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3s? - 27.2 Here the factors are 2 + 3x and 3+ 2x, and =
2 In order that x may be positive must be less than but greater than
3 There are other cases which may be considered when a is a square number, also when c is a square number, and the cases in which the expression cannot be made a square. 2. (1) Since 5x + 7y - 2y=xy, .. 2 =
77–29, and any assumed value of y will give a corresponding value of x. If the equation be restricted to positive values of & and y. Let y=1, 2, 3, 4, 5, 6, 7, &c. It will be found that the corresponding values of x are
Y - 5
6 When y-1, x= ; y=5, x=
x , 2 not being positive integers. (2) From 3y - xy + 4x + 2=0, x =
3y + 2
must be greater than 4 in order
Y-4' to obtain positive values of x and y.
(7) The equation 2.r(x + y) + y2 – [(y - 3)=0 may be reduced to the form y? + 2(x – 3)y = - 18 - 2x2.
(8) x=1, 3, 4, 7; y=39, 19, 15, 9. (9) x=4, 5, 6; y=27, 11, 5. (10) The least values are x=
=70, y=30. 3. Let x, y denote the numbers, then x2 + y2 is to be a square number. Assume
2xy 0:? + y2 =(nx - y)=nor? – 2nxy+y', then x= ; and for all values of X,
** - 1 corresponding values of y may be found. If integral values of x and y only be required. If y=x2 – 1, then x=2n, and by taking x=2, 3, &c., a series of corre. sponding values of y will be found.
Next let x2 - y2 = (.x - ny)?, and by a similar process, if y=2n, then x=a? +1.
8. Let the numbers be x”, y’, z?, then x2 +22=2y. Let x=m+1, and y=m- n1, then x2 +22=2(m2 +12)=2y?, and m2 +122=y’, the question is reduced to find m and n, the sum of whose squares shall be a square. See example 4. The numbers 1, 25, 49, and 4, 100, 196, are two integral solutions of the problem.
9. Let x denote the number of men in the side of the first of the five hollow squares, then x +5, x + 10, x+15, 2+20 are respectively the numbers of men in a siile of the other four hollow squares. Then x2 - (x - 10)2 =number of men in the first hollow
square. (x + 5)2 – (x – 5)2 =
second (x+10)? - *?=
third (x+15)? — (x + 5)2 =
fourth (x+20)? - (x+10)? =
fifth The whole number of men is 100% + 500. If
y denote the number in the side of the solid square, then y2 = 100x +500, and it is required to determine æ, so that 10042 +5) may be a complete square.
ELEMENTARY ARITHMETIC, WITH BRIEF NOTICES OF ITS HISTORY.
In Twelve Sections, demy 8vo.
CONTENTS AND PRICES.
PRICE Sectiox I. Of Numbers, pp. 28
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4d. SECTION XI. Proportion, pp. 32...
4d. SECTION XII. Logarithms, pp. 32
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EACH section of the Arithmetic may be purchased separately; also the twelve sections together, done up in boards, with cloth
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WITH BRIEF NOTICES OF ITS HISTORY.
NATIO, PROPORTION, AND VARIATION.
BY ROBERT POTTS, M.A.,
TRINITY COLLEGE, CAMBRIDGE,
RATIO, PROPORTION, AND VARIATION. Art. 1. Two unequal magnitudes of the same kind may be compared by considering how much (quantum) one of them is greater than the other; or how many times (quot) one contains the other. The former has been named their arithmetical, the latter their geometrical relation; and it is in this latter view the subject of ratio and proportion of quantities is considered.
Ratio has been defined to be the relation which one quantity bears to another of the same kind with respect to quotity, and the comparison is made by considering how many times one quantity is contained in the other, or what multiple part or parts, one is of the other. If a, b denote two quantities of the same kind, a may be
compared with b, or b with a. If a be compared with b, the ratio of a to b is denoted by a :b, and is represented by the quotients, which is called the measure of the ratio a:b, and indicates how many times b is contained in a, or what multiple or fractional part a is of b, and the ratio a: 6 may always be changed into a fraction the fraction into a ratio.
It is also evident that the properties of ratios are dependent on the principles of fractions. The two numbers a, b are called the terms of the ratio a:6, the first is called the antecedent of the ratio, and · the second the consequent. If the antecedent be greater than the consequent, the ratio is said to be one of greater inequality; but if the antecedent be less than the consequent, the ratio is named a ratio of less inequality.
The ratio of two concrete numbers of the same kind is the same as the ratio of two other concrete numbers of a different kind, when the antecedent of the former two contains the consequent or the same part or parts of it, as the antecedent of the latter two, contains the consequent or the like part or parts of it.
Two or more ratios may be compared by reducing their corresponding fractions to a common denominator, and comparing the numerators.
If the antecedents of two or more ratios be multiplied together for a new antecedent, and their consequents for a new consequent, then the resulting ratio is said to be compounded of these ratios, as if a:b, c:d be any two ratios, then the ratio of ac: bd is said to be compounded of the ratios a:b and c:d. And similarly for any number of ratios.