or) 1,094 pence and 3 farthings. Ascending to the pence room, he will change 1,094 pence into shillings, he will then have (1,094 divided by 12, or) 91 shillings and 2 pence 3 farthings; lastly, ascending to the shilling room, he will change 91 shillings into pounds, when he will have altogether £4 11s. 2åd.

This is called Reduction ascending, and the operation may be exhibited thus :

4) 4379 farthings

12) 1094d.

39.

Dividing the farthings by 4, we get 1,c94d. and 3 farthings over; next, dividing 1,094d. by 12, we get 91 shillings and 2d. over; and lastly, dividing 915. by 20, we IIS. get £4 and IIS. Over; therefore, the whole sum is £4 11s. 2åd.

2,0) 9, IS.

£4.

2d.

But if a person wishes to change 359 half-crowns into florins, then, since there is no direct communication between the half-crown and florin room (see Diag. 2), because no number of florins will exactly make one half-crown, he must descend from the half-crown room, and passing through the sixpenny ascend to the florin room. In the half-crown room he will change 359 half-crowns into (359 multiplied by 5, or) 1,795 sixpences, and passing through the sixpenny room he will ascend to the florin room, where he will get (1,795 divided by 4, or) 448 florins and 3 sixpences that is, 448 florins and Is. 6d.

This may be called combined Reduction, since it unites the two processes of descending and ascending Reduction. The operation may be exhibited thus:

359 half-crowns

5

4) 1795 sixpences

448 florins 3 sixpences, or 448 florins, Is. 6d.

From the preceding illustrations we deduce the following rules:

1. Descending Reduction.

Begin with the quantity of the highest name, and multiply by the number which connects it with the next lower denomination, adding in the figures of this denomination from the top line. Apply the same process to this result, and repeat the operation till, step by step, the required denomination is arrived at.

II. Ascending Reduction.

Divide the quantity by the number which connects it with the next higher denomination, and set down the remainder (if any). Apply the same process to the quotient, and repeat the operation till, step by step, the required denomination is arrived at.

III. Combined Reduction.

Multiply the given quantity by the number which connects its denomination with the highest denomination common to the two given quantities, and divide by the number which connects the denomination of the result with the required denomination. Proof. Reverse the operation, and bring the answer back to its original form.

Examples.

1. Reduce £79 15s. 6d. to crowns and sixpences.

£79 15s. 6d. = £79 3 crowns 1 sixpence.

4

319 crowns

IO

3191 sixpences.

2. Reduce 43 lbs. 9 oz. 17 dwts. 21 grains to grains. (Troy weight, see p. 10).

3. Reduce 10,170,684 drams to tons, cwts., &c. (Avoirdupois weight, see p. 11.)

The result is 17 tons 14 cwt. 2 qrs. 25 lbs. 3 oz. 12 drs.

4. Reduce 37 poles 4 yards to feet. (Lineal measure, see p. 11.)

p. yds.

37 4

5

189

181

207 yds.

3 622 ft.

To bring poles to yards, multiply by 5; thus, 5 times 37 is 185, and 4 yards in the top line make 189; now the half of 37 is 18, and adding this to 189 we get 207 yards; next, 3 times 1-half is 3 halves, or 1; set down and carry 1; 3 times 7 is 21 and I are 22, &c.

I

To bring yards to poles, reduce the yards to half yards by multiplying by 2, and divide by the number of half yards (11) in a pole; the remainder (if any) will be half yards.