Math, asked by saleem1838, 10 months ago

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​

Answers

Answered by Anonymous
120

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}}

Answered by unsungwriter
81

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