SIMPLE PROPORTION.

389. Simple Proportion is an equality of two simple ratios, and consists of four terms, any three of which being given, the fourth may readily be found.

When three terms of a proportion are given, the method of finding the fourth is called the Rule of Three.

390. Every question in simple proportion involves the principle of cause and effect.

391. Causes may be regarded as action, of whatever kind, the producer, the consumer, men, animals, time, distance, weight, goods bought or sold, money at interest, etc.

392. Effects may be regarded as whatever is accomplished by action of any kind, the thing produced or consumed, money paid, etc.

393. Causes and effects are of two kinds - simple and compound.

394. A Simple Cause, or Effect, contains but one element; as goods purchased or sold, and the money paid or received for them.

395. A Compound Cause, or Effect, is the product of two or more elements; as men at work taken in connection with time, and the result produced by them taken in connection with dimensions, length and breadth, etc.

396. Causes and effects that admit of computation, that is, involve the idea of quantity, may be represented by numbers, which will have the same relation to each other as the things they represent. And since it is a principle of philosophy that like causes

produce like effects, and that effects are always in proportion to their causes, we have the following proportions:

1st Cause 2d Cause:: 1st Effect: 2d Effect, Or, 1st Effect: 2d Effect:: 1st Cause: 2d Cause.

The two causes, or effects, forming one couplet, must be like numbers, and of the same denomination. Considering all the terms of the proportion as abstract numbers, we may say that

1st Cause: 1st Effect:: 2d Cause: 2d Effect; which will produce the same numerical result.

But as ratio is the result of comparing two numbers or things of the same kind (377), the first form is regarded as more natural and philosophical.

397. Simple causes and simple effects give rise to simple ratios; compound causes and compound effects to compound ratios.

EXAMPLES.

398. 1. If 5 tons of coal cost $30, what will 3 tons cost?

The required term will be denoted by a ( ), and designated "blank."

and the product of the means divided by one of the extremes will give the other, (blank) dollars will be equal to the product of 3 × 30 divided by 5, which is $18.

2. If 15 barrels of flour cost $90, how many barrels

RULE. I. Arrange the terms in the statement so that the causes shall compose one couplet, and the effects the other, putting () in the place of the required term.

II. If the required term is an extreme, divide the product of the means by the given extreme; if the required term is a mean, divide the product of the extremes by the given

mean.

1. If the terms of any couplet are of different denominations, they must be reduced to the same unit value.

2. If the odd term is a compound number, it must be reduced to its lowest unit.

3. If the divisor and dividend contain one or more factors common to both, they should be canceled. If any of the terms of a proportion contain mixed numbers, they should first be changed to improper fractions, or the fractional part to a decimal.

4. When the vertical line is used, the divisor and the required term are written on the left, and the terms of the dividend on the right.

399. There is another method of solving questions in simple proportion, without making the statement, which may be used, by those who prefer it to the one already given.

SECOND METHOD.

Every question which properly belongs to simple proportion must contain four numbers, at least three of which must be given (389).

Of the three given numbers, one must always be of the same denomination as the required number.

The remaining two will be like numbers, and bear the same relation to each other that the third does to the required number; in other words, the ratio of the third to the required number will be the same as the ratio of the other two numbers.

Regarding the third or odd term as the antecedent of the second couplet of a proportion, we find the consequent or required term by dividing the antecedent by the ratio (379).

By comparing the two like numbers, in any given question, with the third, we may readily determine whether the answer, or required term, will be greater or less than the third term.

If greater, then the ratio will be less than 1, and the two like numbers may be arranged in the form of a proper fraction as a divisor; if the answer, or required term, is to be less than the third term, then the ratio will be greater than 1, and the two like numbers may be arranged in the form of an improper fraction, as a divisor.

1. If 4 cords of wood cost $12, what will 20 cords cost?

OPERATION.

4:2012: ().
3
12×20

4

12÷ =

$60.

20

4

3

Or, 20 x 12

$60.

4

SOLUTION.-It will be readily seen in this example, that 4 cords and 20 cords are the like terms, and that $12 is the third term, and of the same denomination as the answer or required term.

If 4 cords cost $12, will 20 cords cost more, or less, than 4 cords? Evidently more: then the answer or required term will be greater than the third term, and the ratio less than 1. The ratio of 4 cords to 20 cords is, or; hence the

ratio of $12 to the answer must be , and the answer will be 5 times $12, which is $60.

Or, after having stated the proportion 4:20::12:(), we find the product of the means and divide by the extreme.

2. If 12 yards of cloth cost $48, what will 4 yards cost?

OPERATION.

12:4::48:( ).

12

4

4

48÷

48 ×

4

12

Or,

4

4 × 48 12

=

$16.

= $16.

SOLUTION. In this example we see that 12 yards and 4 yards are the like terms, and $48 the third term, and of the same denomination as the required answer.

If 12 yards cost $48, will 4 yards cost more or less than 12 yards? Less: then the ratio will be greater than 1, and the divisor an improper fraction. The ratio of 12 yards to 4 yards is 3, hence

the ratio of $48 to the answer is 3, and the answer will be of $48, which is $16. Or, dividing the product of the means by the extreme, we have 4 x 48÷ 12 = $16.

RULE.-I. With the two given numbers, which are of the same name or kind, form a ratio greater or less than 1, according as the answer is to be less or greater than the third given number.

II. Divide the third number by this ratio, and the quotient will be the required number or answer.

1. Mixed numbers should first be reduced to improper fractions, and the ratio of the fractions found according to (378).

2. Reductions and cancellation may be applied as in the first method.

The examples may be solved by either method.

3. If 48 cords of wood cost $120, what will 20 cords cost? Ans. $50. 4. If 6 bushels of corn cost $4.75, what will 75 bushels cost? Ans. $59.37.

5. If 12 horses consume 42 bushels of oats in 3 weeks, how many bushels will 20 horses consume in the same