 2d. An integer by a fraction; 3d. A fraction by a fraction.

CASE I.

241. To multiply a fraction by an integer.

1. Multiply by 6.

SOLUTION.-Six times 13 eighteenths equals 78 eighteenths (78), which equals 13, or 4; or 6 times, or, and if 6 times equals 1,6 times equals 13 times, which is 13, or 43. Therefore, etc.

SOLUTION 2D.-Multiplying the numerator

(Prin. 1), we have 6 times

equals ; or,

dividing the denominator (Prin. 4), we have 6

times 3, or 41.

=

OPERATION.

6:

== 4

Rule. To multiply a fraction by an integer, multiply the numerator, or divide the denominator.

of

OPERATION.

1. Multiply 17 by ; also by 85. SOLUTION.-17 multiplied by equals 5 times 17; 5 times 17 are 85, and of 85 is 25, or 14. Therefore, etc.

17x=14}

SOLUTION.-Multiplying 17 by 5 and dividing by 6, we have 14; and then multiplying 17 by 8 we have 136; adding 136 to 14 we have 150. Therefore, etc.

NOTE-This method of multiplying by a mixed number is more convenient than the one usually presented.

Rule.-Multiply the integer by the numerator of the mul tiplier, and divide the product by the denominator; or divide

243. To multiply a fraction by a fraction. 1. What is the product of by ?

SOLUTION. multiplied by one equals, hence multiplied by equals of, which is, and multiplied by equals 5 times, which are 5. Therefore, etc.

35

OPERATION.

}×1=#

Rule.-Multiply the numerators together and the denominators together, cancelling common factors.

NOTE.-Practically this case is the same as finding a fractional part of a fraction, but theoretically the two cases are entirely distinct.

60

16. 88 of 817 by 57 of 2 of 679

17. 4×5×××21×13ļ.

18. 4×3 of 33×134 of 33×15 of 189.

44 50

Ans. 273.

Ans. 47.

Ans. 685.

Ans. 4300168.

Ans. 32.

Ans. 60.

Ans. 25.

23