Question

If (p)/(q)=4.bar(23) and (x)/(y)=0.bar(67) where p q and x y are co- prime integers and q y!=0 then (py)/(qx) equals
can anybody answe thid in 5 min​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

$\displaystyle \sf{ \frac{p}{q} = 4. \overline{23} \: \: \: and \: \: \: \frac{x}{y} = 0. \overline{23} \: \: }$

where p q and x y are co- prime integers

TO DETERMINE

$\displaystyle \sf{ \frac{py}{qx} \: }$

CALCULATION

Let

$\displaystyle \sf{ k= 4. \overline{23} \: }$

$\implies \: \displaystyle \sf{ k= 4. 232323........ \: \: \: \: \: \: (1)\: }$

Multiplying both sides by 100 we get

$\displaystyle \sf{ 100k= 423. 232323........ \: \: \: \: \: \: (2)\: }$

Equation (2) - Equation (1) produce

$\displaystyle \sf{ 99k= 419 }$

$\implies\displaystyle \sf{k= \frac{419}{99} }$

Therefore

$\displaystyle \sf{ \frac{p}{q} = 4. \overline{23} \: = \frac{419}{99} } \: \: \: \: ....(3)$

Similarly

$\displaystyle \sf{ \frac{x}{y} = 0. \overline{67} \: = \frac{67}{99} }$

$\implies \: \displaystyle \sf{ \frac{y}{x} =\frac{99}{67} } \: \: \: \: \: \: ....(4)$

Now Equation (3) × Equation (4) gives

$\displaystyle \sf{ \frac{py}{qx} = \frac{419}{99} \times \frac{99}{67} \: }$

$\implies \: \displaystyle \sf{ \frac{py}{qx} = \frac{419}{67} \: }$

RESULT

$\boxed{ \: \displaystyle \sf{ \: \: \: \: \frac{py}{qx} = \frac{419}{67} \: } \: \: }$

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