12. Reduce 24 grains to the decimal of an ounce troy. 13. Reduce 5oz. 4dr. avoirdupois to the decimal of a pound troy.

14. Reduce 3cwt. 1qr. 14lb. to the decimal of a ton.

15. Reduce 2qr. 15lb. to the decimal of a hundred-weight. 16. Reduce 5lb. 10oz. 3pwt. 13gr. troy to the decimal of a hundred-weight avoirdupois.

17. Reduce 1qr. Ina. to the decimal of a yard.

18. Réduce 2qr. 3na. to the decimal of an English ell. 19. Reduce 2yds. 2ft. 61in. to the decimal of a mile. 20. What decimal part of an acre is 1R. 37P ?

21. What decimal part of a hogshead of wine is 2 quarts 1 pint?

22. Reduce 3 bushels 3 pecks to the decimal of a chaldron of 36 bushels..

23. What decimal part of a year is 3wk. 6da. 7hr., reckoning 365da. 6hr. a year?

24. Reduce 2.45 shillings to the decimal of a £. 25. Reduce 1.047 roods to the decimal of an acre.

26. Reduce 176.9 yards to the decimal of a mile.

CASE III.

162. To find the value of a denominate decimal in terms

of integers of inferior denominations.

1. What is the value of .832296 of a £?

We first multiply the decimal by 20, which brings it to shillings, and after cutting off from the right as many places for decimals as in the given number, we have 16s. and the decimal .645920 over. This we reduce to pence by multiplying by 12, and then reduce to farthings by multiplying by 4.

OPERATION.

.832296

20

16.645920

12

7.751040

4

3.004160

Ans. 16s. 7d. 3far.

Hence, to make the reduction,

I. Consider how many in the next less denomination make one of the given denomination, and multiply the decimal by this number. Then cut off from the right hand as many places as there are in the given decimal.

II. Multiply the figures so cut off by the number which it takes in the next less denomination to make one of a higheṛ, and cut off as before. Proceed in the same way to the lowest denomination: the figures to the left will form the answer sought

EXAMPLES.

1. What is the value of .625 of a cwt. ?
2. What is the value of .625 of a gallon?

Ans.

Ans.

3. What is the value of .004168lb. troy?

Ans.

4. What is the value of .375 hogshead of beer?

5. What is the value of .375 of a year of 365 days?

6. What is the value of .085 of a £?

Ans.

7. What is the value of .258 of a cut. ?

Ans.

8. What is the difference between .82 of a day and .64 of

an hour?

9. What is the value of 2.078 miles? 10. What is the value of £.3375?

Ans.

Ans.

11. What is the value of .3375 of a ton?

Ans.

12. What is the value of .05 of an acre?

Ans.

13. What is the value of .875 pipes of wine?

14. What is the value of .046875 of a pound, avoirdupois? 15. What is the value of .56986 of a year of 365 days?

16. What is the value of £2.092 ?

17. What is the value of £5.64 ?

Ans.

Ans.

18. What is the value of .36974 of a last, wool weight? 19. What is the value of .827364qr., corn measure ? 20. What is the value of .093765t. ?

Ans.

QUEST.-162. How do you find the value of a denominate decimal in in tegers of inferior denominations? What is the value in shillings of onehalf of a £? In pence of one-half of a shilling?

CIRCULATING OR REPEATING DECIMALS.

163. WE have seen that in changing a vulgar into a decimal fraction, cases will arise in which the division does not terminate, and then the vulgar fraction cannot be exactly expressed by a decimal (Art. 158).

5 12

Let it be required to reduce to its equivalent decimal. We find the equivalent decimal to be

.4166 + &c., giving 6's, as far as we choose to continue the division.

OPERATION.

12)50000 .4166 +

The further the division is continued the nearer the decimal will approach to the true value of the vulgar fraction; and we express this approach to equality of value by saying, that if the division be continued without limit, that is, to infinity, the value of the decimal will then be equal to that of the vulgar fraction. Thus, we also say,

.999, continued to infinity = 1,

because every annexation of a 9 brings the value nearer to 1.

164. Let us now examine the circumstances under which, in the reduction of a vulgar to a decimal fraction, the division will not terminate.

If the vulgar fraction be first reduced to its lowest terms, (which we suppose to be done in all the cases which follow,) there will be no factor common to its numerator and denominator. Now, by the addition of O's to the numerator we may increase its value ten times for every 0 annexed; that is, we introduce into the numerator the two factors 2 and 5 for every

QUEST. 163. Can a vulgar fraction always be exactly expressed by a decimal? Can five-twelfths? If we continue the division, does the quotient approach to the true value? By what language do we express this fact? 164. In annexing a 0 to the numerator, what factors do we introduce into it?

additional 0. But the numerator can never be exactly divided by the denominator, if the denominator contains any prime factor not found in the numerator (Art. 107): hence it can never be so divided, if the denominator contains any prime factor other than 2 or 5. Hence, to determine whether a vulgar fraction in its lowest terms can be expressed by an exact decimal,

Decompose the denominator into its prime factors, and if there are any factors other than 2 or 5, the exact division cannot be made.

QUEST.-Under what circumstances will the numerator be exactly divisible by the denominator? When not so? How do you determine whether a vulgar fraction can be exactly divisible by a decimal?

NOTE.-165. When there are no prime factors in the denominator other than 2 or 5, the division will always be exact, and the number of decimal places in the quotient will be equal to the greatest number of factors among the 2's or 5's.

1 1

7. What is the decimal corresponding to the fraction? 8. What is the decimal corresponding to ? 9. What is the decimal corresponding to 17?

625

128

166. The decimals which arise from vulgar fractions, where the division does not terminate, are called circulating decimals, because of the continual repetition of the same figures. The set of figures which repeats, is called a repetend.

167. A SINGLE REPETEND is one in which only a single figure repeats, as 2.2222+, or 3.3333 +. Such repetends are expressed by simply putting a mark over the first figure; thus, .2222+ is denoted by .2+, and .3333 + by .3+.

168. A COMPOUND REPETEND has the same figures circulating alternately: thus 1.5757+ and 5333.57235723+ are compound repetends, and are distinguished by marking the first and last figures of the circulating period. Thus .5757 + is written 57+, and .57235723 + is written 5723+.

169. A PURE REPETEND is an expression in which there are no figures except the repeating figures which make up the repetend; as 3+, .5+, 473′'+, &c.

170. A MIXED REPETEND is one which has significant figures or ciphers between the repetend and the decimal

QUEST.-165. If there are no prime factors in the denominator other than 2 and 5, will the division be exact? How many decimal places will there be in the quotient? 166. What are the decimals called when the division does not terminate? What is the set of figures which repeats called? 167. What is a single repetend? How is it expressed? 168. What is a compound repetend? How is it expressed? 169. What is a pure repetend? 170. What is a mixed repetend?