When the denominations are to be reduced from a higher denomination to a lower, it is called Reduction Descending; but when they are to be reduced from a lower to a higher denomination, it is called Reduction Ascending.

REDUCTION DESCENDING.

Let it be required to reduce £7 5s. 10d. 3 far. to farthings.

OPERATION.

7 the number of pounds.

Multiply by 20, the number of shillings in one pound. 140 product in shillings.

Add 5 shillings.

Multiply by

145 the number of shillings.

12, the number of pence in one shilling.

290

145

1740 product in pence.

Add 10 pence.

1750 the number of pence.

Multiply by 4, the number of farthings in one penny.

Add

7000 product in farthings.

3 farthings.

7003 the number of farthings sought.

From the above operation, we readily deduce this general

RULE.

Multiply the number in the highest denomination by the number indicating how many of the next lower make one in

that higher; to this product add the number, if any, belonging to this lower denomination; we shall thus obtain an equivalent value in the next lower denomination.

II. Proceed in a similar way for all the successive denominations; the last result will be the number sought.

What is Reduction? When is it called Descending? And when Ascending? Repeat the rule for Reduction Descending.

REDUCTION ASCENDING.

81. Let it be required to reverse the last example, that is, to find the number of pounds, shillings, pence, and farthings, in 7003 farthings.

We must obviously perform a reverse operation to that performed under Reduction Descending.

Collecting results, we have 7003 farthings, equivalent

to £7 5s. 10d. 3 far.

EXPLANATION.

First, we divide the number of farthings, 7003, by 4, because 4 farthings make one penny; the quotient is 1750 pence, and 3 farthings remaining.

Secondly, we divide the number of pence, 1750, by 12, because 12 pence make one shilling; the work being performed by Long Division, we get for the quotient 145 shillings, and 10 pence remaining.

Thirdly, we divide the number of shillings, 145, by 20, because 20 shillings make one pound; cutting off the cipher from the right of 20, and the right-hand figure from the dividend, (ART. 30,) we perform the work by Short Division, and obtain the quotient, 7 pounds, and 5 shillings remaining.

We may, therefore, deduce this general

RULE.

I. Divide the given number by as many of its denomination as make one of the next higher; write down the quotient and remainder, if any.

II. Divide the quotient by as many of its denomination as make one of the next higher; write this new quotient and the remainder as before.

III. Proceed in this way through all the denominations to the highest, and the quotient last found, together with the several remainders, if any, will give the value sought.

Repeat the Rule for Reduction Ascending.

EXAMPLES.

1. In £47 5s. 2d. 1 far., how many farthings?

2. In 118567 farthings, how many pounds, shillings,

Hence, 118567 farthings are equal to £123 10s. 1d. 3 far.

3. Reduce £75 to shillings.

4. Reduce 19s. 6d. to pence.

5. Reduce 15s. 3d. 2 far. to farthings.

Ans. 1500s.

Ans. 234d.

Ans. 734 far.

6. In 48926 grains, Troy Weight, how many pounds, ounces, pennyweights, and grains?

Ans. 8lb. 5oz. 18pwt. 14gr.

7. In 3605 pennyweights, how many pounds, ounces, and pennyweights? Ans. 15lb. Ooz. 5pwt. 8. In 1000 ounces, Troy Weight, how many pounds and ounces ?

Ans. 83lb. 4oz.

9. In 4lb. 6oz. 13pwt. 5gr. how many grains?

10. In 100lb. 1gr. how many grains?

Ans. 26237gr.

Ans. 576001gr.

11 In 4 lb 53 1 3 how many drams? Ans. 425 3. 12. In 1000 grains, Apothecaries' Weight, how many ounces, drams, scruples, and grains? Ans. 23 03 29. 13. In 11521 grains, Apothecaries' Weight, how many pounds? Ans. 2 03 03 09 1 gr. 14. In 873450 drams, Avoirdupois Weight, how many tons? Ans. 1T. 10cwt. 1qr. 23lb. 14oz. 10dr. 15. Reduce 5cwt. 21lb. 4oz. to ounces.

16. Reduce IT. 1cwt. 1dr. to drams.

Ans. 9300 ounces.

Ans. 602113 drams.

17. Reduce 856702 drams to tons.

Ans. 1T. 9cwt. 3qr. 14lb. 7oz. 14dr.

18. In 4355 inches, how many yards?

19. In 248 miles, how many

Ans. 120yds. 2ft. 11in. inches?

Ans. 15713280 inches.