6. If when the quartern loaf cost 6d. the threepenny loaf weigh 2lb. 2oz. 12dr. what ought it to weigh when the quartern costs 8d.? Ans. 1lb. 10oz. 1dr.

7. If 2 cwt. be carried 204 miles for a certain sum, how much ought to be carried 24 miles for the same money ? Ans. 17cwt.

8. How much must be cut off from a board 8 inches wide, to make a top to a stool 12 inches long and 12 wide? Ans. 18 inc. 9. How many crowns are equal in value to 200 half-guineas? Ans. 420.

10. A field of wheat can be reaped by 20 men in 4 days; now supposing only 8 men can be hired, how long will they require to reap the same? Ans. 10 days.

11. A person drinks 100 bottles of wine, at 2s. 6d. per bottle, in a year; now supposing the price of the wine increases to 4s. 3d. the bottle, how much may he drink without increasing the expence ? Ans. 58 bottles, 42 rem.

12. A family takes 12 loaves, at 1s. 2d. per loaf, in a week, but bread rising, they make shift with 9 loaves, which cost exactly as much; what is the increased value of the loaf? Ans. 1s. 6d., 6 rem.

136. COMPOUND PROPORTION.

When five terms are given to find a sixth, this rule is called The Double Rule of Three, or The Rule of Five; also, whatever number of terms is given, it is usually called by the general name of Compound Proportion.

137. RULE I. Let that term be put in the second place which is of the same kind with the answer required.

II. Place the terms of supposition one above another in the first place, and the terms of demand one above another in the third, so that the first and third terms in each row may be of the same kind; let them be reduced to the same denomination, and the second to the lowest mentioned in it.

III. Examine each stating separately (using the second term in common for each) by saying, If the first term give the second, does the third term require more or less than the second? if more, mark the less extreme for a divisor; but if less, mark the greater extreme.

III. Multiply the unmarked numbers together for a dividend,

and the marked ones together for a divisor; divide the dividend by the divisor, and the quotient will be the answer in the same denomination the second term was brought into TM.

EXAMPLES.

1. If 6 men spend 154 shillings in 7 days, what sum will 8 men spend in 9 days?

Here 154 shillings is of the same kind with the answer, and is therefore made the second term; 6 and 7 are evidently the terms of supposition, and therefore put in the first place; 8 and 9 being terms of demand are put in the third. I then say, If 6 men spend 154 shillings, 8 men will spend more ; I therefore mark the less extreme 6 for a divisor: also, If 7 days spend 154 shillings, 9 days will require more; I therefore again mark the less extreme 7. Next I multiply the three unmarked numbers, viz. 154, 8, and 9, together for a dividend, and the product is 11088; and also the two marked numbers 6 and 7 for a divisor, and the product is 42. Dividing the former by the latter, the quotient is 264 shillings, which divided by 20 gives 137. 4s. for the answer. 2. If 4 gallons of beer serve 5 persons 6 days, how many days will 7 gallons last 8 persons?

If 4 gallons serve 6 days, 7 gallons will serve more; I therefore mark the less 4. Again, if 5 persons are supplied 6 days, 8 persons will be supplied (with the same quantity) less than 6 days. I therefore mark the greater extreme 8, and proceed as before.

m The truth of this conclusion may be shewn by working the first example according to the rules of simple Proportion, namely, by employing two statings.

Thus, 6 men: 154 shil. : : 8 men:

:: 9 days :

154 X 8 X 9
6x7

154 X 8
6

shil. Then 7 days:

shil.

154 X 8 6 264 shil. 137. 4s. the answer as above. In the

same manner every example in the rule may be proved; and it will furnish a profitable exercise for the industrious student to prove all his operations in this rule by two single rules of three statings.

3. If 100l. in 12 months gain 57. interest, what sum will Sol. gain in 10 months?

4. If 8 men in 9 weeks earn 751. 128. how many weeks must 11 men work to earn 100%. ?

Here 75l. 12s. = 1512 shil. 100l.=2000 shil.

5. If a carrier receive 21. 2s. for the carriage of 3cwt. 150 miles, how much must be paid for the carriage of 7cwt. 3qr. 14lb. 100 miles? Ans. 31. 13s. 6d.

6. If 10 acres of grass be mowed by 2 men in 7 days, how many acres can be mowed by 24 men in 14 days? Ans. 240 acres. 7. If a man earn 5 shillings a day, what sum will 64 men earn in 12 days? Ans. 2001.

8. If 100l. in 12 months gain 37. interest, in what time will 751. gain 11. 13s. 9d.? Ans. 9 months.

9. If 13cwt. be carried 20 miles for 21.

I have carried 50 miles for 31. 3s. 6d. ? 162 rem.

10s. what weight can Ans. 6cwt. 2qr. 11lb.

10. If 150l. gain 31. 7s. 6d. in 9 months, what sum will gain 31. in 12 months? Ans. 100l.

11. If 3 horses eat 12 bushels of oats in 16 days, how many quarters will 200 horses eat in 24 days? Ans. 150 quarters.

12. If 24 men can build a wall in 36 days, how many men would be required to do 5 times as much work in 3 days? Ans. 1440 men.

PRACTICE.

137. Practice is an easy method of solving such rule of three questions as have unity for their first term. The operations consist of Compound Multiplication and Division; and the latter is performed by the given price, &c. taken in aliquot parts of an unit of some superior denomination.

138. One number is said to be an aliquot part of another, when the former is contained some number of times exactly in the other; that is, when the former will divide the latter without leaving any remainder.

a The rules of Practice are particularly useful to merchants and traders, in computing the value of their commodities. These rules are likewise useful in the Mathematics; and it is on account of their ready and convenient application to the computations, which occur in almost every branch of science, that they are introduced in this place.

139. Preparatory direction. Write down the given number, namely, the number which expresses the quantity of goods of which you want to find the value, and to the left of it draw three perpendicular lines sufficiently wide apart to contain a column of figures in each of the two intervals; then, having (if necessary) broken the given price into sums which are respectively aliquot parts of some greater whole, and of one another, you are to place these sums in their proper order under one another in the left hand interval; and opposite each, in the right hand interval, place the number expressing what aliquot part it is; then, work according to the directions given in the rule to which the particular question belongs.

140. When the price is an aliquot part of a penny.

RULE I. Having drawn the lines, and placed the price and the aliquot part as above directed, divide the given number by the aliquot part, and the quotient will be the answer in pence; these (if there be a sufficient number) must be divided by 12 and 20, to reduce them into pounds.

II. If when you have divided by the aliquot part there be a remainder, it will be pence, and must be reduced to farthings, which being divided by the aliquot part, will give farthings.

EXAMPLES.

1. What is the value of 3571 yards of tape, at d. per yard?

OPERATION.

3571

12 1785

20 148 9

Answer 71. 88. 9d4

Explanation.

Here the given price (a halfpenny) is one half of a penny; I put the halfpenny in the left hand column, and the half in the right; I then divide by 2, and the quotient is 1785, with 1 remaining; this 1 penny I make 4 farthings, which divided by 2 gives; I then divide successively by 12 and 20.

• Nothing can be easier to understand than the rules of Practice. In the rule here given let us take Example 1; where, if 3571 yards had cost a penny each, they would evidently have amounted to 3571 pence; but as they cost only. each, it is plain that they will amount to half that number of pence. And in Example 2, 9867 pears at 7. each will, it is plain, amount to one quarter that number of pence: wherefore this rule is evident.