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3. Edward Doton owes Daniel Stetson, 1855, May 1, for merchandise, $ 500 ; May 15, for timber, $ 400 ; June 14, for a horse, $ 300; July 24, for bill of labor, $ 100. Stetson owes Doton, 1855, March. 7, for a pleasure-boat, $ 400; April 2, for merchandise, $ 200; May 6, for merchandise, $ 300, June 13, for a carriage, $ 120. Allowing all the items to be on 6 months' credit, when will the balance of the account become due ?
Ans. April 27, 1856.
234. 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, and the last the Consequent. Thus, in the example given, 6 is the antecedent, and 3 the consequent.
A Simple Ratio is that having but one antecedent and one consequent.
The Terms of a ratio are its antecedent and consequent.
235. 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, f.
The terms of a ratio must be of the same kind, or such as may be reduced to the same denomination. Thus, shillings have a ratio to shillings; but shillings have not a ratio to gallons, nor pounds to days.
236. 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 f, and the inverse ratio of 6 to 3 is , or 2.
234. What is ratio ? How many numbers are necessary to form a ratio ? What are the antecedent and consequent called ? - 235 What two ways are there of expressing a rativ ? - 236. What is a direct ratio ? An inverse ratio ?
The DIRECT ratio of one 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.
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 ?
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 ?
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. f. Of 27 to 81? Of 16 to 48 ? Of 72 to 9? Of 11 to 88 ?
5. What is the direct ratio of 2£ 5s. to 93. ? Ans. 5. Of Sin. to 1ft. 6in. ?
237. A Compound Ratio is the product of two or more ratios. Thus the ratio compounded of the ratios of 8:4 and 12:3 is of x = if = 8, or 8 X 12:4 X 3 = 8. A compound ratio is generally expressed by writing the ratios
8:4 of which it is composed, one above the other. Thus,
12:3 presses a compound ratio.
One quantity is said to vary directly as another, when both increase or decrease together in the same ratio.
One quantity is said to vary inversely as another, when the one increases in the same ratio as the other decreases.
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.
237. What is a compound ratio? What a duplicate ratio? What a triplicate ratio? What is the effect of multiplying or dividing the terms of a ratio ?
238. Proportion is an equality of ratios. Thus 9:3= 12:4 expresses a proportion.
Proportion is usually written with four dots (::), instead of the sign of equality between the ratios ; thus, 9:3::12:4 expresses a proportion, and is read, the ratio of 9 to 3 is equal to the ratio of 12 to 4, or 9 is to 3 as 12 is 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.
239. 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 ratio to 3 that 12 has to 4, because 9 contains 3 as many times as 12 contains 4.
240. In a proportion, if the antecedents or consequents, 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.
241. In a proportion, the product of the extremes is equal to the product of the means. Thus, the proportion, 14:7::18:9 may be expressed fractionally, 14 = Now, if we reduce these fractions to a common denominator, we have the = t; but in this operation we multiplied together the two extremes of the proportion, 14 and 9, and the two means, 18 and 7; thus, 14 X 9
18 x 7.
238. What is proportion ? How is proportion written ? What are the numbers called that form a proportion? Which are the extremes ? Which the means ? 239. When are numbers said to be in proportion to each other ? — 240 What is the effect of dividing the antecedents or consequents of a proportion ? Of multiplying them ? 241. How does the product of the extremes compare with that of the means ?
242. 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
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 x 8
128 ; and 1.28 • 4 32.
224. Simple Proportion is an equality between two simple ratios.
Simple Proportion is sometimes called the Rule of Three, from three terms being given to find a fourth.
245. To state and solve questions in Simple Proportion.
Ex. 1. If 7lb. of sugar cost 56 cents, what will 361b. cost ?
Ans. $ 2.88.
Since 7lb. have the same ratio to Extreme. Mean.
Mean. 3615. as 56 cents, the cost of the 7 lb.: :3 6 lb. : : 5 6 cts.
former, have to the cost of the latter, 36
we have the first three terms of a
proportion given, namely, one of the 3 3 6
extremes and the two means. Now, 1 68
to arrange the given numbers in the
order of a proportion, or state the 7) 20.1 6
question, we make 56 cents the third $ 2.8 8 Extreme.
term, because it is of the same kind,
and has the same ratio to the required answer, or fourth term, as the first has to the second. From the nature of the question, since the answer will be more than 56 cents, or the third term, the second term must be larger than the first; we make the 36 the second term, and the 7 the first, and then the product of the means divided by the given extreme, gives the required extreme. (Art. 243.)
242. If the extremes and one of the means are given, how can the other mean be found ? - 243. When the means and one of the extremes are given, how can the other extreme be found ? — 244. What is simple proportion ? How many terms are given in questions in simple proportion?
BY ANALYSIS. --If ylb. cost 56 cents, 1 pound will cost of 56 cents, or 8 cents. Then, if ilb. cost 8 cents, 361b. will cost 36 times 8 cents, or $ 2.88.
OPERATION. bar. bar.
Ex. 2. If 76 barrels of flour cost $ 456, what will 12 barrels cost?
Ans. $ 72. We state this question by making
$ 456 the third term, because it is of 7 6:12: :4 5 6
the same kind of the required term. 12
Then, since the answer must be less 7 6 5 4 7 2 ( $72
than $ 456, because 12 barrels will
cost less than 76 barrels, we make 12 5 3 2
barrels, the smaller of the two terms, 1 5 2
the second term, and 76 barrels the 152
first term, and proceed as before.
BY ANALYSIS. - If 76 barrels cost $ 456, 1 barrel will cost to of $ 456, or $ 6. Then, if i barrel cost $ 6, 12 barrels will cost 12 times $ 6, or $ 72.
Ex. 3. If 3 men can dig a well in 20 days, how long will it take 12 men ?
Ans. 5 days. Since the required answer is days, we make men. days.
20 days the third term. And as 12 men will 12:3:: 20 dig the well in less time than 3 men, the 3
answer must be less than 20 days. There
fore we make the 3 the second term, and the 12) 60
12 the first, and proceed as in the other examples.
BY ANALYSIS. — If 3 men can dig the well in 20 days, it will take one man 3 times 20 days, or 60 days. Again, If one man can dig the well in 60 days, 12 men can dig it in the of 60 days, or 5 days.
RULE. Write for the third term that number which is of the same kind as the required fourth term.
Of the other two numbers, write the larger for the second term, and the less for the first, when the answer should exceed the third term; but write the less for the second term, and the larger for the first, when the answer should be less than the third term.
Multiply the second and third terms together, and divide their product by the first.
245. 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 ?