Ex. 2. Find the ratio of 1 yd. 2 ft. to 1 ft. 3 in.

SOLUTION.-1. Since, according to (668), only numbers of the same denomination can be compared, we reduce both terms to inches, giving 60 in. and 15 in.

2. Dividing 60 in. by 15 in. we have 60 in. ÷ 15 in. 4; that is, 60 in. is 4 times 15 in. Hence the ratio of 1 yd. 2 ft. to 1 ft. 3 in. is .

6. Of 20 T. 6 cwt. 93 lb. to 25 cwt. 43 lb. 5 oz.

682. PROB. II.-To reduce a ratio to its simplest terms.

Reduce the ratio 15 to its simplest terms.

SOLUTION.-Since, according to (679—III), the value of the ratio 15 is not changed by dividing both terms by the same number, we divide the antecedent 15 and the consequent 9 by 3, their greatest 15 ÷ 3 common divisor, giving 9 ÷ 3

5

=

3

But having divided 15 and 9 by

their greatest common divisor, the quotients 5 and 3 must be prime to each other. Hence (677) § are the simplest terms of the ratio 15.

Express in its simplest terms the ratio (see 668)

7. Of 96 T. to 56 T.

8. Of ft. to 2 yd.

9. Of 8s. 9d. to £1.

10. Of 3 pk. 5 qt. to 1 bu. 2 pk.

684. PROB. III.-To find a number that has a given ratio to a given number.

How many dollars are § of $72 ?

SOLUTION.—The fraction § denotes the ratio of the required number to $72; namely, for every $8 in $72 there are $5 in the required number. Consequently we divide the $72 by $8, and multiply $5 by the quotient. Hence, first step, $72 ÷ $8 = 9; second step $5 × 9 $45, the required number.

=

Observe, that this problem is the same as PROB. VIII, 501, and PROB. II, 274. Compare this solution with the solution in each of these problems.

685. Solve and explain each of the following examples, regarding the fraction in every case as a ratio.

1. How many days are of 360 days?

2. A man owning a farm of 243 acres, sold of it; how many acres did he sell?

3. James has $796 and John has as much; how much has John?

4. A man's capital is $4500, and he gains of his capital; how much does he gain?

5. Mr. Jones has a quantity of flour worth $3140; part of it being damaged he sells the whole for of its value; how much does he receive for it?

686. PROB. IV.-To find a number to which a given number has a given ratio.

$42 are of how many dollars?

SOLUTION.-The fraction denotes the ratio of $42 to the required number; namely, for every $7 in $42 there are $4 in the required number. Consequently we divide the $42 by $7 and multiply $4 by the quotient. Hence, first step, $42 ÷ $7 = 6; second step, $4 × 6 = $24, the required number.

Observe, that this problem is the same as PROB. IX, 502. Compare the solutions and notice the points of difference.

687. Solve and explain each of the following examples, regarding the fraction in every case as a ratio.

1. 96 acres are of how many acres?

2. I received $75, which is of my wages; how much is still due ?

3. James attended school 117 days, or of the term; how many days in the term?

4. Sold my house for $2150, which was of what I paid for it; how much did I lose?

5. Henry reviewed 249 lines of Latin, or of the term's work; how many lines did he read during the term? 6. 48 cd. 3 cd. ft. of wood is

much did I buy?

of what I bought; how

7. Mr. Smith's expenses are of his income. He spends $1500 per year; what is his income?

8. 4 gal. 3 qt. 1 pt. are of how many gallons?

9. A merchant sells a piece of cloth at a. profit of $2.50, which is of what it cost him; how much did he pay for it?

688.

PROB. V.-To find a number to which a given number has the same ratio that two other given numbers have to each other.

To how many dollars have $18 the same ratio that 6 yd. have to 15 yd.?

SOLUTION.-1. We find by (680—1) the ratio of 6 yd. to 15 yd., which is, according to (677).

2. Since denotes the ratio of the $18 to the required number, the $18 must be the antecedent; hence we have, according to (686), first step, $18 ÷ $2 = 9; second step, $5 × 9 = $45, the required number.

Observe, that in this problem we have the antecedent of a ratio given to find the consequent. In the following we have the consequent given to find the antecedent.

689. PROB. VI.-To find a number that has the same ratio to a given number that two other given numbers have to each other.

How many acres have the same ratio to 12 acres that $56 have to $84 ?

SOLUTION.-1. We find by (680—1) the ratio of $56 to $84, which is, according to (677).

2. Since denotes the ratio of the required number to 12 acres, the 12 acres must be the consequent; hence we have, according to (684), first step, 12 acr. ÷3 acr. = 4; second step, 2 acr. × 4= 8 acres, the required number.

690. The following are applications of PROB. V and VI. 1. If 12 bu. of wheat cost $16, what will 42 bu. cost? Regarding the solution of examples of this kind, observe that the price or rate per unit is assumed to be the same for each of the quantities given.

Thus, since the 12 bu. cost $16, the price per bushel or unit is $1.25, and the example asks for the cost of 42 bu. at this price per bushel. Consequently whatever part the 12 bu. are of 42 bu., the $16, the cost of 12 bu., must be the same part of the cost of 42 bu. Hence we find the ratio of 12 bu. to 42 bu. and solve the example by PROB. V.

2. What will 16 cords of wood cost, if 2 cords cost $9? 3. If a man earn $18 in 2 weeks, how much will he earn in 52 weeks?

4. If 24 bu. of wheat cost $18, what will 36 bu. cost?

5. If 24 cords of wood cost $60, what will 40 cords cost? 6. Bought 170 pounds of butter for $51; what would 680 pounds cost, at the same price?

7. Two numbers are to each other as 10 to 15, and the less number is 329; what is the greater?

8. At the rate of 16 yards for $7, how many yards of cloth can be bought for $100 ?

PROPORTION

DEFINITIONS.

691. A Proportion is an equality of ratios, the terms of the ratios being expressed.

=

Thus the ratio is equal to the ratio ; hence is a proportion, and is read, The ratio of 3 to 5 is equal to the ratio of 12 to 20, or 3 is to 5 as 12 is to 20.

692. The equality of two ratios constituting a proportion is indicated either by a double colon (::) or by the sign (=). Thus, &, or 3: 4 = 9: 16, or 3: 4 :: 9:16.

=

693. A Simple Proportion is an expression of the equality of two simple ratios.

Thus,, or 8: 12: 32: 48, or 8: 1232: 48 is a simple proportion. Hence a simple proportion contains four terms.

694. A Compound Proportion is an expression of the equality of a compound (674) and a simple ratio (673).

Thus,

2:31
6:5

:: 48: 60, or

=

48, is a compound proportion. It

is read, The ratio of 2 into 6 is to 3 into 5 as 48 is to 60.

695. A Proportional is a number used as a term in a proportion.

Thus in the simple proportion 2 : 5 :: 6 : 15 the numbers 2, 5, 6, and 15 are its terms; hence, each one of these numbers is called a proportional, and the four numbers together are called proportionals.