CHAPTER XXIX.

RATIO.

444. Ratio is the relation that one quantity bears to another of the same kind. It is represented by the quotient arising from dividing one by the other. The ratio of 8 to 2 is 4.

445. Two quantities are necessary to form a ratio; these are called its Terms.

The Antecedent is the first term of a ratio; the Consequent, the second.

446. A ratio is either Direct or Inverse. It is Direct, when the antecedent is divided by the consequent; Inverse, when the consequent is divided by the antecedent. When the word ratio is used alone, a direct ratio is meant. The direct ratio of 8 to 2 is 4. The inverse ratio of 8 to 2 is . In either case, 8 is the antecedent, and 2 the consequent.

447. Ratio is expressed in two ways:-1. By two dots, in the form of a colon, between the terms; as, 8:4. 2. In the form of a fraction; as, .

The two dots and the fractional line both come from the sign of division. When the two dots are used, the line between is omitted; when the fractional line is used, the two dots are omitted.

8:4 is read the ratio of 8 to 4.

448. A ratio being expressed by a fraction, of which the antecedent is the numerator and the consequent the denominator, it follows that the principles which apply to the terms of a fraction, § 137, apply also to the terms of a ratio. That is,

Multiplying the antecedent multiplies the ratio, and dividing the antecedent divides the ratio.

444. What is Ratio? By what is it represented ?-445. How many quantities are necessary to form a ratio? What are they called? Which is the Antecedent? Which, the Consequent ?-446. What is the difference between Direct and Inverse Ratio? Give an example.-447. In how many ways is ratio expressed? Describe them. What is the origin of the two dots and the fractional line?-448. State the three principles that apply to multiplying or dividing the terms of a ratio.

Multiplying the consequent divides the ratio, and dividing the consequent multiplies the ratio.

Multiplying or dividing both terms by the same number does not alter the ratio.

449. Fractions having a common denominator are to each other as their numerators.

16:7:9, or 3. For, as we have just seen, dividing both terms of the second ratio, 7 and 9, by the same number, 10, does not alter their ratio. The ratio between two fractions that have not a common denominator, may be found by reducing them to others that have, and taking the ratio of their numerators.

450. There is no ratio between quantities of different kinds; as, 8 yd. and 4 lb. But a ratio subsists between

quantities of the same kind, though of different denomi

nations.

Thus, the ratio of 8 yd. ( 24 ft.) to 4 ft. is 6. In such cases, to find the ratio, the terms must be brought to the same denomination.

451. A Simple Ratio is one into which but two terms enter. A Compound Ratio is the product of two or more simple ratios, the first term being the product of the antecedents, the second that of the consequents.

Simple Ratios, 8:4

9:3

2:6

The ratio compounded of these three simple ratios is

8 x 9 x 2: 4 × 3 × 6.

EXERCISE.

1. Express the ratio of 27 to 9; of 7 to 16; of 43 to 100.

2. Read the following ratios:—

3. Find the value of the above ratios, when direct.

4. Find the value of the above ratios, when inverse.

449. What ratio do fractions having a common denominator sustain to each other? Prove this. Hence, how may the ratio between two fractions that have not a common denominator be found?-450. How may we find the ratio between two quantities of the same kind, but different denominations ?-451. What is a Simple Ratio? What is a Compound Ratio? Give an example.

CHAPTER XXX.

PROPORTION.

452. Proportion is an equality of ratios.

The ratio of 8 to 4 is 2; the ratio of 6 to 3 is also 2. Hence the proportion, 8:46:3.

453. Proportion is expressed in two ways:-1. By the sign of equality between the ratios. 2. By four dots, in the form of a double colon, between the ratios.

8:46:3 Or, the ratio of 8 to 4 equals the ratio of 6 to 3.

454. Four quantities forming a proportion are called Proportionals. The first two are called the First Couplet; the last two, the Second Couplet. The first and fourth are called the Extremes; the second and third, the Means.

In the proportion 8: 4 :: 6: 3, 8 and 4 form the first couplet, 6 and 3 the second. 8 and 3 are the extremes, 4 and 6 the means.

