Question

the denominator of a rational number is greater than its numerator by 7 if the numerator is increased by 17 and the denominator is decreased by 6 the new number become 12 find the original number​

Answer 1

Answer -

Original number is 5/82

$\rule{200}2$

Explanation -

Let the numerator be N and denominator be D.

$\therefore$ $\bold{\sf{Fraction\:=\:\dfrac{N}{D}}}$

The denominator of a rational number is greater than its numerator by 7.

If we add 7 to numerator then, both numerator and denominator becomes equal.

i.e.

$\implies\:\sf{D\:=\:N\:+\:7}$ ...(1)

If the numerator is increased by 17 and the denominator is decreased by 6 the new number become 12.

Now,

• Numerator = N + 17
• Denominator = D - 6

Fraction of both numerator (N + 17) and denominatior (D - 6) is equal to a number i.e. 12.

$\implies\:\sf{\dfrac{N\:+\:17}{D\:-\:6}\:=\:12}$

We can also, write it like -

$\implies\:\sf{\dfrac{N\:+\:17}{D\:-\:6}\:=\:\dfrac{12}{1}}$

Cross-multiply them

$\implies\:\sf{1(N\:+\:17)\:=\:12(D\:-\:6)}$

$\implies\:\sf{N\:+\:17\:=\:12D\:-\:72}$ ...(2)

Substitute value of (1) in (2)

$\implies\:\sf{N\:+\:17\:=\:12(N\:+\:7)\:-\:72}$

$\implies\:\sf{N\:+\:17\:=\:12N\:+\:84\:-\:72}$

$\implies\:\sf{N\:-\:12N\:=\:12\:-\:17}$

$\implies\:\sf{-\:11N\:=\:-\:5}$

$\implies\:\sf{N\:=\:\dfrac{5}{11}}$

Substitute value of N in (1)

$\implies\:\sf{D\:=\:\dfrac{5}{11}\:+\:7}$

$\implies\:\sf{D\:=\:\dfrac{5\:+\:77}{11}}$

$\implies\:\sf{D\:=\:\dfrac{82}{11}}$

Now,

$\Rightarrow\:\sf{Fraction\: = \:\dfrac{N}{D}}$

$\Rightarrow\:\sf{Fraction\:=\:\dfrac{\frac{5}{11}}{\frac{82}{11}}}$

$\Rightarrow\:\sf{Fraction\:=\:\dfrac{5}{82}}$

•°• Original number is 5/82.

$\rule{200}2$

Verification -

From the above calculations we have -

• Numerator = N = 5/11
• Denominator = D = 82/11

Substitute value of N and D in equation (1)

=> $\sf{\dfrac{82}{11}\:=\:\dfrac{5}{11}\:+\:7}$

=> $\sf{\dfrac{82}{11}\:=\:\dfrac{5\:+\:77}{11}}$

=> $\sf{\dfrac{82}{11}\:=\:\dfrac{82}{11}}$

Similarly, substitute value of N and D in equation (2)

=> $\sf{\dfrac{5}{11}\:+\:17\:=\:\dfrac{82}{11}\:-\:72}$

=> $\sf{\dfrac{5\:+\:187}{11}\:=\:\dfrac{12(82)\:-\:792}{11}}$

=> $\sf{\dfrac{192}{11}\:=\:\dfrac{984\:-\:792}{11}}$

=> $\sf{\dfrac{192}{11}\:=\:\dfrac{192}{11}}$

Answer 2

ANSWER IS IN THE ATTACHMENT