TO REDUCE ANY SIMPLE OR COMPOUND QUANTITY TO THE FRACTIONAL PART OF ANY OTHER SIMPLE OR COMPOUND QUANTITY OF A LIKE KIND.

Art. 91. Suppose we wish to express what part 7 inches is of 9 inches.

As every fraction expresses the part which its numerator is of its denominator, we have only to write the number which the question requires to be made the fractional part of the other for the numerator of a fraction, and the other for the denominator; the fraction thus formed will express the fractional part required. Then writing 7 inches for the numerator of a fraction, and 9 inches for the denominator, thus, we find that 7 inches is 7 ninths of 9 inches.

Again; suppose we wish to reduce 13s. 4d. to the fractional part of £1.

13s. 4d. are equal to 160 pence. £1 is equal to 240 pence. Writing 160 pence for the numerator of a fraction, and 240 pence for the denominator, thus, 18, we find that 13s. 4d. is 160 two hundred fortieths, or 2 thirds, of £1. From these illustrations we derive the following

RULE. Reduce the given quantities to the lowest denomination in either of them; then write the number which the question requires to be made the fractional part of the other for the numerator of a fraction, and the other number for the denominator; the fraction thus formed will express the required part.

1. Reduce 12 cents and 5 mills to the fraction of a dollar.

3. What fractional part of 1

yard is 1 foot 9 inches?

2. Reduce 4s. 6d. to the fraction of a dollar.

4. What fractional part of 5 acres is 3 acres 2 roods?

5. Reduce 7 oz. 4 pwts. to the fraction of a pound.

Ans. of a pound.

6. What fractional part of a cwt. are 3 qrs. 14 lbs.?

Ans. of a cwt.

7. Reduce 6 fur. 26 rds. 3 yds. 2 ft. to the fraction of a

mile.

Ans. of a mile.

8. What fractional part of an acre are 1 rods?

rood and 30 Ans. 7 of an acre.

9. Reduce 3 hhds. 31 gals. 2 qts. to the fraction of a ton. Ans. of a ton. 10. What fractional part of a month are 3 w. 1 d. 9 h. Ans. of a month.

36 m.?

11. What fractional part of 10 dollars is $6.47?

647 Ans. of 10 dollars.

12. What fractional part of £5. 16s. is 18s. 6d. ?

Ans. of £5. 16s. 23/2

13. What fractional part of 15 acres is 24 acres 3 roods?

14. What fractional part of 12 cwt.

Ans. 23 of 15 acres.
is 7 cwt. 3 qrs. 14 lbs. ?
Ans. of 12 cwt.

15. What fractional part of 25 m. 5 fur. is 17 m. 3 fur. 20 rods?

Ans. 18 of 25 m. 5 fur.

TO REDUCE THE FRACTION OF ANY HIGHER DENOMINATION OF MONEY, WEIGHT, OR MEASURE, TO ITS VALUE IN WHOLE NUMBERS OF LOWER DENOMINATIONS.

Art. 92. Suppose we wish to find the value of of £1. in shillings and pence.

of a pound is equal to 20 times as great a fraction of a shilling, or of a shilling, and 40 of a shilling is equal to 13 shillings.

Again, of a shilling is equal to 12 times as great a fraction of a penny, or 12 of a penny, and 12 of a penny is equal to 4 pence. We have thus found that of £1. is equal to 13s. 4d. Hence the following

RULE. Reduce the given fraction to a fraction of the next lower denomination by multiplying its numerator by that number of the lower denomination which is equal to a unit of the higher, and write the product over the denominator; if this product be an improper fraction, reduce it to a whole or mixed number by dividing the numerator by the denominator. If the quotient be a mixed number, reduce the fractional part to a fraction of the next lower denomination, as before, and thus proceed through all the lower denominations; the several quotients will express the value of the given fraction in the lower denominations.

1. Reduce of a dollar to its value in cents and mills. 3. What is the length of of a yard, in feet and inches?

5. Reduce g of a lb. troy to tions.

2. Reduce of a dollar to its value in shillings and pence.

4. What number of roods and rods in of an acre? its value in lower denominaAns. 7 oz. 4 pwts.

6. In of a cwt., how many quarters and pounds?

Ans. 3 qrs. 14 lbs.

7. Reduce & of a mile to its value in whole numbers of lower denominations. Ans. 6 fur. 26 rds. 3 yds. 2 ft. 8. In of an acre, how many roods and rods?