455. Three quantities are in proportion when the 1st is to the 2d as the 2d to the 3d. 8: 4 :: 4 : 2.

A term so repeated is called a Mean Proportional between the other two. 4 is a mean proportional between 8 and 2.

456. The product of the extremes, in every proportion, equals the product of the means. Thus, in the last proportion, 8 x 2 = 4 × 4. Hence the following rules :

457. RULES.-I. To find an extreme, divide the product of the means by the given extreme.

II. To find a mean, divide the product of the extremes by the given mean.

452. What is Proportion ?-453. In how many ways is proportion expressed? Describe them.-454. What are four quantities forming a proportion called? What are the first two called? The last two? Which are the Extremes? Which, the Means?-455. When are three quantities in proportion? What is meant by a Mean Proportional?-456. What principle holds good in every proportion ?-457. Give the rule for finding an extreme. For finding a mean.

Ex. 1.-Find the 4th term of the proportion 8. 4:: 26: ?

Find the product of the means: 4 x 26
Divide by the given extreme :

104.

1048 13. Ans.

Ex. 2. Find the 2d term of the proportion 8: ? :: 26: 13.

Find the product of the extremes: 8 x 13 = 104.

Divide by the given mean :

104 26 4. Ans.

Simple Proportion, or Rule of Three.

458. A Simple Proportion expresses the equality of two simple ratios. Simple Proportions may be used to solve many questions in which three proportionals are given and the fourth is required.

As three terms are given, the rule for Simple Proportion is often called the Rule of Three.

Ex. 1.-If 8 yd. of cloth cost $40, what will 24 yd. cost?

The terms of a couplet must be of the same kind. Hence, in forming a proportion from the above question, as the answer, or fourth term, is to be dollars, we take $40 for the third term. Then, since 24 yd. will cost more than 8 yd., we arrange the other two numbers so as to form an inverse ratio greater than 1, by taking 24, the greater, for the second term, and 8, the less, for the first. The proportion then stands,

8 yd. 24 yd.:: $40, the cost of 8 yd. : the cost of 24 yd. The 4th term is required; we find it by Rule 1, § 457.

24 × 40 960

9608 120

Ans. $120.

453. What does a Simple Proportion express? To what questions do Simple Proportions apply? What is the rule often called? Explain Ex. 1.-459. How may cancellation be brought to bear ?-460. Recite the rule.

459. In solving questions in Proportion, equal factors, if there are any, in the 1st and 2d, or 1st and 3d terms, should be cancelled. Thus, in Ex. 1:—

$ yd. : 24 yd. :: $40.

3

$40 x 3 $120 Ans.

460. RULE.-1. Take for the third term the number that is of the same kind as the answer. Of the two remaining numbers, make the larger the second term, when from the nature of the question the answer should exceed the third term; when not, make the smaller the second term.

2. Cancel equal factors in the first and second terms, or the first and third. Then multiply the means together, and divide their product by the given extreme.

The first and second terms must be of the same denomination. If · the third term is a compound number, it must be reduced to the lowest denomination it contains, and this will be the denomination of the answer.

1. What cost 8 cords of wood, if 2 cords cost $9?

Ans. $36.

2. If 25 lb. of coffee cost $4.50, what cost 312 lb.? Ans. $56.16. 3. If a railroad car goes 17 miles in 45 minutes, how far will it go in 5 hours at the same rate? Ans. 1131 mi.

4. How long will it take $100 to produce $100 interest, if it produces $7 in one year?

5. If 15 men can build a wall 12 ft. high in 1 wk., how many will be needed to raise it 20 ft. in the same time? How long would it take 5 men to raise it 20 ft.?

6. What cost 9 hats, if 5 hats cost £4 5s.? 7. If 7 tons of coal, of 2000 lb. each, last 3 each, how much will be consumed in 3 weeks?

Last ans. 5 wk.

Ans. £7 13s. months, of 30 days

8. If 9 bu. 2 pk. of wheat make 2 barrels of flour, how many bushels will be required to make 13 barrels ?

9. If 5 bu. of potatoes last 8 adults and 2 children 40 days, how long, at the same rate, will they last 18 adults and 9 children, each adult consuming as much as 2 children? Ans. 16 days.