EXERCISES

In Analysis and Cancellation.

1. Allowing of a yard of cloth to cost $6, what should be paid for of a yard at the same rate? Ans. $2. 2. If 5 men can accomplish a certain work in 20.5 days, in what time ought 25 men to perform 3.5 times as much work? Ans. 14.35 days.

3. What should be paid for of the same amount to $18?

of a ton of hay, when 24 tons Ans. $3.00.

4. Allowing a person to walk of a mile in 12 minutes, what distance would he walk in 40 minutes ? Ans. 21 miles. 5. If of of an acre of ground be worth $16, what is the value of 2 acres at the same rate? Ans. $106.

6. A teamster hauled 25 cut. of iron 30.5 miles for a certain sum of money. How far then ought he to haul 6.5 cut. for the same sum?

Ans. 117.307 miles. 7. How long ought 20 men to subsist on a stock of provisions which would suffice 18 men for 300 days?

Ans. 270 days. 8. A person who owned & of a merchant ship, sold of his share for $400; what was the whole ship worth at that rate? Ans. $4000.

9. Allowing a person to perform of a certain work in of a day, in what time ought he to perform the entire work?

10. The distance from A to B, which is the distance from B to C: how far then is it

11. If of of a yard of cloth be worth of of a yard worth, at the same rate?

Ans. 1 day. 40 miles, is of from B to C?

12. A contributed towards building a was of the sum contributed by B, who gave What was the amount of C's contribution? 13. If 4.5 cords of wood sell for $13.5, what for 5 loads, each containing 1.25 cords?

Ans. 50 miles. $1.25, what is

Ans. $2.083'. church $200, which as much as C. Ans. $571. should be given Ans. $18.75. worth as much as

14. Allowing 3 bushels of wheat to be 6 bushels of corn, how many bushels of corn are equal in value to 12 bushels of wheat?

Ans. 26 bu. 15. A has as much money as B, and as much as C, who has as much as D, and he has $1800. What sums are owned

by A, B, and C, respectively?

Ans. A, $1200; B, $1600; C, $1500.

RATIO.

§ 212. The Ratio of one number or quantity, called the antecedent, to another of the same kind, called the consequent, is the quotient of the former divided by the latter.

Thus the ratio of 15 to 5 is 3, since 15 contains 5, 3 times; and the ratio of 4 to 9 is , since 4 is of 9.

In these examples 15 and 4 are the antecedents; 5 and 9 the consequents.

The antecedent and consequent together are called the terms of the ratio.

What is the ratio of 12 to 3? the consequent? Of 3 to 12?

Of 3 yards to 6 yards?

What is the ratio of 10 miles to 5 miles? Of 6 hours to 13 hours? Of 25 pounds to 8 pounds? Of 9 s. to 20 s.?

Ratio of Monomials and Polynomials.

§ 213. To find the ratio between two quantities, the antecedent and consequent must be in the same denomination.

Thus the ratio of 2 ft. to 5 yd. is the ratio of 2 ft. to 15 ft. =1's.

What is the ratio of 3 in. to 2 ft.? Of 4 yd. to 6 ft. Of 2 hr. to 1 da.? Of 10 s. to 2£. 10 s.? Of 5 m. to 3 L. 3 m. Of 1 A. 2 R. to 5 A. 4 R.?

Two quantities of different kinds, that is, such that one can form no part of the other, have no ratio to each other; as 2 ft. and 5 hours.

Sign of Ratio.

§ 214. A colon (:) placed between two numbers, signifies that the numbers are taken as the antecedent and consequent of a ratio.

Thus 3: 5 signifies the ratio of 3 to 5.

Ratio is also expressed by making the antecedent the numerator, and the consequent the denominator, of a fraction; thus 3:5 is .

What fraction expresses the ratio of 3 to 13? Of 25 to 17? Of 4 to 19? Of 21 to 7? Of 3 quarters to 7 yd. 3 qr.? What fraction expresses the ratio of 5 to 17? 25? Of 7 to 100? Of 5 bushels to 7 bu. 3 pk.

Of 4 to 19? Of 5 to

Direct and Inverse Ratio.

$215. The direct ratio of the first of two quantities to the second, is the quotient of the first divided by the second.

The inverse ratio of the first to the second, is the same as the direct ratio of the second to the first.

For example, the direct ratio of 6 to 3 is §=2; the inverse ratio of 6 to 3 is the direct ratio 3 to 6=}=}.

216. The inverse ratio of the first of two quantities to the second, is equal to the direct ratio of the reciprocals of those quantities.

