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§ XXXIII. RATIO. Art. 236. Ratio is the relation, in respect to magnitude or value, which one quantity or number has to another of the same kind, or the quotient arising from the division of one number by another. Thus, the ratio of 6 to 3 is 2.

Of the two numbers necessary to form a ratio, the first is called the antecedent, the last the consequent. Thus, in the example given, 6 is the antecedent and 3 the consequent.

When there is but one antecedent and one consequent, the ratio is called a simple ratio. The antecedent and consequent are also called the terms of the ratio.

Art. 237. A ratio may be expressed in two ways. The ratio of 6 to 3 may be expressed by two dots between the terms, thus, 6:3; or in the form of a fraction, by making the antecedent the numerator and the consequent the denominator, thus, .

The terms of a ratio must be of the same kind, or such as may be reduced to the same denomination, in order that they may have a ratio to each other. Thus, shillings have a ratio to shillings, and shillings to pounds, &c.; but shillings have not a ratio to gallons, nor pounds to days, because they are not commensurable.

Art. 238. A ratio may be either direct or inverse. A direct ratio is when the antecedent is divided by the consequent; an inverse ratio is when the consequent is divided by the antecedent. Thus, the direct ratio of 6 to 3 is , and the inverse ratio of 6 to 3 is , or 1.

The direct ratio of one quantity or number to another is found by dividing the number whose ratio is required, which is the antecedent, by the number with which it is compared, which is the consequent. The inverse ratio is found by reversing this process.

QUESTIONS. — Art. 236. What is ratio ? How many numbers are neces sary to form a ratio? What is the first called? What the second ? What is a simple ratio ? What are the antecedent and consequent called ?- Art. 237. What two ways are there of expressing a ratio ? How must the terms of a ratio compare? Have pounds any ratio 10 days? Why?- Art. 238. What is a direci ratio ? - What an inverse ratio ? How is the direct ratio of one number to another found ? How the inverse ratio ?

EXAMPLES FOR PRACTICE. 1. What is the direct ratio of 9 to 3? Ans. 3. Of 18 to 6? Of 16 to 4? Of 24 to 12? Of 20 to 5? Of 15 to 3 ? Of 100 to 25 ? Of 144 to 12 ?

2. What is the direct ratio of 7 to 21 ? Ans. . Of 4 to 28? Of 6 to 30 ? Of 9 to 11 ? Of 9 to 99? Of 30 to 90 ? Of 12 to 108 ?

3. What is the direct ratio of 60 to 12 ? Of 132 to 11 ? Of 40 to 120 ? Of 32 to 96 ? Of 200 to 50? Of 144 to 1728? Of 360 to 60 ?

4. What is the inverse ratio of 10 to 5 ? Ans. 1. Of 27 to 81 ? Of 16 to 48? Of 72 to 9? Of 11 to 88?. Of 7 to 35 ? Of 150 to 75 ?

5. What is the direct ratio of 2£. 5s, to 9s. ? Ans. 5. Of 9 in. to 1 ft. 6 in. ?

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Art. 239. A compound ratio consists of two or more simple ratios, whose corresponding terms are to be multiplied together. Thus, The simple ratio of

8:4 is 2 And


12 : 3 is 4
The compound ratio of 8 X 12:4 X 3 is 2 X 4

of 96 : 12 is 8 When a compound ratio is composed of two equal ratios, it is alled a duplicate ratio; when of three, it is called a triplicate ratio, &c. The simple ratio of

4:2 is 2 of

6:3 is 2 of

8:4 is 2 The triplicate ratio of 4 X 6 X 8:2X3 X 4 is x 2X2 X 2 Or

66 of

192 : 24 is 8

Art. 240. If the terms of a ratio are both multiplied or divided by the same number the ratio is not altered. Thus, the ratio of 8 : 2 is 4; the ratio 8 X 2:2 X 2 is 4; and the ratio of 8;2:2; 2 is 4.

QUESTIONS. - Art. 239. What is a compound ratio ? What a duplicate ratio ? What a triplicate ratio ? – Art. 240. What is the effect of multiply. ing or dividing the terms of a ratio ?

$ XXXIV. PROPORTION. Art. 241. PROPORTION is the equality of ratios. Thus the ratios 9:3 and 12 : 4 are equal, and when united form a proportion.

Proportion is usually expressed by four dots between the two ratios; thus, the proportion in the preceding example is written 9:3:: 12 : 4, and is read, 9 is to 3 as 12 to 4.

The numbers, which form a proportion, are called proportionals. The first and third are called antecedents, the second and fourth are called consequents ; also, the first and last are called extremes, and the remaining two the means.

Art. 242. Any four numbers are said to be proportional to each other when the first contains the second as many times as the third contains the fourth ; or when the second contains the first as many times as the fourth contains the third. Thus, 9 has the same proportion or ratio to 3 that 12 has to 4, because 9 contains 3 as many times as 12 contains 4.

