PROBLEM 10.

148. To reduce a fraction of a higher denomination to a fraction of a lower denomination.

Ex. 1. Reduce of a penny, to the fraction of a farthing.

As 1 penny is equal to 4 farthings, so any fraction of a penny will be 4 times as great a fraction of a farthing; .. . }d.=4 times tqr.qr., Ans.

2. Reduce of a shilling to the fraction of a farthing. As 1s. is equal to 12d., so

d.4 times qr.qr., Ans.

s. 12 times d. d., and

=

Hence,

=

RULE. Multiply the fraction by such numbers as are necessary to reduce the given to the required denomination.

3. Reduces. to the fraction of a farthing.

×

s. (d. 12)=3d. (=3qr. X 4)=2&qr., Ans. ; or, 7 X 12 X 4 7 × 12 × 4 28

= qr., Ans., as before.

3

NOTE 1. The sign of multiplication, in these examples, is written only between the numbers which are given before the canceling is begun; thus, in Ex. 3, no sign is written between 36 and 3, for they are not to be multiplied together, but the 3 is obtained by canceling 12 in 36. So in Ex. 4, the 12 comes from canceling 20 in 240, and the 3 from canceling 4 in 12.

4. Reduce of a ton to the fraction of a dram.

7 × 20 × 4 × 25. X 16 X 16

44800
3

dr., Ans.

of a rod to the fraction of a barleycorn.

NOTE 2. In the first statement of Ex. 5, the 16, in the numerator, is equal to 33, and, in the second statement, the 33 is retained in the numera tor as a factor in the dividend, and the 2 is put in the denominator as a fac tor in the divisor.

148. Rule for reducing a fraction from a higher to a lower denomination? Explanation? How is Ex. 5 solved?

grain.

6. Reduce of a pound, Troy Weight, to the fraction of a Ans. 188. 7. Reduce of a pound, Apothecaries' Weight, to the frac tion of a grain. Ans.

8. Reduce of a day to the fraction of a second.

188.

Ans. 144.

9. Reduce of a bushel to the fraction of a pint.

10. Reduce

11. Reduce
12. Reduce

13. Reduce

14. Reduce

15. Reduce

16. Reduce

17. Reduce

18. Reduce

Ans. 128.

of a gallon to the fraction of a gill.
40c. yd. to the fraction of a cubic inch.
of a sign to the fraction of a second.
ʊʊsq.m. to the fraction of a rod.

Ans. 38.

Ans. 14.

fur. to the fraction of a link. of an acre to the fraction of a square yard. yd. of cloth to the fraction of an inch. circ. to the fraction of a second.

of a ton to the fraction of an ounce. 19. Reduce 3024 of a day to the fraction of a second. 20. Reduce £ to the fraction of a farthing.

21. Reduce of a bushel to the fraction of a pint.

PROBLEM 11.

149. To reduce a fraction of a lower denomination

to a fraction of a higher denomination.

Ex. 1. Reduce of a barleycorn to the fraction of an inch. In 15 barleycorns there is only of 15 inches, so in % of a barleycorn there is only of of an inch of an inch, Ans. 2. Reduce of a gill to the fraction of a quart.

As 1 gill is of a pint, so fgi. is of pt. pt. and, for a like reason,pt. is of qt.qt., Ans. Hence,

RULE. Divide the given fraction by such numbers as are required to reduce the given to the required denomination.

149. Rule for reducing a fraction from a lower to a higher denomination! Explanation?

1

3. Reduce 2qr. to the fraction of a shilling.

28qr. (=2&d.÷4)=3d. (=3s.÷ 12)=s., Ans.; or,

28 7

3 X 4 X 12

7

= s. Ans., as before.
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36

4. Reduce 44800dr. to the fraction of a ton. 44800 2800 175 7

3 × 16 × 16 × 25 × 4 × 20

5. Reduce 1982b. c. to the fraction of a rod.

1980

10

1980 180 30 2

7 X 3 X 12 X 3 × 57 × 3 × 12 × 3 × 11

6. Reduce 188gr. to the fraction of a pound, Apothecaries'

Weight.

Ans. 1200

7. Reduce 1gr. to the fraction of a pound, Troy Weight. 8. Reduce 25200sec. to the fraction of a day.

9. Reduce in. to the fraction of a yard, Cloth Measure.
10. Reduce 188sec. to the fraction of a week.

11. Reduce 432sq. in. to the fraction of a yard.
12. Reduce 450 links to the fraction of a furlong.
13. Reduce 33yd. to the fraction of an acre.
14. Reduce 120 seconds to the fraction of a sign.
15. Reduce gills to the fraction of a gallon.

