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digits is 15; and if 31 be added to their product, the digits will be inverted ?

Ans. 78. Ex. 19. What two numbers are those whose product is 54, and quotient is 6 ?

Ans. 3 and 18. Ex. 20. The product of two numbers is a, and their quotient is b. What are the numbers ?

Ans. Vab and V. Ex. 21. The sum of the squares of two numbers is 260, and the difference of their squares 132. What are the numbers ?

Ans. 8 and 14. Ex. 22. The sum of the squares of two numbers is a, and the difference of their squares is b. What are the numbers ?

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Ans. V and V 2 Ex. 23. Find two numbers which are to each other as 3 to 4, and the sum of whose squares is 400.

Ans. 12 and 16. Ex. 24. Find two numbers which are to each other as 2 to 3, and the difference of whose squares is 125.

Ans. 10 and 15. Ex. 25. Divide the number 16 into two parts, such that the product of the two parts, added to the sum of their squares, may be equal to 208.

Ans. 4 and 12. Ex. 26. What two numbers are those whose product is 255, and the sum of whose squares is 514 ?

Ans. 15 and 17.

Ex. 27. What two numbers are those whose differ. ence is 8, and the sum of whose squares is 544 ?

Ans. 12 and 20. Ex. 28. What two numbers are those whose sum is 41, and the sum of whose squares is 901 ?

Ans. 15 and 26. Ex. 29. What two numbers are those whose product is 120; and if the greater be increased by 8, and the less by 5, the product of the two numbers thus obtained shall be 300 ?

Ans. 12 and 10, or 16 and 71. Ex. 30. Divide the number 100 into two such parts that the sum of their square roots may be 14.

Ans. 36 and 64. Ex. 31. From two places at a distance of 720 miles, two persons, A and B, set out at the same time to meet each other. A traveled 12 miles a day more than B, and the number of days in which they met was equal to half the number of miles B went in a day. How many miles did each travel per day?

Ans. A 36 miles, and B 24 miles. Ex. 32. A tailor bought a piece of cloth for $120, from which he cut four yards for his own use and sold the remainder for $120, gaining one dollar per yard. How many yards were there, and what did it cost him per yard ?

Ans. 24 yards at $5 per yard. Ex. 33. The fore wheel of a carriage makes five revolutions more than the hind wheel in going 60 yards; but if the circumference of each wheel be increased one yard, it will make only three revolutions more than the hind wheel in the same space. Ro. quired the circumference of each.

Ans. Fore wheel 3 yards, and hind wheel 4 yards.

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SECTION XIII.

RATIO AND PROPORTION. (174.) NUMBERS may be compared in two ways, either by means of their difference, or by their quo. tient. We may inquire how much one quantity is greater than another, or how many times the one contains the other. One is called Arithmetical, and the other Geometrical Ratio.

(175.) The difference between two numbers is called their Arithmetical Ratio. Thus the arithmetical ratio of 9 to 7 is 9–7 or 2; and if a and b designate two numbers, their arithmetical ratio is designated by a-b.

(176.) Numbers are more generally compared by means of quotients; that is, by inquiring how many times one number contains another. The quotient of one number divided by another is called their Geometrical Ratio. The term Ratio, when used without any qualification, is always understood to signify a geometrical ratio ; and we shall confine our attention to ratios of this description.

(177.) By the ratio of two numbers, then, we mean the quotient which arises from dividing one of these numbers by the other.

Thus the ratio of 12 to 4 is represented by or 3.

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QUEST.-In how many ways may numbers be compared ? What is Arithmetical Ratio ? What is Geometrical Ratio ?

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If a and b represent two quantities of the same kind, the ratio of a to b is the quotient arising from dividing a by b, and may be represented by writing them a:b, or The first term, a, is called the antecedent of the ratio ; the last term, b, is called the consequent of the ratio.

Hence it appears that the theory of ratios is included in the theory of fractions; and a ratio may be consid. ered as a fraction whose numerator is the antecedent, and whose denominator is the consequent.

(178.) Proportion is an equality of ratios. Thus, if we take four numbers,

3, 4, 9, 12, such that the quotient of the first, divided by the seo ond, is equal to the quotient of the third divided by the fourth, the numbers are said to be proportional, and the proportion may be written

3 9

= 12

3:4::9:12. In general, if a, b, c, d are four quantities such that a, when divided by b, gives the same quotient as c when divided by d, then a, b, c, d are called propor. tionals, and we say that a is to b as c is to d; and this is expressed by writing them thus: .

a:b::c:d,
a:b=c:d,

a C

ī ā QUEST.—Define the terms antecedent and consequent. What is a proportion ?

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(179.) In ordinary language, the terms ratio and proportion are confounded with each other. Thus two quantities are said to be in the proportion of 3 to 5 instead of the ratio of 3 to 5. In strictness, however, a ratio subsists between two quantities, a proportion only between four. Ratio is the quotient arising from dividing one quantity by another; two equal ratios form a proportion. (180.) In the proportion

a:6::c:d, a, b, c, d are called the terms of the proportion. The first and last terms are called the extremes, the second

first antecedent, the second term the first consequent, the third term the second antecedent, and the fourth term the second consequent. The last term is said to be a fourth proportional to the other three taken in order.

(181.) The word term, when applied to a proportion, is used in a slightly different sense from that explained in Art. 12. The terms of a proportion may be polynomials. Thus

a+b:c+d::e+f:g+h. (182.) Three quantities are said to be in proportion when the first has the same ratio to the second that the second has to the third, and then the middle term is said to be a mean proportional between the other two. For example,

2:4::4:8, where 4 is a mean proportional between 2 and 8.

QUEST.-Explain the difference between a ratio and a proportion? How are the terms of a proportion distinguished? When are three quantities said to be proportional ?

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