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SIMPLE RATIO AND SIMPLE

PROPORTION.

Art. 126. THE part which one number is of another is sometimes called the ratio of the one to the other; thus, 3 is g of 5, and the ratio of 3 to 5 is 3÷ 5 = §. Hence, the ratio of one number to another is expressed by the quotient obtained by dividing the one by the other.

The two given numbers are called the terms of the ratio; the first number or term is called the antecedent, and the second, the consequent.

The ratio of one number to another is expressed or written in two different forms.

The first is by a fraction, writing the antecedent for the numerator, and the consequent for the denominator; thus, the ratio of 8 to 12 is

The second is by placing two points indicating division between the two terms of the ratio; thus, the ratio of 8 to 12 is written 8: 12.

1. What is the ratio of 4 to 8? 3. What is the ratio of 7 to 9? 5. What is the ratio of 15 cents to 17 cents?

7. What is the ratio of 17 shillings to £2?

9. What is the ratio of $1.25 to 5 dollars?

2. What is the ratio of 8 to 4? 4. What is the ratio of 9 to 7? 6. What is the ratio of 19 pounds to 16 pounds?

8. What is the ratio of 2 gallons to 3 quarts?

10. What is the ratio of of a of a yd.?

yd. to

Art. 127. When we have four quantities or numbers given, two of them having the same name, the other two having also a like name, and the ratio of the first to the second being equal to the ratio of the third to the fourth, these four quantities or numbers are said to be in proportion. Hence, proportion is the equality of two ratios.

ILLUSTRATION. There are two pieces of cloth: one measures 5 yards, and is worth 15 dollars; the other measures 8 yards, and is worth 24 dollars. Comparing 8 yards with 5 yards, we perceive that 8 yards is g of 5 yards. Comparing 24 dollars with 15 dollars, we perceive that 24 dollars is

of 15 dollars. Hence, it is plain that 24 dollars is the same proportional part of 15 dollars that 8 yards is of 5 yards; also, that the ratio of 5 yards to 8 yards is equal to the ratio of 15 dollars to 24 dollars; therefore, these four quantities or numbers constitute a proportion.

The usual method of indicating that the two ratios are

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equal, and that the four terms are in proportion, is, by placing two short parallel lines between the two ratios, and two points indicating division between the two terms of each ratio; thus, 5:8 15:24. The whole expression is read thus the ratio of 5 to 8 is equal to the ratio of 15 to 24. A more convenient method of indicating the equality of two ratios is by expressing each of the ratios by a fraction, and placing the sign of equality between them; thus, &= This expression is read thus: 5 divided by 8 equals 15 divided by 24.

A third method is by placing four points between the two ratios, and two points between the two terms of each ratio; thus, 5:8: 15:24; and this expression is usually read thus: as 5 is to 8, so is 15 to 24.

Art. 128. The first and fourth numbers or terms of a proportion are called the extremes, and the second and third numbers or terms are called the means. In the other method of writing the terms, the numerator of the first fraction and the denominator of the second are the extremes, and the denominator of the first fraction and the numerator of the second are the means.

Since the quotient of the first term divided by the second is equal to the quotient of the third divided by the fourth, it follows that the product of the extremes is equal to the product of the means.

Take the proportion 5:8=15:24, or §=. The product of the extremes 5 × 24=120. The product of the means 8 X 15: 120.

=

The product of the extremes in every proportion being equal to the product of the means, it follows that if the product of the extremes be divided by either of the means, the quotient will be the other mean; also, if the product of the means be divided by either extreme, the quotient will be the other extreme.

Art. 129. In every practical question in simple proportion there are three numbers or terms given, to find a fourth term, or answer to the question; two of which have the same name, or are of a like kind, and the other has the same name, or is of the same kind, as the required fourth term or

answer.

ILLUSTRATION. Suppose 6 barrels of flour to be worth 36 dollars, how many dollars are 8 barrels worth?

In this question we have two quantities of flour given, 6 barrels and 8 barrels, which numbers are the first and

second terms of a proportion; we have given also 36 dollars, the value of 6 barrels, which is the third term, and we are required to find the value of 8 barrels, or the fourth term.

Multiplying the third term $36, by the second term 8, the product is $288; dividing this product by the first term 6, the quotient is $48, the fourth term or value of 8 barrels. Art. 130. Arranging the three given numbers or terms proper form or order, is called stating the question. From the preceding remarks and illustrations we obtain the following rule.

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RULE. Write that number for the third term which has the same name, or is of the same kind, as the required fourth term

or answer.

