£. S. d. £. s.

d.

As there are no numbers, 57) 12 16 6(0 4 6 which by multiplying, will pro

20

)256 shillings.

228

28

12

)342 pence.

342

000

duce 57, we must divide by the whole number. But 57 is not contained in 12 the number of pounds; hence we reduce the pounds to shillings, by multiplying them by 20, and add the 16 s. to the product, which makes 256 s. in which 57 is contained 4 times and 28 s. remain. Reduce these to pence, and add the 6 d.; the amount is 342 d. in which 57 is contained exactly 6 times.

In some instances, the divisor is not given directly, but must be ascertained from the condition of the question.

EXAMPLE.

How

Divide 189 dollars among 3 men, A, B, and C ; give B twice as much as A, and C 3 times as much as B. much will each receive?

1 A 1 part.

2 B 2 parts.

6 C 6 parts.

9)189

21 A's share.
2

We first give one dollar to A, 2 dollars, (or twice as much) to B, and 6 dollars, (or 3 times as much as B receives,) to C. Thus 1+2

+6=9 dollars are disposed of. We now wish to know how many times we can repeat this process. This is ascertained by dividing the 180 dollars by 9, which gives the quotient of 21. Now as A receives 1 dollar at each division, it is evident that 21 dollars is his share. B receives twice as much, 21 × 2=42; and C 3 times as much as B, 42 ×3=126. This is proved to be correct by adding the several shares together; thus, 21 +42+ 126 = 189 dollars. Hence we deduce the following rule,

42 B's share.
3

126 C's share.

,

RULE.

"When the shares of partners are unequal, find how many of the least share are contained in the whole number of shares.

Divide the given sum by the whole number of shares, and the quotient will be the value of the least share.

Multiply the quotient by the number of shares belonging to each partner separately; and the product will be the required answer."

METHOD OF PROOF.

Compound Multiplication and Division mutually prove

each other.

EXAMPLES.

What is the weight of 6 loads of hay, each weighing 18 cwt. 3 qrs. 15 lbs. ?

Proof.

COMBINATION OF THE PRECEDING RULES.

Reduction teaches the method of bringing numbers of one denomination into another retaining the same amount of value.

"Reduction Descending"* is the bringing of a higher into a lower denomination, as pounds into shillings, pence, and farthings; and is performed by multiplication.

"Reduction Ascending," is the bringing of a lower into a higher denomination, and is performed by division. It is exactly the reverse of the former.

When a question to be reduced contains numbers of different denominations, we may reduce them all to one denomination, and then perform the operation by the simple rules. If the question is, How many pence in 6 £. 14 s. and 7 d. we reduce the whole sum to pence, by multiplying the pounds by 20, and the shillings by 12.

To reduce a sum of different denominations to the lowest contained in it, multiply the highest by that number which it takes of the next lower denomination, to make one of that number, and to the product add the number belonging to the lower. Proceed with the other denominations in the same manner till the whole is reduced to the denomination required.

How many inches are there in 2 rods, 6 ft. and 4 inches? Ans. 472.

If we divide the 1615 pence obtained in the first example by 12, we shall evidently reduce them to shillings; and by dividing these by 20, we shall reduce them to pounds.

*The terms Reduction Ascending, and Descending are objectionable; but as they are common, it is thought better to use them. Reduction combines several of the other rules, and involves no different principles.

When lower denominations are to be brought into higher, divide the lower by that number which it takes to make one of the next higher. Proceed in the same manner through all the denominations. The remainder after each division will be of the same denomination as the dividend.

In 21758 grains how many pounds, &c.?

Change the order of the question and divide the last product by the last multiplier; or if the sum is in Reduction Ascending, multiply the last quotient by the last divisor, and so on, till all the denominations are divided or multiplied.

EXAMPLES.

In 6 rods, 10 feet, and 9 inches, how many inches?

In some instances, the conditions of the question to be answered require a preparation to be made before the direct answer can be obtained.

EXAMPLE. I have 72 shillings which I am to divide between John, George, and William, so that George shall have 4 cents as often as John has 2, and William 6.

In 72 s. there is part of 72 dollars, because 6 shillings make one dollar.

6)72(12 which is the number of dollars to be divided. In one dollar are 100 cents; 100 × 12 =1200 cents, the number to be divided.

4 0 0 Geo.'s share. 6 0 0 William's share.

SECTION XV.

FRACTIONS.

Fractions denote a part or parts of a unit; though sometimes the term fraction denotes numbers larger than unity. They are of two kinds, Vulgar and Decimal.

To multiply by,, &c. is only to take,, &c. of the multiplicand.