Question

the numerator of a fraction is 3 less than its denominator if 1is added to the denominator the fraction is decrased by 1/15find the fraction
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Answer 1

AnswEr:-

Fraction = 2/5 or 6/9

$\rule{200}{2}$

Case 1:-

• Numerator = Denominator - 3

Let the denominator be n & numerator be (n - 3)

∴ Fraction = (n - 3)/n

Case 2:-

• 1 added to denominator, fraction is decreased by 1/15

According to question:-

⇒ (n - 3)/(n + 1) = [(n - 3)/n] - 1/15

⇒ (n - 3)/(n + 1) = [15(n - 3) - n]/15n

⇒ (n - 3)/(n + 1) = [15n - 45 - n]/15n

⇒ (n - 3)/(n + 1) = [14n - 45]/15n

⇒ 15n(n - 3) = (n + 1)(14n - 45)

⇒ 15n² - 45n = 14n² - 45n + 14n - 45

⇒ 15n² - 45n = 14n² - 31n - 45

⇒ 15n² - 14n² - 45n + 31n + 45 = 0

⇒ n² - 14n + 45 = 0

⇒ n² - 5n - 9n + 45 = 0

⇒ n(n - 5) - 9(n - 5) = 0

⇒ (n - 5)(n - 9) = 0

⇒ n = 5 or n = 9

So, denominator of fraction = 5 or 9 .

Numerator:- (when denominator is 5)

⇒ Numerator = n - 3

⇒ Numerator = 5 - 3

⇒ Numerator = 2

∴ Fraction = 2/5

Numerator:- (when denominator is 9)

⇒ Numerator = n - 3

⇒ Numerator = 9 - 3

⇒ Numerator = 6

∴ Fraction = 6/9

Therefore,as both fractions fulfills the desires of question:-

Fraction is 2/5 or 6/9 .

Answer 2

Given:-

numerator is 3 less than the denominator of a fraction.

To find:-

The fraction.

Solution:-

Let the denominator be 'x' and the numerator be (x - 3).

The fraction becomes (x - 3)/x

A/q,

1 is added to the denominator and the fraction is decreased by 1/15.

⇒ $\bold{\frac{(n-3)}{(n+1)} = \frac{n-3}{n} -\frac{1}{15} }$

⇒$\bold{\frac{(n-3)}{(n+1)} =\frac{15n-45-n}{15n} }$

⇒$\bold{\frac{(n-3)}{(n+1)} =\frac{14n-45}{15n} }$

⇒$\bold{15n^2-45n=14n^2-45n+14n-45}$

⇒$\bold{15n^2-14n^2-45n+45n-14n+45=0}$

⇒$\bold{n^2-14n+45=0}$

⇒$\bold{n^2-9n-5n+45=0}$

⇒$\bold{n(n-9)-5(n-9)=0}$

⇒$\bold{(n-9)(n-5)=0}$

⇒$\bold{(n-9)=0\ or\ (n-5)=0}$

⇒$\bold{n=9\ or\ n=5}$

So, we find two values for the denominator which are 9 and 5.

If denominator is 9, numerator = (n - 3) = (9 - 3) = 6

So, the fraction is 6/9 (or 2/3).

if denominator is 5, numerator = (n - 3) = (5 - 3) = 2

So, the fraction is 2/5.

∴Hence, the fraction is 2/5 or 6/9 (2/3).