Arithmetic Simplified

Change a unit of the order to which the sum is to be reduced, to units of the same order as the numerator, and place it for the denominator.

EXAMPLES.

Reduce 6 oz. 4 pwt. to the fraction of a pound Troy. Reduce 3 days, 6 hours, 9 minutes to the fraction of a month.

Reduce 2 cwt. 2 qrs. 16 lbs. to the fraction of a ton.
Reduce 2 lb. 4 oz. to the fraction of a cwt.

REDUCTION OF A COMPOUND NUMBER TO A DECIMAL FRACTION.

It is often convenient to change a compound number to a decimal fraction.

Thus we can reduce 1 oz. 10 pwt. to a decimal of the pound order.

Let the figures be placed thus, and the process will be explained below. The 10 pwts. are first written, and then the 1 oz. set under.

20)10'0 pwt.
12) 1'5 oz.
125 lb.

We first change the lowest order (10 pwts.) to an improper decimal, thus, 10'0. Now as 20 pwts. make an oz. there are but one twentieth as many ounces in a sum as there are penny weights.

For the same reason, in any sum there are but one twentieth as many tenths of an ounce as there are tenths of a penny weight.

As there are then 100 tenths of a put. in this sum, if we take one twentieth of them, we shall find how many tenths of an oz. there are.

We therefore divide the 100 pwts. by 20, and the

What is the rule for reducing units of one order to fractions of another order?

amount is ,5. This,5 is placed (beside the 1 oz. of the sum) under the 10'0 pwts., and thus, instead of reading the sum as 1 oz. 10 pwts., we read it as 1,5 oz., or 1 oz. and 5 tenths of an oz.

As the pwts. are thus reduced to the decimal of an oz. we now reduce the 1,5 oz. to the decimal of a lb. in the same way.

We make the 1,5 an improper decimal, thus, 1'5 (15 tenths) of an oz.

oz.

Now as there are 12 oz. in a lb. there are but one twelfth as many tenths of a lb. in a sum, as there are tenths of an We therefore divide the 15 tenths of an oz. by 12, and the answer is,1 of a lb. and 3 left over. This 3 is reduced to hundredths by adding a cipher and dividing it again. The quotient is 2 hundredths. The next remain. der is changed to thousandths in the same way, and the answer is,125 of a lb.

RULE FOR CHANGING A COMPOUND NUMBER TO A DECI. MAL.

Change the lowest order to an improper decimal.

Divide

it by the number of units of this order, which are required to make a unit of the next higher order, and set the answer beside the units of the next higher order. Repeat this process till the sum is brought to the order required.

EXAMPLES.

Reduce 10s. 4d. to the decimal of a lb.
Reduce 8s. 6d. 3 qrs. to the decimal of a £.
Reduce 17 hrs. 16 min. to the decimal of a day.
Reduce 3 qrs. 2 na. to the decimal of a yd.
Reduce 32 gals. 4 qts. to the decimal of a hogshead.
Reduce 10d. 3 qrs. to the decimal of a shilling.

What is the rule for changing a compound number to a decimal ?

REDUCTION OF A DECIMAL TO UNITS OF COMPOUND ORDERS.

The preceding process can be reversed, and a decimal of one order, be changed back to units of other orders. Thus, if we have ,125 of a lb. Troy, we can change it to units of the oz. and pwt. order.

In performing the process, we place the figures thus

,125 lb.
12

1,500 oz.
20

10,000 pwt.

We reason thus. In,125 of a lb. there must be 12 times as many thousandths of an oz. (for 12 oz. = 1 lb.) We therefore multiply by 12, and point off according to rule, and the answer is 1 oz. and 500 thousandths of

an oz.

Now as we have found how many oz. there are, we must find how many pwts. there are in the ,500 of an oz. There must be 20 times as many thousandths of a pwt. as there are thousandths of an oz. therefore multiply the decimal only, by 20, and point off according to rule, and we find there are 10 pwts.

We have thus found that in,125 of a lb. there are 1 oz. and 10 pwts.

RULE FOR CHANGING A DECIMAL OF ONE Compound ORDER, TO UNITS OF OTHER ORDERS.

Multiply the decimal by the number of units of the next lower order which are required to make one unit of the order in which the decimal stands.

What is the rule for changing a decimal of one compound order to units of other orders ?

Point off according to rule, and multiply the decimal part of the answer in the same way, pointing off as before. Thus till the sum is brought into the order required. The units of each answer make the final answer.

In,1257 of a £ how many shillings, pence and farthings?

What is the value of‚2325 of a ton?
What is the value of ,375 of a yard?

What is the value of ,713 of a day?

What is the value of,15334821 of a ton?

REDUCTION OF CURRENCIES.

There are few exercises in Reduction, of more prac tical use than the Reduction of Currencies, by which a sum in one currency is changed to express the same val. ue in another currency.

An example of this kind of reduction occurs, when the value of $1 is expressed in British currency thus, 4s. 6d. The necessity for using this process in this country, re. sults from the following facts.

Before the independence of the U. States, business was transacted in the currency of Great Britain. But at various times, the governments of the different States put bills into circulation, which constantly lessened in value, until they became very much depreciated. For example, a bill which was called a pound or twenty shillings, British currency, was reduced to be worth only fifteen shillings, in the New England states.

This depreciation was greater in some states than it was in others, and the result is, that pounds, shillings and pence have different values in different states.

12 pence make a shilling, and 20 shillings make a pound in all cases, but the value of a penny, a shilling, or a pound, depends upon the currency to which it be. longs.

The following table shows the relative value of the several currencies, by showing the value of one dollar in each of the different currencies.

VALUE OF ONE DOLLAR IN EACH OF THE DIFFERENT CUR

·RENCIES.

$1 equals 6s. New England currency.

$1 CO