Question

The numerator of fraction is one more than its denominator. If it's reciprocals added to it then it's sum is 61/30 then find the fraction

Answer 1

$\bf{\huge{\underline{\boxed{\tt{\purple{Answer:}}}}}}$

Given:-

• the numerator of fraction is one more than its denominator
• when it's reciprocals added to it then it's sum is 61/30

To find :-

• The fraction

Solution :-

Let the denominator of the fraction be x

As per the first condition,

• the numerator of fraction is one more than its denominator

Break up the statement to get it done easier.

The numerator = N [Assuming]

Is one more than = +1

Its denominator = x

Get it into linear state,

•°•Numerator of the fraction = x + 1

Original fraction = $\bf\large\frac{x+1}{x}$

As per the second condition,

• when it's reciprocals added to it then it's sum is 61/30

Reciprocal => $\bf\large\frac{x}{x+1}$

Their sum => $\bf\large\frac{x+1}{x}$ + $\bf\large\frac{x}{x+1}$ = $\bf\large\frac{61}{30}$

Cross multiplying and multiplying the terms with each other,

$\bf\large\frac{x(x+1) 1(x+1) + (x) (x) }{x (x+1)}$ = $\bf\large\frac{61}{30}$

$\bf\large\frac{x^2+x + x +1 +x^2}{x^2 + x}$ = $\bf\large\frac{61}{30}$

$\bf\large\frac{2x^2 + 2x +1}{x^2+x}$ = $\bf\large\frac{61}{30}$

Cross multiplying,

30 (2x² + 2x + 1) = 61 ( x² + x)

60x² + 60x + 30 = 61x² + 61x

61x² + 61x = 60x² + 60x + 30

61x² + 61x - 60x² - 60x - 30 = 0

61x² - 60x² + 61x - 60x - 30 = 0

x² + x - 30 = 0

x² + 6x - 5x - 30 = 0

x (x + 6) -5(x + 6) = 0

(x + 6) (x - 5) = 0

x + 6 = 0 OR x - 5 = 0

x = - 6 OR x = 5

When, x = - 6

Denominator = x = - 6

Numerator = x + 1 = - 6 + 1 = -5

Fraction = $\bf\large\frac{-5}{-6}$

When, x = 5,

Denominator = x = 5

Numerator = x + 1 = 5 + 1 = 6

Fraction = $\bf\large\frac{6}{5}$

$\bf{\huge{\underline{\boxed{\rm{\red{Verification:}}}}}}$

When the fraction = $\bf\large\frac{-5}{-6}$

For first condition :-

• The numerator of fraction is one more than its denominator.

Numerator = - 5

Denominator = -6

-5 is greater than -6 by 1

Hence, the first condition is satisfied.

For second condition :-

• when it's reciprocals added to it then it's sum is 61/30

Reciprocal = $\bf\large\frac{-6}{-5}$

$\bf\large\frac{-5}{-6}$ + $\bf\large\frac{-6}{-5}$ = $\bf\large\frac{61}{30}$

$\bf\large\frac{(-5)(-5) + (-6)(-6)}{(-6) (-5)}$ = $\bf\large\frac{61}{30}$

$\bf\large\frac{25 + 36}{30}$ = $\bf\large\frac{61}{30}$

$\bf\large\frac{61}{30}$ = $\bf\large\frac{61}{30}$

LHS = RHS!

When the fraction = $\bf\large\frac{6}{5}$

Reciprocal = $\bf\large\frac{5}{6}$

The sum of fraction and reciprocal = 61/30

6 × 6 + 5 × 5 / 6 × 5 = 61/30

$\bf\large\frac{36 + 25}{30}$ = $\bf\large\frac{61}{30}$

$\bf\large\frac{61}{30}$ = $\bf\large\frac{61}{30}$

LHS = RHS.

Hence verified.

I will suggest you to go with the second fraction i.e $\bf\large\frac{6}{5}$ it would be easier and a compatible one.