The supposition in this question is, that 7 C. of pepper costs £21 and the two terms in it are 7 C. and £21; the demand is, how much will 5 C. cost? and the term in it is 5 C.

The number sought is the price of 5 C. and the term in the supposition of the same kind is the price of 7 cwt. viz. £21, which I place in the middle. The two remaining terms are extremes, and of the same kind, viz. quantities of pepper.

C. £. C.
As 7 21:5

5

7)105

£15 Answer.

It is obvious, that the answer must be less than the middle term; for 5 C. will cost less than 7 C. and therefore the greatest extreme, viz. 7 C. is the divisor.

Accordingly, I place the divisor 7 C. on the the left hand, and the other extreme, 5 C. on the right; and having multiplied the second and third terms, I divide their product by the first term, and the quotient 15 is the answer, of the same name with the middle term, viz. £. sterling.

And because the divisor happens to be the extreme in the supposition, the proportion is direct.

When there happens to be a remainder, it may be reduced to the next inferior denomination, and the operation continued; and in this case the quotient will consist of two or more parts.

N. B. The remainders ar e always of the same name with the middle term.

3. If 14 lb. of tobacco cost 27 shillings, what will 478 lb. cost at that rate ?

4. If 18 cwt. of sugar cost £54, what will 7 cwt. 3 qrs. 14 lb. cost at that rate?

When any or both of the extremes happen to be complex, or mixed, and so consist of two or more parts, reduce both extremes to the lowest of the parts; for in the operation both extremes must be equally low, or of the same name. .Thus,in this example the extreme on the right hand, is complex,consisting of three parts, viz. cwt. qrs. lbs. and so I reduce both this extreme. and also the other on the left hand, to pounds, the lowest of the parts.

brought up. 12096

Ans. £23 12 6

5. If the early rent of a house be £73, how much is that per

day?

Days. £. Day.

As 365: 73: : 1

20

1460(4s.
1460

When the third term happens to be 1, you have only to divide the middle term by the first, and the quotient is the answer. But in this case, the middle term being less than the first, it must be reduced to some lower denomination, viz.

till the middle term thus reduced may be divided by the first.

6. A merchant bought 14 pieces of broad cloth, each piece containing 28 yards, at 13s. 6d. per yard; how much did the 14 pieces

cost?

£260 8 4 Answer.

The middle term here being complex, is reduced to half pence, and accordingly the product, or answer, comes out in half pence, which I reduce to pounds.

CONTRACTIONS OR CANCELLING.

1. If the first term be an aliquot part of the second or third, divide the said second or third term by the first, and the product of the quotient and the other term is the answer. Thus if 9 27 : : 42, then divide 27 by 9, and the quotient 3×42-126 the answer.

Again, if 85: 64, divide 64 by 8, and the quotient 8X5= 40, the answer.

EXAMPLE. 1. If 20 lb. of sugar cost 20 dollars, what will 70 lb come to at that rate?

lb.

As ·2·0

jb.

dolls.
::2.0: 70: 70 dollars, answer.

EXPLANATION. Because 20 measures 20, and bears the same proportion to each other, as 1 does to 1, therefore I cancel each, and 70 stands for the answer required.

2. A. B. C. and D. have £100 to be divided among them, in such a manner, that for every £3 A receives, B must have 5, C 7, and D 10; that is, their shares are to be as the numbers 3, 5, 7, 10. Required their several shares ?

Add the proportional numbers, thus: 3+5+7+10=25, then say : 3X12 A's share. 5X 20 B's share. 7X28 C's share. 10X 40 D's share.

As 2.5 1.0·0: :

100 proof.

Because 25 measures 100, I cancel each, and multiply the quotient 4 into all the extremes on the right hand, and the products are the respective shares or answers.

2. If the first term be a multiple of the second or third, divide the first term by the said second or third; and the remaining term, divided by the quotient, gives the answer.

EXAMPLE 3. If 60 yards cost £20, what will 45 yards cost?
£. Yds.

Yds.

As 6.0

.3

£15 Answer.

Here the first term 60 is a multiple of the second term 20, and 60÷÷ 20=3, and 45÷÷3=15£. Answer.

4. If 18 yards cost £12, what will 6 yards cost?

Yds. £. Yds.

As 1.8 1.2 ·6

⚫3 £4 Ans.

Here the first term 18, is a multiple of the third term 6, and 186=3, and 12÷3=4£. Ans. So 18 bears the same proportion to 6, as 3 does to 1, therefore 6 destroys 18=3, and 3 reduces 12=£4. N. B. Sometimes a question of the rule of three may be performed by multiplication only.

Sometimes by division only.

Sometimes by two multiplications.
Sometimes by two divisions.

And sometimes by a smaller multiplication and division, than your given numbers allow of.

If at any time your divisor, with either of the other numbers, may be severally divided by some common measure, without any remainder, and your divisor come to be an unit, the answer will be given by multiplication only; as in the following example

5. If 6 yards of velvet lace cost 15 dollars, how many dollars will 48 yards cost at that rate?

$120 Answer.

Here the first term 6, bears the same proportion to the third term 48, as 1 8-therefore I multiply 15, the middle term, by 8, which gives $120 for the answer.

But if either of the other numbers comes to be an unit, the work may be performed by division only, as in the following example : If 18 yards cost 12 dollars, what will 6 yards cost?

As

Yds. 1.8 +3

$

Y ds.

1.2 :: ·6 $4 Ans.

Here the first term 18 is cancelled by the third term 6=3, and 3 cancels 12=4 dollars, Answer. If your divisor or first term be exactly contained in both the second and third, the question may be answered by two multiplications; as in the following example : If 3 yards cost 9 dollars, what will 12 yards cost?

Yds. $ Yds.

36 dollars, Ans.

Here the third part of 9 is 3, of 12 is 4; multiply 3 by 4 gives 12, and that by 3 gives 36, the Ans.

If you would use two divisions, divide your divisor by your second term, and the third by that quotient, which gives the answer. EXAMPLE. If 9 gives 3, what will 45 give?

As 9 3: : 4.5

So dividing 9 by 3, the quotient is 3, and by that dividing 45, quotes 15 the answer.

Sometimes you must use both multiplication and division, yet but small ones.

RULE.

Suppose the second or third terms cannot be measured by the first term, but the whole, viz. 1st, 2d and 3d terms to be surd numbers; in this case, divide the second or third terms by the first, and multiply the quotient and the other term together, the product will give the

answer.

EXAMPLE. If 5 yards of cloth cost 6 dollars, what will 11 yards cost at that rate?