151. A Simple Number is a number expressed in units of one denomination; as, 5 books, 7 pens.

152. A Compound Number is a number expressed in units of two or more denominations, but of the same nature; as, 5 pounds 6 ounces of sugar, 3 years 2 months 4 days of time. 153. Reduction is the process of changing the denomination of numbers without altering their value.

154. Reduction Descending is the process of changing numbers to numbers of equal value in lower denominations; thus, 1 dollar 100 cents.

155. Reduction Ascending is the process of changing numbers to numbers of equal value in higher denominations; thus, 100 cents 1 dollar.

156. Compound numbers express Currency, Weight, and Measure.

CURRENCY.

Every nation has its own currency.

That of the United

States has already been given (Art. 68), but the table will be inserted here for the sake of uniformity.

157. FEDERAL MONEY.

The denominations are eagles, dollars, dimes, cents, and mills. The legal coins in circulation are as follows:

Copper and nickel Cent, and Two Cent pieces.

ΝΟΤΕ.

copper and silver

- The gold coin is hardened by an alloy of (the silver not to exceed the copper). The silver coin is hardened by copper. The cent coined since 1856 has 88 parts of copper to 12 of nickel. The two-cent piece, coined 1864, has 95 parts copper to 5 of tin and zinc. TABLE.

NOTE.-Mill is derived from the Latin mille, one thousand, because

1000 mills 1 dollar, the unit of computation; cent from Latin centum, one hundred, because 100 cents 1 dollar; dime from the French dime,

=

a tenth, as a dime is one tenth of a dollar; dollar from the German thaler, dollar, dollars having been first coined in Germany.

EXERCISES.

1. Write 3 E. $2. 7 d. 5 c. 2 m. as it is usually written.

2. Write 162 E. $8. 3 d. 9 c. 8 m. as it is usually written.

Ans. $32.752.

9. Add the answers of the last six examples, and give the amount in mills.

Ans. 42,017,798 mills

158. ENGLish Money.

The denominations are pounds, shillings, pence, and farthings.

TABLE.

4 farthings (qr. or far.) = 1 penny, marked d.

NOTE.

-

-The guinea of 21 s., and the crown of S., are also used. The

soin which represents the £ value is called a sovereign.

240d, a £

159. REDUCTION DESCENDING.

ILL EX. Reduce 3 £ 11 s. 8 d. 2 far. to farthings.

OPERATION.

3 £ 11 s. 8 d. 2 far.

20

71 s. 12

860 d.

4

3442 farthings, Ans.

As 20 s. 1 £, we shall have 20 times

(20 × 3) s.

60 s.;

1 s., we

as many s. as £.
60 s. 11 s. 71 s. As 12 d.
shall have 12 times as many d. as

s.; (12 × 71).d. 852 d.; 852 d. +8 d. = 860 d. As 4 far. 1 d., we shall have 4 times as many far. as d.; (4 × 860) far. = 3440 far; 3440 far. +2 far. 3442 farthings. Hence the

RULE FOR REDUCTION DESCENDING. Multiply the number of the highest denomination by the number which it takes of the next lower denomination to make one of that higher, and to the product add the given number of the next lower denomination. Multiply that sum in like manner, and thus proceed till the number is reduced to the required denomination.

EXAMPLES.

1. Reduce 7 £ 8 s. 3 d. 3 far. to farthings. 2. Reduce 30£ 2 s. 0 d. 2 far. to farthings. 3. Reduce 8£ 0 s. 3 d. to farthings.

4. Reduce 9 s. 1 d. 2 far. to farthings.

5. Reduce 368 £ 17 s. 2 d. to pence.

6. Reduce 25 crowns 3 s. 2 d. to farthings. 7. Reduce 43 crowns 4 s. 8 d. to pence.

8. Reduce 209 guineas to pence.

Ans. 7119 far.

Ans. 28898 far.

9. What will be the number of farthing candles that may be bought for 2 s. 6 d.?

160. REDUCTION ASCENDING.

ILL. EX. Reduce 3579 farthings to an equivalent value in higher denominations.

4) 3579

OPERATION.

12) 894 d. +3 qr.

20) 74 s. 6 d.

3£ 14 s. 6 d. 3 qr., Ans.

As 4 qr. = 1 d., we shall have Į as many pence as farthings, or 894 d. and 3 qr. remaining; as 12 d.: = 18., we shall have as many shillings

as pence, or 74 s. and 6 d. remaining. As 20 s. 1 £, we shall have as many £ as s., or 3£ and 14 s. remaining, making the entire result 3 £ 14 s. 6 d. 3 qr.

the

Hence

RULE FOR REDUCTION ASCENDING. Divide the given number by the number which it takes of its denomination to equal one of the next higher, and note the remainder. Divide the quotient thus obtained as before, and thus proceed till the required denomination is attained. The last quotient, with the several remainders, will be the required result.

PROOF. As reduction ascending is the converse of reduction descending, either process may be proved by the other.

EXAMPLES.

1. Reduce 3681 farthings to an equivalent value in higher denominations.

In the same manner reduce,

2. 36875 farthings.

3. 4328 pence.

Ans. 3 £ 16 s. 8d. 1 qr.

Ans. 38 £ 8 s. 2 d. 3 far.
Ans. 18 £0 s. 8.

4. 39818 shillings.

5. 86347 farthings.

6. 298721 farthings.

WEIGHT.

162. TROY WEIGHT.

Gold, silver, and precious stones are weighed by this system. The denominations are pounds, ounces, pennyweights, and grains.

24 grains (gr.) 1 pennyweight, marked pwt.

20 pwt.
12 oz.

TABLE.

ILL. EX. Reduce 2 lb. 9

oz. 18 pwt. 3 gr. to grains.

OPERATION.

2 lb. 9 oz. 18 pwt. 3 gr.

12

33 oz.

20

678 pwt.

24

2715

1356

16275 gr., Ans.

ILL. EX.

to numbers of higher denomina

tions.

OPERATION.

24) 16275 gr.

210) 6718 pwt. +3 gr.

12) 33 oz. +18 pwt.

2 lb. +9 oz.

Ans. 2 lb. 9 oz. 18 pwt. 3 gr.

EXAMPLES.

1. Reduce 18 lb. 11 oz. 5 pwt. 17 gr. to grains.

2. Reduce 48 lb. 2 oz. 0 pwt. 3 gr. to grains. 3. Reduce 1 oz. 23 gr. to grains.

Ans. 109097 gr.

4. Reduce 3681 lb. 9 oz. 1 pwt. to pennyweights.

Reduce to equivalent values in higher denominations,

5. 928641 pwts.

6. 3786541 grs.

7. 9042028 grs.

Ans. 3869 lb. 4 oz. 1 pwt.

8. What is the value of 2 lb. 8 oz. of gold, at $1600 an Ounce ?

5740gr to lb. toy