Art. 243. If the antecedents or consequents of a proportion, or both, are divided by the same number, they are still proportionals. Thus, dividing the antecedents of the proportion 4 : 8::10 : 20 by 2, we have 2:8::5:20; dividing the consequents by 2, we have 4:4:: 10 : 10; and dividing both the consequents and antecedents by 2, we have 2:4::5:10; each of which is a proportion, since if we divide the second term of each by the first, and the fourth by the third, the two quotients will be equal. The effect is the same when the terms are multiplied by the same number.

Art. 244. The product of the extremes of a proportion is equal to the product of the means. Thus, the proportion 14:7 :: 18: 9 may be expressed fractionally, 4 = Now, if we reduce these fractions to a common denominator we have

26; but in this operation we multiplied together the


QUESTIONS. — Art. 241. What is proportion? How is proportion expressed ? What are the numbers called that form a proportion? Which are called the antecedents? Which the consequents? Which the extremes? Which the means ? — Art. 242. When are numbers said to be in proportion to each other? - Art. 243. What is the effect of dividing the antecedenis or consequents of a proportion? Of multiplying them ? - Art. 244. How does the product of the extremes compare with that of the means ? How is it shown that the product of the extremes is equal to that of the means ?

two extremes of the proportion, 14 and 9, and the two means, 18 and 7, thus 14 X 9=18 x 7.

Art. 245. If the extremes and one of the means are given the other mean may be found by dividing the product of the extremes by the given mean. Thus, if the extremes are 3 and 24, and the given mean 6, the other mean is 12; because 24 X 3

72; and 72 = 6= 12.

Art. 246. If the means and one of the extremes are given, the other extreme may be found by dividing the product of the means by the given extreme. Thus, if the means are 8 and 16, and the given extreme 4, the other extreme is 32; because 16 X8= 128; and 128 ; 4= 32.


Art. 247. SIMPLE PROPORTION is an expression of the equality between two simple ratios. Note. - Simple Proportion is sometimes called the Rule of Three.

Art. 248. Method of stating and solving questions in Simple Proportion.

Ex. 1. If 7lb. of sugar cost 56 cents, what will 36lb. cost?

Ans. $ 2.88.

Since 7b. have the same ratio Extreme.


to 36lb. as 56 cents, the cost of 7 lb. : 36 lb. :: 56 cts. the former, have to the cost of the


latter, we have the first three terms

of a proportion given, viz., one of 336

the extremes and the two means. 168

Now, to ascertain which of these


terms are the means, and which the 7) 20.16

extreme, we arrange them in the $ 2.88 Extreme.

order of a proportion, or state the

question, by making 56 cents the third term, because it is of the same kind, and has the same proportion to the required answer or fourth term as the first has to the second.

QUESTIONS. Art. 245. If the extremes and one of the means are given, how can the other mean be found ? - Art. 246. When the means and one of the extremes are given, how can the other extreme be found ? — Art. 247. What is simple proportion ? By what other name is it sometimes called ? Art. 248. How many terms are given in questions in simple proportion ? What are they?

And from the nature of the question, since the answer will be more than 56 cents, or the third term, the second term must be greater than the first; we therefore make 36lb. the second term, and 7lb. the first, and then proceed as in Art. 246.

BY ANALYSIS. - If 71b. cost 56 cents, llb. will cost 7 of 56 cents, which is 8 cents. Then, if ilb. cost 8 cents, 36]b. will cost 36 times as much ; that is, 36 times 8 cents, which are $2.88, Ans. as before. Ex. 2. If 76 barrels of flour cost $ 456, what will 12 barrels

If 76 cost?

Ans. $ 72. OPERATION. bar. bar.

We state this question by making 76 :12:: 456

$ 456 the third term, because it is of 12

the same kind of the required answer.

Then, since the answer must be less 76)5472 ($ 72 than $ 456, because 12 barrels will cost 532

less than 76 barrels, we make 12 bar

rels, the smaller of the other two terms, 152

the second term, and 76 barrels the first 152

term, and proceed as before.

BY ANALYSIS. - If 76 barrels cost $ 456, 1 barrel will cost to of $ 456, which is $6. Then, if i barrel cost $ 6, 12 barrels will cost 12 times as much, that is, $ 72, Ans. as before.

Ex. 3. If 3 men can dig a well in 20 days, how long will it take 12 men ?

Ans. 5 days.

men. days.


Since the required answer is days, 12 : 3::20

we make 20 days the third term. And 3

as 12 men will dig the well in less time

than 3 men, the answer must be less 12)60

than 20 days. Therefore we make 3

men the second term and 12 men the first, 5 days.

and proceed as in the other examples. BY ANALYSIS. — If 3 men dig the well in 20 days, it will take one man 3 times as long, that is, 60 days. Again, we say, If one man dig the well in 60 days, 12 men would dig it in the of 60 days, that is, 5 days, Ans. as before.

From the preceding examples we deduce the following

QUESTIONS. What is meant by stating the question ? Which of the terms given in the example do you make the third ? Why? Which the second ? Why? Which the first? Why? After the question is stated, how do you obtain the answer ?

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