Question

the sum of the numerator and denominator of the certain fraction is 10. if 1 is subtracted from both the numerator and denominator, the fraction is decreased by 2 ÷21 . find the fraction

Answer 1

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Here is your answer.
Let the Denominator is x
So Numerator is 10- x
Fraction is ( 10-x)/x

Now 1 is Subtracted in each
Numerator is 10-x - 1 = 9-x
Denominator is x - 1

ATQ
$\frac{10 - x}{x} - \frac{2}{21} = \frac{9 - x}{x - 1} \\ \\ \frac{21(10 - x) - 2x}{21x} = \frac{9 - x}{x - 1} \\ \\ (x - 1)(210 - 21x - 2x) = 21x(9 - x) \\ \\ x(210 - 23x) - 1(210 - 23x) = 189x - 21 {x}^{2} \\ \\ 210x - 23 {x}^{2} - 210 + 23x - 189x + 21 {x}^{2} = 0 \\ \\ - 2 {x}^{2} + 44x - 210 = 0 \\ \\ {x}^{2} - 22x + 105 = 0 \\ \\ {x}^{2} - 15x - 7x + 105 = 0 \\ \\ x(x - 15) - 7(x - 15) = 0 \\ \\ (x - 15)(x - 7) = 0 \\ \\ x = 7 \: \: and \: \: 15$
Hence
Numerator is 10 - 7 = 3
and 10 - 15 = -5

Therefore fraction is
$\frac{3}{7} \: \: and \: \: \frac{ - 5}{15}$

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