men.

16. A certain building was raised in 8 months by 120 workHow many workmen could have done the same amount of labor in 2 months? Ans. 480 men. 17. How much in length that is 16 rods in width will it take to make an acre? Ans. 10 rods. 18. There is a cistern having a pipe which will empty it in 6 hours. How many pipes of the same capacity will empty it in 20 minutes? Ans. 18 pipes. 19. If 30 men can perform a piece of work in 11 days, how many men will accomplish another piece of work, 4 times as large, in a fifth part of the time? Ans. 600 men.

COMPOUND PROPORTION.

Art. 181.—When a proportion is formed by the combination of two or more simple proportions, it is called Compound Proportion, or Double Rule of Three.

1. If 8 men consume 24 bushels of wheat in 5 months, how many bushels will 4 men consume in 15 months?

In this question, the number of bushels consumed depends on two circumstances-the number of men, and the time. We may consider the circumstances separately, and solve the question by two statements in the Single Rule of Three. First, the number of men. If 8 men consume 24 bushels in 5 months, how many bushels will 4 men consume in the same time?

Operation 1st.
2)

28:4: 24(12 Ans.

Secondly, the time. If 4 men consume 12 bushels in 5 months, how many bushels will the same number of men consume in 15 months?

The first operation is in Simple Proportion, because we employed but one simple ratio as a multiplier upon 24 bushels, the name of the answer, viz., the ratio of 4 men to 8 men. The second operation is also in Simple Proportion, for the The ratio employed is the ratio of 15 months to

same reason.

5 months.

We may now unite these two statements in one, applying the rule already given in Simple Proportion. Thus,

Here we have two terms of demand, viz., 4 men and 15 months; and two terms of condition, 8 men and 5 months.

The ratio of 4 to 8 is, and the ratio of 15 to 5 is 3. If we multiply these two simple ratios together, we have a compound ratio, which, multiplied into 24 bushels, gives the answer. 3, and 24×3=36, the answer. It is the use of a compound ratio which constitutes Compound Proportion.

All questions in Compound Proportion may be solved by two or more statements in Simple Proportion, or they may be analyzed thus: If 8 men consume 24 bushels in 5 months, 1 man would consume of 24=3 bushels, and in 1 month of 3=3 of 1 bushel. Then 4 men would consume 4 times 3=12 in 1 month, and in 15 months 15 times 12=36 bushels, the

answer.

5

Art. 182.-Compound Proportion teaches to solve by one statement questions which would require two or more by Simple 'Proportion.

OBS. 1.-The student should be required, first, to solve the question by analysis, then by proportion.

2. If a man build 27 rods of wall in 3 days, when the days are 12 hours long, how many rods can he build in 9 days, when the days are 16 hours long?

If a man in 3 days build 27 rods, in one day he would build of 279 rods. If in one day, 12 hours long, he build 9 rods, in one hour he would build of a rod, and in 16 hours, ×16=14412 rods. If in 1 day, 16 hours long, he build

12

QUESTIONS.-1. What does Compound Proportion teach? 2. What constitutes Compound Proportion?

12 rods, in 9 days he would build 12×9=108 rods, the

answer.

36

27

In this example, 3 days, 12 hours long, are equal to 12 × 3 = 36 hours; and 9 days, 16 hours long, are equal to 16 x 9=144 hours. We have, then, this proportion; 36 h. 144 h. :: 27 rds. 108 rods, for 144=4, and 108=4. The ratio of the time in the demand, to the time in the supposition, is the same as the ratio of the term sought to the rods in the conditions of the question. That is, the ratio of 144 hours to 36 hours, expresses how many more rods can be built in 9 days, 16 hours long, than in 3 days, 12 hours long.

It will be perceived, that the ratio of the time in the demand, to the time in the supposition, is the product of two simple ratios. It is a ratio of the ratio of days to days, and hours to hours, (a ratio produced by the multiplication of simple ratios is called a compound ratio.) Thus, if a man in 3 days, 12 hours long, build 27 rods of wall, the amount of wall built in 9 days, (the days being of equal length,) is expressed by the ratio of 9 to 3, 2=3; that is, he could build 3 times the number of rods, 27×3=81 rods; but the days are 16 hours long; this circumstance, again, affects the result, and is expressed by the ratio of 16 to 12, 1913; that is, the amount of labor performed in 16 hours is greater by than the labor performed in 12 hours; of 81 rods=27, and 27+81=108 rods. If we multiply these simple ratios, 3×13 =4, we have a compound ratio, the same as above, and 27 x 4 108 rods, the same answer. The ratios of the days to days, and hours to hours, may be expressed thus. If, in 3 days, 12 hours long, 27 rods of wall are built, how many rods days. hours. 6

36

can be built in of 19? 3 of 15=144=4, and 27 ×4=108, the answer.