Thus, the inverse ratio of 6 to 3 is &=;

and the direct ratio of the reciprocals and is 1÷÷}=}=}. The inverse ratio of 4 to 2 equals the direct ratio of what fractions? Of 3 to 5? Of 4 to 9 of to? Of to ? Off to 10?

The inverse ratio of 3 to 8 equals the direct ratio of what fractions? Of 5 to 7 Of 11 to 4? Of to ? Of & to 5? Of to?

Hence inverse is sometimes called reciprocal ratio. ratio, when used alone, always means direct ratio.

Ratio of Fractions having a Common Term.

The term

§ 217. The ratio of two fractions having a common denominator, is the same as the ratio of their numerators; and

The ratio of two fractions having a common numerator, is the same as the inverse ratio of their denominators.

Thus, the ratio of to÷==2, which is the ratio of the numerators 4 and 2.

And the ratio of to %==6=2, which is the inverse ratio of the denominators 8 and 16.

is equal to the ratio of what integral numbers ? to 1 Of 4 to? Off to 8? to is equal to the ratio of what integral numbers? Ofto? Of 1 to 8 of 2 to ff?

The ratio of Of 1 to fz?

to

Of

Of to

The ratio of

PROPORTION.

§ 218. PROPORTION consists in an equality of ratios. Four quantities are in proportion, when the first has the same ratio to the second, that the third has to the fourth.

Thus, the numbers 6, 3, 8, 4, are in proportion; since the ratio of 6 to 3 is §=2, and the ratio of 8 to 4 is §=2.

The first and third terms, 6 and 8, are the antecedents of the ratios; the second and fourth are the consequents.

The first and fourth terms are called the two extremes; the second and third, the two means.

The fourth term is called a fourth proportional to the other three taken in order; thus, 4 is a fourth proportional to 6, 3, and 8.

What is the fourth proportional to 9, 3, and 12; that is, the number to which 12 has the same ratio that 9 has to 3? Which are the antecedents, and which the consequents? the extremes, and which the means?

What is the 4th proportional to 16, 4, and 20?

To 3, 6, and 8?

To 2, 1, and 10?

To 4, 12, and 10?

To 2, 3, and 4?

Which are

To 4, 8, and 10?
To 5, 20, and 12?
To, 1, and 2?

Direct and Inverse Proportion.

§ 219. A direct proportion consists in an equality between two direct ratios. An inverse or reciprocal proportion consists in an equality between a direct and an inverse ratio. (§ 215).

Thus, the numbers 6, 3, 8, 4, are in direct proportion, since the direct ratio of 6 to 3 is equal to the direct ratio of 8 to 4.

The same numbers, in the order 6, 3, 4, 8, are in inverse proportion, since the direct ratio of 6 to 3 is equal to the inverse ratio of 4 to 8.

What is the inverse fourth proportional to 12, 6, and 8; that is, the number to which 8 has the inverse ratio of 12 to 6?

What is the inverse fourth proportional to 8, 2, and 3? To 3, 9, and 15? To 4, 20, and 30? To 24, 3, and 4?

The term proportion, used alone, always means direct proportion.

Variation or General Proportion.

§ 220. VARIATION expresses the dependence of one term or quantity on another, according to some constant ratio, whatever new value either of the terms may assume.

One term varies directly as another, when both increase or decrease together in the same ratio.

Thus the value of a particular commodity varies directly as the quantity, since the value will be multiplied by 2, or 3, &c., if the quantity be multiplied by the same number. More briefly, we say, the value is directly as the quantity; or, the value is as the quantity.

One term varies inversely as another, when one of them increases in the same ratio in which the other decreases.

Thus the time required for a laborer to earn a given sum, varies inversely as his rate of wages, since the time will be multiplied by 2, or 3, &c., if his rate of wages be divided by the same number. More briefly, we say, the time is inversely as his rate of wages.

Say whether the two italicised terms would vary directly, or inversely, with each other, in each of the following instances:

1. The value and the quantity of a piece of cloth ?-The time, and the number of men required for a given amount of labor?

2. The weight and the number of gallons of water ?—The number of men and the amount of provisions that will serve them for a given period of time?

3. The weight of an article and the distance it may be carried for a given sum of money?—The length and the breadth of a garden to contain a given area?

4. The weight of the five-cent loaf of bread and the price of flour?— A sum of money and the number of laborers that may be hired with it for a given time?

Variation is sometimes called General Proportion: being, indeed, an abridgement of a proportion containing four terms without respect to any particular values of those terms.

Thus when we say that the weight of water is as the number of gallons, it is understood that,

The weight of any number of gallons is to the weight of any other number of gallons, as the first of those numbers is to the second.