Ans. 1 rood 30 rods. 9. Reduce of a tun of wine to its value in whole numbers of lower denominations. Ans. 3 hhds. 31 gals. 2 qts. 10. In of a month, how many weeks, days, hours, and minutes? Ans. 3 w. 1 d. 9 h. 36 m. 11. What is the value of 3 of a £. in shillings, pence, and farthings? Ans. 15s. 6d. 23qrs. of a lb. avoirdupois, in ounces Ans. 10 oz. 10 drs. of a cwt., in quarters, pounds, Ans. 3 qrs. 3 lbs. 1 oz. 12 drs. of a mile, in furlongs, rods,

12. What is the value of and drams?

13. What is the value of ounces, and drams?

14. What is the value of yards, feet, and inches?

Ans. 4 fur. 22 rds. 4 yds. 2 ft. 14 in.

ADDITION OF VULGAR FRACTIONS.

Art. 93. Addition of vulgar fractions is the method of finding the sum of two or more fractional numbers or quantities.

Suppose we wish to find the sum of 1, 2, and 1⁄2, of a yard.

When fractions have a common denominator, and are of the same denomination, we can find the sum of them by adding the numerators and writing their sum over the common denominator; thus, ++√=}= 14 yards.

When fractions have not a common denominator, they must be changed to fractions having a common denominator before we can add them.

When fractions are of different denominations, they must either be reduced to the same denomination, or to their value in whole numbers of lower denominations, before they can be added.

Compound and complex fractions must be reduced to simple fractions. Mixed numbers may either be reduced to improper fractions, or the whole numbers and fractional parts may be added separately.

From the preceding remarks we derive the following

RULE. Reduce compound and complex fractions to simple fractions, mixed numbers to improper fractions, fractions of

different denominations to the same, and all of them to their lowest terms.

If the fractions have different denominators, change them to fractions having the least common denominator, then add the numerators and write their sum over the common denominator. If the result be an improper fraction, reduce it to a whole or mixed number.

5. What is the sum of of and g of §?

7. What is the sum of,

of a shilling?

Ans. 102. Ans. 118.

8. What is the sum of,,, and ?
9. What is the amount of of 1 and 3 of 4?

10. What is the sum of 14, 24, 31, 4, and 51?

3960

Ans. $.

Ans. 16.

415

54, and 8?

11. What is the amount of

6

9

Ans. 248.

12. What is the sum of of a dollar, of a cent, and 3 of

a mill?

15. What is the amount of of a week,

an hour, and of a minute?

of a day, of

Ans. 2 days 2 hours 30

min. 45 sec.

16. What is the sum of, of 44, and of 6% ?

17. What is the amount of 819?

18. What is the sum of 4 of 18

Ans: 31818.

of §, 1 of 51, 409%, and Ans. 1232. and 11 of 3 of 16? Ans. 17.2309 27027

19. What is the sum of of a ton, of a cwt., of a qr., and of a lb. ? Ans. 16 cwt. 3 qrs. 2 lbs. 14 oz.

20. A trader purchased 4 pieces of silk; the first measured 35 yds.; the second, 36 yds.; the third, 37 yds.; and the fourth, 387 yds. What number of yards did he purchase? Ans. 148 yards.

SUBTRACTION OF VULGAR FRACTIONS.

Art. 94. Subtraction of vulgar fractions is the method of finding the difference between any two fractional numbers or quantities.

When two fractions have a common denominator, and are of the same denomination, we can find their difference by taking the less numerator from the greater, and writing their difference over the common denominator; thus, the difference between of a yard and of a yard is or of a yard. When the two fractions have not a common denominator, they must be reduced to a common denominator before subtraction can be performed.

When the two fractions are of different denominations, they must either be reduced to the same denomination, or to their value in whole numbers of lower denominations.

Compound and complex fractions must be reduced to simple fractions. Mixed numbers may be reduced to improper fractions, or the fractional parts and whole numbers may be subtracted separately. Hence the following

RULE. Reduce compound and complex fractions to simple fractions, mixed numbers to improper fractions, fractions of different denominations to the same, and both of them to their lowest terms.

If the two fractions have different denominators, change them to fractions having the least common denominator, then subtract the less numerator from the greater, and write the remainder over the common denominator.

1. What is the difference 2. What is the difference between and ? between and ?? 3. What is the difference 4. What is the difference between and ? between and ?

5. What is the difference

6. What is the difference

between of and of? between of and of ?

7. What is the difference

8. What is the difference

between 12 yards and 15% between 15 lbs. and 21 lbs.? yards?

9. What is the difference between 4 and 14?

Ans. 2.

10. What is the difference between 1 of 21 and

of 1? Ans. 1.

11. What is the difference between 93 and 2 of 61?

Ans. 7.

Ans. 81.

12. What is the difference between 457 and 367? 10