Thus, if the antecedent is represented by a, and the consequent by a and the ratio by r, we have a ÷ c=r, or =r.

a

C

2. The antecedent is equal to the product of the consequent and ratio.

a

For, since =r, multiplying by c, we have a = c × r.

с

3. The consequent is equal to the quotient of the antecedent divided by the ratio.

a

For, since =r, a=cX r, from which we see that c==

a
с

MENTAL AND WRITTEN EXERCISES.

What is the ratio of

SOLUTION.-This compound ratio equals (2: 4) × (3:9), which equals

12. The antecedent is 24, the consequent 8; what is the ratio?

Ans. 3. 13. The consequent is 8 and ratio 9; what is the antecelent? Ans. 72. 14. The antecedent is 36 and ratio 4; what is the consequent? Ans. 9. 15. The consequent is 15 and ratio ; what is the ante. cedent? Ans. . 16. The antecedent is 15 and ratio ; what is the conse quent? Ans. 1.

17. Can you express the ratio between $24 and 6 lb.? Why not?

18. The antecedents of a ratio are 5 and 6, and the conse quents 10 and 14; what is the ratio? Ans.

SIMPLE PROPORTION.

556. A Proportion is the expression of equality between equal ratios, the terms of the ratios being indicated.

557. The Symbol for proportion is the double colon, (:), which expresses an equality of ratios; thus, 8:4:: 6:3, means the same as 8:4 6:3.

558. A Proportion is read in two ways; thus, 8:4:: 6:3 is read "the ratio of 8 to 4 equals the ratio of 6 to 3;" or "8 is to 4 as 6 is to 3."

559. The Terms of a proportion are the four numbers used in the comparison. The first and fourth terms are

the Extremes; the second and third are the Means.

560. The Couplets are the two ratios compared. The first couplet consists of the first and second terms. The second couplet consists of the third and fourth terms.

561. Proportion may be Simple or Compound. In Simple Proportion both the ratios compared are simple; in Compound Proportion one or both of the ratios are compound.

562. A Simple Proportion is the expression of the equality of two simple ratios.

563. The Principles of proportion are the truths relating to a proportion. They enable us to find any one term when the other three are given.

PRINCIPLES.

1. In every proportion the product of the means equals the product of the extremes.

=

, and multiplying

In any proportion, as 6:3::8: 4, we have : these equals by 4 and 3 we have 6 × 4 =8 × 3; that is, the product of the two means 8 and 3, equals the product of the two extremes 6 and 4. 2. Either extreme equals the product of the means divided by the other extreme.

For, from the proportion 6:3::8:4, we have 6 × 4: 3 x 8; hence, 6=3 x 84, or 4 = 3 × 8÷÷6. Therefore, etc.

3. Either mean equals the product of the extremes divided by the other mean.

For, from the proportion 6:3:: 8:4, we have 6 × 4: 86x48, or 8=6× 43. Therefore, etc.

=

3 x 8; hence,

4. The first term of a proportion equals the second term multiplied by the ratio of the third to the fourth.

For, from the proportion 8:6:: 12: 9, we have ; hence, 8= × 6, or 12:9 multiplied by 6. Therefore, etc.

5. The fourth term of a proportion equals the third term divided by the ratio of the first to the second.

For, from the proportion 8:6:12:9, we have 8 x 9

6 x 12, or 9

6 x 128, which equals 12 X, which equals 12÷ %, or 12 (8:6). Therefore, etc.

NOTES.-1. Let the pupils be required to demonstrate these principles by using symbols of any numbers; that is, by letters. French authors usually represent the unknown term by x; the same is done in this work. . Principle 1 may be demonstrated by showing that in a proportion we have 2d term X ratio : 2d term: 4th term X ratio : 4th term; in which we see the factors in the means are the same as the factors in the extremes.

MENTAL EXERCISES.

1. Write a proportion and point out the different terms and couplets Write a proportion and show that the ratios are equal.