OBS. 2.-The teacher will now call upon some member of the class, to select the terms, and form first a simple proportion, and then a compound, in the following manner, and illustrate as he proceeds; 39: 27: This statement involves Simple Proportion. The days are considered of equal length. There is but one circumstance that affects the answer, viz.: the difference in the number of the days. This affects it in a threefold ratio. The ratio of 9: 33, which shows how many more rods could be built in 9 days, than in 3 days. But the days are not of equal length. This circumstance must also be considered. We will therefore introduce into the statement another simple ratio. The ratio of 16 to 12, which shows how many more rods could be built in 16 hours, than in 12.

Multiplying antecedents and consequents by antecedents and consequents, the ratios are compounded, and thus the question becomes Compound Proportion.

T. How does it appear that antecedents and consequents have been multiplied?

S. Introducing a factor, multiplies by that factor, and cancelling equal factors from antecedents and consequents, does not affect the ratios. In this example, the factors are all cancelled but 4 and 1. 4 is therefore the compound ratio.

T. Does this question involve Direct or Inverse Proportion? How do you know? How many simple ratios are there? Upon how many circumstances does the answer depend? What are they? In what ratio does the first circumstance affect the answer? The second? In what both combined? What is this ratio called? What is a compound ratio? Does it differ, in itself considered, from a simple ratio?

Question 2.-If a man build 27 rods of wall in 3 days, when the days are 12 hours long, in how many days can he build 108 rods, when the days are 16 hours long?

Let the student state and illustrate this question, in the following

manner:

This question involves Inverse Proportion-more requires less. It would require a less number of days to perform the same amount of labor, when they are 16 hours long, than when they are 12. The number of the days will be inversely as their length. We therefore place the 16 hours in the demand, for the first term, and the 12 hours in the condition for the second. We reject the factor 4, which is common to 16, the antecedent, and to 12, the consequent. Then 4×27--108, which is an antecedent and consequent, and therefore may be rejected. The ratios are now compounded, and the proportion reads, 1: :: 3 : 9 the answer. 7. By what rule are the foregoing questions stated? S. By the rule given for Simple Proportion.

3. If 3 men can build 360 rods of wall in 24 days, how many rods can 8 men build in 27 days?

4. How many men will it take to build a wall, 75 rods long, 8 feet high, 3 feet thick, in 6 days, working 9 hours each day, if 20 men can build a wall 100 rods long, 6 feet high, 4 feet thick, in 12 days, working 12 hours each day?

How many men? 75 5×8=40 men Ans.

5 100 8

3 63
412 3

612 2

920 men.

40 men Ans.

5. If of a yard of cloth, yd. wide, cost £3, what is the value of yard, 13 yard wide, of the same quality?

6. If a man travel 240 miles in 12 days, when the days are 12 hours long, how far can he travel in 27 days, when the days are 16 hours long?

Art. 183.-In Proportion, both Simple and Compound, the terms in the supposition and demand may be distinguished by cause and effect, or producing and produced terms. That which causes any thing, or produces an effect, as men, time, length, breadth, depth, etc., may be denominated a producing term. Thus, in the foregoing question, among the terms of supposition, one man, 12 days, 12 hours long, are the joint cause, or producing rms, and miles the effect, or produced term. Among the terms of demand, 27 days, 16 hours long, are the joint cause, or the producing terms, and the rods required are the effect, or the produced term. In all questions in Proportion the answer required will be either cause or effect, (a producing or produced term.) Hence,

Art. 184. When the term required is a produced term,

Draw a perpendicular line, and place all the terms of demand on the right of the line, and all the corresponding terms of the condition on the left, closing the statement by placing the name of the answer on the right.

QUESTIONS.-1. What is meant by producing and produced terms? 2. Rule, when the term required is a produced term?