Then ascertain whether the fourth term or answer must be greater or less than the third term; if the fourth term must be greater than the third, write the greater of the two remaining numbers for the second term, or numerator of a fraction, and the smaller for the first term, or denominator; but if the fourth term or answer must be less than the third, write the smaller of the two remaining numbers for the second term, or numerator of a fraction, and the greater for the first term, or denominator.

Multiply the third term by the numerator, or second term, and divide the product by the denominator, or first term; the quotient will be the fourth term, or answer, in the same denomination as the third term.

When the first and second terms are compound numbers, or when they are of different denominations, reduce them to the lowest denomination mentioned in either of them; and when the third term is a compound number, reduce it to the lowest denomination mentioned in it.

1. The first term of a proportion is 6, the second 8, the third 12; what is the fourth?

3. If 6 yards of cloth cost 12 dollars, how many dollars will 8 yards cost?

5. If a man travel 12 miles in 3 hours, how many miles will he travel in 5 hours?

7. If 10 bushels of potatoes last a family 8 months, what number of bushels will last the same family 12 months?

2. The first term of a proportion is 8, the second 6, the third 16; what is the fourth?

4. If 8 yards of cloth cost 16 dollars, how many dollars will 6 yards cost?

6. If a man travel 20 miles in 5 hours, how many miles will he travel in 3 hours?

8. If a family consume 15 bushels of apples in 12 months, how many bushels will the same family consume in 8 months?

9. If 10 barrels of flour are 10. If 15 barrels of flour are worth 60 dollars, how many dol-worth 90 dollars, how many dollars are 15 barrels worth?

11. If 12 men can perform a piece of work in 15 days, how many men will perform the same work in 30 days?

13. If 8 men can dig a cellar in 10 days, what number of men will be required to dig the same in 5 days?

lars are 10 barrels worth?

12. If 6 men can perform a piece of work in 30 days, how many men will perform the same work in 15 days?

14. If 16 men can build the walls of a house in 5 days, how many days will it take 8 men to build the same walls?

1. If 5 yards of broadcloth cost 30 dollars, how lars will 8 yards of the same cloth cost?

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many dol

X30-249-48. Ans.

In this question the required fourth term or answer is to be in money; hence we write 30 dollars for the third term. 8 yards being of 5 yards, 8 yards will cost of 30 dollars; therefore, we write 8 for the second term or numerator of a fraction, and 5 for the first term or denominator. We obtain the fourth term, or answer, by multiplying the third term by the second, or numerator, and then dividing the product by the first term, or denominator.

2. If 48 dollars will pay for 8 yards of broadcloth, how many yards of the same cloth can be purchased with 30 dollars?

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In this question, the required fourth term, or answer, is to be in yards; hence we write 8 yards for the third term. 30 dollars being only of 48 dollars, we can purchase only g of 8 yards with 30 dollars; therefore, we write 30 for the second term, or numerator of a fraction, and 48 for the first term, or denominator.

3. If 20 men can perform a piece of work in 15 days, how many men will be required to perform the same work in 10 days?

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As the required fourth term, or answer, is to be a number of men, we write the 20 men for the third term. As the same work is to be performed in 1% of the time in which 20 men can perform it, it will require 13 of 20 men to per

form it in 10 days; therefore, we write 15 days for the second term, or numerator of a fraction, and 10 days for the first term, or denominator.

4. If 30 men can perform a piece of work in 10 days, how many men will be required to perform the same work in 15 days?

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As the answer is to be a number of men, we write 30

men for the third term. As it will require only 19 as many men to perform the same work in 15 days, we write 10 days for the second term, or numerator of a fraction, and 15 days for the first term, or denominator.

Art. 131. When the first and second terms, or numerator and denominator, have a common factor, the operation may be abridged by dividing each term by that common factor, or by reducing the fraction to its lowest terms.

5. If 40 sheep yield 120 pounds of wool at a shearing, how many pounds will be obtained from 320 sheep?

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6. If 75 acres of land are worth 1500 dollars, what is the value of a farm containing 300 acres? Ans. 6000 dollars. 7. If 500 men consume 50 barrels of provisions in 6 months, how many barrels will 350 men consume in the same time? Ans. 35 barrels.

8. If 200 dollars gain 12 dollars interest in one year, how much interest will 750 dollars gain in the same time? Ans. 45 dollars. 9. Purchased 75 tons of coal, for which I paid 525 dollars; what must I pay for a cargo containing 325 tons? Ans. 2275 dollars. 10. If a post 6 feet in height casts a shadow 8 feet in length, what must be the height of a tree that casts a shadow 90 feet in length at the same time? Ans. 67 feet. 11. How many days will it take a ship to sail round the

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