2 If we multiply the antecedent of one couplet, what must we dc to the other couplet to make the ratios equal?

3. If we divide the antecedent of one couplet, wha must we do to the other couplet to make the ratios equal?

4. Write a proportion and illustrate Prin. 1; Prin. 2; Prin.3; Prin. 4; Prin. 5.

5. Show that if we change the two means one for the other, or the two extremes, the four numbers will still form a proportion.

6. Take some proportion and show that we may invert the terms of the couplets, and the four terms will still be in proportion.

WRITTEN EXERCISES.

Find the terms denoted by x in each of the following pro portions:

APPLICATION OF SIMPLE PROPORTION.

564. Simple Proportion is employed for the solution of problems in which three of four quantities are given, so related that the fourth may be determined from them, by equality of the ratios.

565. The required quantity must bear the same relation to a given quantity of the same kind that one of the remaining quantities does to the other. We can then form a proportion containing one unknown quantity, and find the unknown term by the principles of proportion.

NOTE.-Proportion was formerly called the "Rule of Three." Some of the old arithmeticians thought so highly of it that they called it "The Golden Rule of Three."

1. What will 20 yards of cloth cost, if 5 yards cost $15 ?

Rule.-I. Write the required quantity for the first term and the similar known quantity for the second term, and place the other two quantities for the third and fourth terms, so that the two ratios will be equal.

II. Find the first term by dividing the product of the second and third terms by the fourth.

SOLUTION 2d.-It is evident that the relation of 5 yd. to 20 yd. is the same as the relation of the cost of 5 yd. to the cost of 20 yd.; hence, we have the proportion,5 yd. is to 20 yd. as $15 is to the cost of 20 yd., from which, by Prin. 2, we have the cost of 20 ́yd. equals $60.

yd. yd.

OPERATION.

$

5 2015: cost of 20 yd. 20×15 Cost of 20 yd.= 5

Rule 2d.-I. Write the number which is of the same kind as the required quantity for the third term.

II. Place the other two numbers in the first and second ms, the greater in the second term when the result is to be

greater than the third term, and the less in the second term when the result is less than the third term.

III. Find the fourth term by dividing the product of the second and third terms by the first.

NOTES.-1. The author believes that the simplest method of using proportion is to put the unknown quantity in the first term. He gives the old method also, for teachers who prefer it. See Brooks's Philosophy of arıtk metic.

2. Pupils should be required to put the unknown quantity, which they may represent by x, in different terms, that they may thoroughly under stand the subject.

WRITTEN EXERCISES.

2. What cost 78 hhd. of molasses, if 13 hhd. are worth $250? Ans. $1500. 3. How many yards of cloth will $144 buy, if 28 yd. cost $112? Ans. 36. 4. What cost 132 acres of land, if 110 acres are worth $8250 ? Ans. $9900. 5. If.$100 gains $6 in a year, how much will $250 gain in a year? 6. If 16 horses eat 26 bundles of hay in a week how many will 36 horses eat in the same time? Ans. 58.50.

Ans. $15.

7. If 75 horses cost $9000, how many horses can be bought for $16200?

Ans. 135.

8. If there are 84 privates in each company, how many companies in a brigade of 3360 men? Ans. 40.

9. If 25 oxen eat 36 acres of grass in a month, how many oxen would 468 acres keep the same time? Ans. 325. 10. If 79 men earn $395 in a week, how many men can earn $675 in the same time? Ans. 135 men.

11. How much will 34 lb. of tea cost, if 8 lb. 8 oz. of the

aame kind of tea cost $43?

12. If 19 bu. of rye make 4 bar. of flour, how

els will it require to make 19 barrels?

Ans. $18.

many bush.

Ans. 904 bu.

13. How much will 28 cwt. 75 lb. of sugar cost at the rate Ans. $155.25.

of 7 cwt. 50 lb. for $40.50?

14. In what time will the cars go from Lancaster to Phil adelphia, 68 miles, at the rate of 5 miles in 10 min. 45 sec. 1 Ans. 2 h. 26 min.