But since the fractions are equal, and have the same denominators, their numerators must be equal, viz. :

24 X 8 = 32 × 6 ; that is,

In any proportion, the product of the extremes is equal to the product of the means.

239. Since, in any proportion, the product of the extremes is equal to the product of the means, it follows that,

1st. If the product of the means be divided by one of the extremes, the quotient will be the other extreme.

Thus, in the proportion,

4 : 16 : : 6 : 24, and 4 × 24 = 16 × 6 = 96; then, if 96, the product of the means, be divided by one of the extremes, 4, the quotient will be the other extreme, 24; or, if the product be divided by 24, the quotient will be 4. 2d. If the product of the extremes be divided by either of the means, the quotient will be the other mean.

Thus, if 4 x 24 = 16 × 6 = 96 be divided by 16, the quotient will be 6; or if it be divided by 6, the quotient

will be 16.

NOTE. We shall denote the required term of a proportion by the letter x

235. When the antecedent is less than the consequent, what does the ratio express ? What does it express when the antecedent is greater han the consequent ?-236. To what is the consequent equal, In any ratio? To what is the antecedent equal?-237. What is a simple proportion ?--238. Which are the extremes of a proportion? Which the means? What is the product of the means equal to ?239. If the product of the means be divided by one of the extremes, what is the quotient? If the product of the extremes be divided by one of the means, what is the quotient?

Examples.

Find the required term in each of the following examples:

5. The first three terms of a proportion are 5, 10, and 19.

what is the fourth term?

6. The first three terms of a proportion are 6, 24, and 14. what is the fourth term?

7. The first, second, and fourth terms of a proportion are 9, 12, and 16: what is the third term?

8. The first, third, and fourth terms of a proportion are 16, 8, and 20: what is the second term?

DIRECT AND INVERSE PROPORTION.

240. It often happens, that two numbers which are compared together, may undergo certain changes of value, in which case they represent variable and not fixed quantities. Thus, when we say that the amount of work done, in a single day, will be proportional to the number of men employed, we mean, that if we increase the number of men, the amount of work done will also be increased; or, if we diminish the number of men, the work done will also be diminished. This is called Direct Proportion.

If we say that a barrel of flour will serve 12 men a certain time, and ask how long it will serve 24 men, the time will be less that is, the time will decrease as the number of men is increased, and will increase as the number of men is decreased. This is called, Inverse Proportion; hence,

1. Two numbers are directly proportional, when they in or decrease together; in which case their ratio is always the same.

crease

2. Two numbers are inversely proportional, when one increases as the other decreases; in which case their product is always the same.

NOTE. This is sometimes called, Reciprocal Proportion.

First Illustration.

If we refer to the numeration table of integra. and decimal numbers (Art. 190), we see that the unit of the first place, at the left of 1, is 1 ten; that is, a number ten times as great as 1. The unit of the first decimal place at the right, is 1 tenth, a number only one-tenth of 1. The unit of the second place, at the left, is one hundred times as great as 1; while the unit of the second place, at the right, is only one hundredth of 1; and similarly for all other corresponding places. Hence,

The units of place, taken at equal distances from the unit 1, are inversely proportional.

Second Illustration.

The floor of a room is 20 feet long: what must be its breadth, in order that it may contain 360 square feet?

OPERATION.

360

18 ft. breadth.

20

ANALYSIS. The length of the floor, multiplied by its breadth, will give the area or contents; hence, the area, divided by the length, will give the breadth. If the contents remain the same, the length will increase as the breadth diminishes; and the reverse, Hence, when the contents are the same, the length and breadth are inversely proportional.

COMPOUND PROPORTION.

241. A COMPOUND PROPORTION is the comparison of the terms of two equal ratios, when one or both are compound.

Any compound proportion may be reduced to a simple one, by multiplying the elements of each term together; thus, by multiplying together the elements of the last proportion, we have,

20 : 24 :: 15 : 18.

Hence, in any compound proportion,

The product of the extremes is equal to the product of the means; and the required term may be found as in Art. 239.

What are the required terms in the following proportions?

242. If an element is unknown, denote it by x. Then, if all the parts are known except one element, as in the follow. ing proportion,

that element is equal to the product of the means divided by the product of all the elements of the first term and the known elements of the fourth term; thus,

1. What is the required element in the proportion,

240. When are two numbers directly proportional? When are two numbers inversely proportional? What is then said of their product? Give the first illustration of inverse proportion. Give the second.241. What is a compound proportion? What is the product of the extremes equal to ?-242. How do you find the unknown element ?

SINGLE RULE OF THREE.

243. THE SINGLE RULE OF THREE is the process of finding from three given numbers, a fourth, to which one of them shall have the same ratio as exists between the other two.

1. If 8 barrels of flour cost $56, what will 9 barrels cost, at the same rate.

NOTE. We shall denote the required term of the proportion by the letter x.

ANALYSIS.-The condition, "at the same rate," requires that the quantity, 8 barrels of flour, have the same ratio to the quantity, 9 barrels, as $56, the cost of 8 barrels, to x dollars, the cost of 9 barrels.

STATEMENT.

bar. bar. $ $

89: 56: x.

x =

56 × 9

8

= $63.

NOTE. It is plain that 8 barrels of flour will cost less than 9 bar rels hence, the 3d term is less than the 4th, and these terms are directly proportional.

2. If 36 men, in 12 days, can do a certain work, in what time will 48 men do the same work?

ANALYSIS.-Write the required term, x, in the 4th place, and the term 12, having the same unit value, in the third place.

OPERATION.

48: 36: 12: x.

36 × 12

x =

= 9 days.

48

Then, analyzing the question, we see that 48 men will do the work in a less time than 36 men; therefore, the 3d term will be greater than the 4th, which requires that the 1st shall be greater than the 2d. This brings 48 men into the first place, and 36 men into the 2d. The reason of this is obvious: for, the work done by 36 men in 12 days, is a fired quantity. If a less number of men were employed, it will require more time to do the work: if a greater number were employed, it will require less time; hence, the men and time are inversely pro portional.

243. What is the single rule of three? Give the rule.