PROBLEM 12.

Ans.

150. To reduce a fraction of a higher denomination to whole numbers of lower denominations.

Ex. 1. Reduce £ to shillings and pence.

£(s. X 20)=s.

4d.;.. £3s. 4d., Ans.

3s.; again s. (=d. X 12)=
×
Hence,

RULE. Reduce the given fraction to a fraction of the next lower denomination (Art. 148); then, if the fraction is improper, reduce it to a whole or mixed number (Art. 140). If the result is

150. Rule for reducing a fraction of a higher denomination to integers of lower denominations? Explanation?

a mixed number, reduce the fractional part of it to the next lower denomination, as before, and so proceed as far as desirable.

NOTE. If, at any time, the reduced fraction is proper, there will be no whole number of that denomination.

2. Reduce £ to whole numbers of lower denominations.

4. Reduce of a furlong to rods, yards, etc.

5. Reduce

6. Reduce

of a week to days, etc.

Aus. 18rd. 3yd. 2ft.

of a rod, Long Measure, to yards, etc.

7. Reduce 1887 of a circumference to signs, etc.

8. Reduce

of a ton to hundred weights, etc.

9. Reduce 17lb to ounces, drams, scruples, etc.

10. Reduce 30 circ. to signs, degrees, etc.

368

of a pound Troy?

of a bushel?

11. Reduce of a civil year (365 days) to days, etc.
12. What is the value of
13. What is the value of
14. What is the value of
15. What is the value of

Weight?

of a gallon?

of a pound, Apothecaries'

16. Reduce of a mile to furlongs, chains, etc.

17. Reduce of a cord to cord feet, cubic feet, etc.

18. Reduce

of a yard to quarters, nails, etc.

PROBLEM 13.

151. To reduce whole numbers of lower denominations to the fraction of a higher denomination.

Ans.

Ex. 1. One farthing is what part of a penny ? Since 4 farthings make a penny, 1 farthing is of a penny. 2. Six pence and 1 farthing are what part of a shilling? 6d.1qr.25qr; and 1s. 48qr.; .. 6d. and 1qr.gs., Ans.

To determine what part one thing is of another, considered as a unit or whole thing, the part is always made the numerator of a fraction, and the unit or whole thing is put for the denominator; thus, the fraction expresses the part that 3 miles is of 5 miles. Before the comparison can be made, the part and the whole must be of the same kind or denomination; thus, 3 pecks is not of 5 bushels, but, reducing the 5 bushels to 20 pecks, we have 3 pecks equal to of 20 pecks, i. e. 2 of 5 bushels. Hence,

RULE 1. Reduce the given quantity to the lowest denomination it contains, for a numerator; and reduce a unit of the higher denomination to the same denomination as the numerator, for a denominator.

3. Reduce 6rd. 5ft. 9in. to the fraction of a furlong. 6rd. 5ft. 9in. 1257in. and 1fur.

7920in.

.. 6rd. 5ft. 9in.fur.fur., Ans.

4. Reduce 7oz. 4dwt. to the fraction of a pound.

Ans..

5. Reduce 9 rods, 1 foot, and 6 inches to the fraction of a

furlong.

9rd. 1ft. 6in. 1800in. and 1fur. —

= 7920in.;

.. 9rd. 1ft. 6in.=4898fur.fur., Ans. (a) In Ex. 5, 6in. = ¿ft.; 1}ft. — Hyd.=rd. and 9rd. frd.=fur., Ans., as by Rule 1. Hence,

=

RULE 2. Divide the number of the lowest denomination given by the number required to reduce it to the next higher denomination, and annex the fractional quotient so obtained to the given number of that higher denomination; divide the mixed number so formed by the number required to reduce it to the NEXT higher denomination, annex the quotient to the given number of that denomination, and so proceed as far as necessary.

NOTE 1. This rule is frequently preferable to the 1st, because it enables us to use smaller numbers and gives the result in lower terms.

151. Rule for reducing the lower denominations of a compound number to a fraction of a higher denomination? Explanation? Principle? Second rule for reducing integers of lower denominations to the fraction of a higher denomination? Explanation? Why preferable to Rule 1?