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The Difference between the Numerator and the Denominator of a Fraction is 5. If 5 is Added to its Denominator, the Fraction is Decreased by 5/4. Find the Value of the Fraction?

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Explanation

  • The difference between the numerator and the denominator of a fraction is 5.
  • If 5 is added to its denominator, the fraction is decreased by 5/4.
  • The value of the fraction can be figure out in this way.

Let suppose y shows numerator and z shows denominator in a fraction.

The difference between y (numerator) and z (denominator) is 5.

y – z = 5 ________ (i)

The fraction (y/z) decreased by 5/4 when 5 is added to z (denominator).

y/(z +5) = y/z – 5/4 ________ (ii)

By solving both equations simultaneously, we can easily find out the fraction (y/z = 9/4).

To Find

Fraction = ?

Solution

Let suppose

Let suppose y shows numerator and z shows denominator in a fraction.

According to first given condition

y – z = 5 ________ (i)

y = 5 + z ________ (A)

According to second given condition

y/(z +5) = y/z – 5/4 ________ (ii)

y/(z + 5) = (4y – 5z)/4z

4yz = (4y – 5z)(z + 5)

4yz = 4yz + 20y – 5z2 – 25z

5z2 + 25z – 20y = 0

z2 + 5z – 4y = 0 ________ (B)

Put y = z + 5 in equation (B)

z2 + 5z – 4(z + 5) = 0

z2 + 5z – 4z – 20 = 0

z(z + 5) -4(z + 5) = 0

(z + 5)(z – 4) = 0

z + 5 = 0             &                z – 4 = 0

z = -5 (not possible)              &                z = 4 (possible)

Put z = 4 in equation (A)

y = 4 + 5

y = 9

Fraction = 9/4 answer

Conclusion

The difference between the numerator and the denominator of a fraction is 5. If 5 is added to its denominator, the fraction is decreased by 5/4. The fraction is 9/4.

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