use of these operations by familiar questions involving low numbers. For instance, he may be called to find how many apples are wanted in order to give 4 a-piece to 16 persons; or called to divide 96 apples equally among 4 persons. Reduction descending and ascending will furnish a variety of useful examples for exercising the student in multiplication and division.

REDUCTION DESCENDING.

To reduce a quantity from a higher to a lower denomi

nation.

34. Rule. Multiply the number which expresses the quantity by the number which shows how many of the lower denomination make one of the higher; and if any part of the given quantity be already of the lower denomination, add it to the product.

Federal money, or the money of the United States.

35. The denominations of this money are eagle, dollar, dime, cent, and mill.

Hence the following particular Rule for reducing federal

money from a higher to a lower denomination.

Eagles multiplied by 10 give dollars.

Dollars

1. In 350 eagles and 6 dollars, how many cents?

2. How many mills in 365 dollars, 37 cents, and 3 mills?

:

Here it is proper to observe, that instead of multiplying the dollars by 100, and multiplying the product thus arising, after adding 37 cents, by 10, we may write down the dollars, cents, and mills, as we do whole numbers; because this money increases anddecreases according to the decimal notation this is evident from the above example. But, if the cents be less than 10 or, which amounts to the same thing, if there be no dimes, then 0 must be put in the place of dimes. For instance, if it were required to reduce 47 dollars and 7 cents to cents; we must put 0 in the dimes', or tens' place, and the cents will be expressed; thus, 4707 cents. And if there are no cents, two ciphers are used. F

It may also be observed, that as one dollar is the unit of federal money, to which all other denominations of the same money are compared, it is usual to separate the dollars from the dimes, cents, and mills, by a point, called a separatrix : for instance, 375 dollars, 29 cents, is written thus:

$375.29,
cts.

and sometimes thus; 375-29, although it is not necessary to place the mark (cts.) of cents over the cents, when they are separated from the dollars by a point.

In like manner, if it were required to write dollars, cents, and mills, the cents are usually separated from the mills by a point for instance, 375 dollars, 29 cents, and 3 mills, is written thus,

$375.29.3,

cts. m.

and sometimes thus; 375 29.3.

When the marks of the several denominations are placed over the quantity, as in the last instance, it is not necessary to separate the dollars, cents, and mills, by points; and it is usually written thus,

$ cts. m.

375 29 3

It is plain that any sum of this money may be considered as mills, or as cents and mills, by removing the point or points, without the operation of reduction: for instance,

cts. m.

$375 29.3 is reduced to 37529 3, or 37529 cents and 3 mills, by removing one point; and again, by removing both points, it is reduced to 375293 mills.

This money is usually read and written in dollars and cents, because accounts in the United States are kept in dollars and cents, the mills being generally neglected, or, if they are considered, parts of a cent are used instead of them; for instance, 5 mills, or any number less, is written half a cent, orcent; for any number of mills between 5 and 10 a cent is usually counted; sometimes one-fourth (marked 4) and three-fourths (marked 2) are reckoned.

The eagles and dimes are seldom mentioned in reading this money, the former being considered as tens of dollars, and the latter as tens of cents.

3. Reduce 37 eagles, 7 dollars, and 7 mills to mills.

Ans. 377007 mills.

4. In 47 dollars, how many cents and mills?
Ans. 4700 cents and 47000 mills.

5. Reduce 37 cents to mills.
6. Reduce 375 dollars and 3 mills to mills.

Ans. 370 mills.

Ans. 375003 mills.

7. Reduce 976 dollars and 9 cents to cents.

Ans. 97609 cents.

3. In 36 eagles, 17 dollars, and 7 cents, how many cents? Ans. 37707 cents.

9. In 760 dollars, 3 cents, and 7 mills, how many mills? Ans. 760037 mills.

English Money.

35. The denominations of English money are pound, shilling, penny, halfpenny, and farthing.

Table.

4 farthings, (gr., 4) or 2 halfpence, ()=1 penny, (d.)

make

= shilling (s.) =1 pound (£)*

Hence the following Rule for reducing this money from a higher to a lower denomination.

Example 1. In 76 pounds, 13 shillings, and 9 pence, how many pence?

* L. s. d. and q. are the initials of the Latin words libra, solidi, denarii, and quadrantes, signifying pounds, shillings, pence, and far things.

Here it may be observed that 13 may be added mentally to the product of 76 by 20 according as we perform the operation of multiplication, and only the sum is put down; and so on for the shillings and pence: this will abridge the work, and the operation of the preceding example will stand thus ;

£8. d.

76 13 9

20

1533 s. in £76 138.

12

Ans. 18405 pence, as before.

Here, in multiplying 76 by 20, a cipher must be in the unit's place; therefore, in adding 13 to the product, 3 must be in the unit's place of the sum; hence we put 3 in the unit's place again, 6×2 tens gives 12 tens, to which 1 ten must be added, and the sum will be 13 tens. We must, therefore, put 3 in the place of tens, and carry 1 to the place of hundreds, and then proceed as in whole numbers. In like manner, the multiplication of 1533 by 12, and the addition of 9 to the product, is performed mentally: the same method may be used in working the following examples :2. In £3974 how many farthings? Ans. 3815040. 3. In £99 how many shillings, pence, and farthings? Ans. 1980s. 23760d. and 95040qrs.