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11 Dec 2016, 12:26
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1/x + 1/y = 1/9; What may be the greatest difference between x and y
a. 70
b. 80
c. 90
d. 100
e. Cannot be determined
a. 70
b. 80
c. 90
d. 100
e. Cannot be determined
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11 Dec 2016, 12:46
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\(\frac{1}{x} + \frac{1}{y} = \frac{1}{9}\)
\(\frac{x + y}{xy} = \frac{1}{9}\)
\(9x + 9y = xy\)
\(9x + 9y - xy - 81 = - 81\)
\((x - 9)(y - 9) = 81\)
Factors of \(81\) ----> (\(1*81)(3*27)(9*9)(27*3)(81*1)\) and negative counterparts
Intuitively 1 and 81 will give max difference
\(x - 9 = 1\) ---> \(x = 10\)
\(y - 9 = 81\) -----> \(y = 90\)
max \((y - x) = 90 - 10 = 80\)
Answer B
\(\frac{x + y}{xy} = \frac{1}{9}\)
\(9x + 9y = xy\)
\(9x + 9y - xy - 81 = - 81\)
\((x - 9)(y - 9) = 81\)
Factors of \(81\) ----> (\(1*81)(3*27)(9*9)(27*3)(81*1)\) and negative counterparts
Intuitively 1 and 81 will give max difference
\(x - 9 = 1\) ---> \(x = 10\)
\(y - 9 = 81\) -----> \(y = 90\)
max \((y - x) = 90 - 10 = 80\)
Answer B
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11 Dec 2016, 12:59
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vitaliyGMAT
\(\frac{1}{x} + \frac{1}{y} = \frac{1}{9}\)
\(\frac{x + y}{xy} = \frac{1}{9}\)
\(9x + 9y = xy\)
\(9x + 9y - xy - 81 = - 81\)
\((x - 9)(y - 9) = 81\)
Factors of \(81\) ----> (\(1*81)(3*27)(9*9)(27*3)(81*1)\) and negative counterparts
Intuitively 1 and 81 will give max difference
\(x - 9 = 1\) ---> \(x = 10\)
\(y - 9 = 81\) -----> \(y = 90\)
max \((y - x) = 90 - 10 = 80\)
Answer B
\(\frac{x + y}{xy} = \frac{1}{9}\)
\(9x + 9y = xy\)
\(9x + 9y - xy - 81 = - 81\)
\((x - 9)(y - 9) = 81\)
Factors of \(81\) ----> (\(1*81)(3*27)(9*9)(27*3)(81*1)\) and negative counterparts
Intuitively 1 and 81 will give max difference
\(x - 9 = 1\) ---> \(x = 10\)
\(y - 9 = 81\) -----> \(y = 90\)
max \((y - x) = 90 - 10 = 80\)
Answer B
Why must \((x - 9)(y - 9) = 81\) have integer roots? Why wouldn't \((x - 9)(y - 9) = (810)(.1) = 81\) and \(x = 819, y = 9.1\) be appropriate?
