Question 15 Marks

Express $0.6+0.\overline7+0.4\overline7$ in the form $\frac{\text{p}}{\text{q}},$ where $p$ and $q$ are integers and $\text{q}\neq0.$

Answer

We have $0.6=\frac{6}{10}\ ...(\text{i})$
Let $\text{x}=0.\overline7=0.777...\ \ ...(\text{ii})$
Multiply $10$ on both side of the equation.
$10x = 7.777777... ...(iii)$
Subtracting $(ii)$ from $(iii),$
we get $9\text{x}=7\Rightarrow\text{x}=\frac{7}{9}\text{ or }0.\overline7=\frac{7}{9}$
Now, let $y = 0.4777777777... ...(iv)$
Multiply 10 on both side of the equation.
$10y = 4.77777777777... ...(v)$
again Multiply 10 on both side of the equation.
$100y = 47.77777777777... ...(vi)$
Subtracting $(v)$ from $(iv),$
we get $90y = 43$
$\Rightarrow\ \text{y}=\frac{43}{90}$
Now, $0.6+0.\overline7+0.4\overline7$
$=\frac{6}{10}+\frac{7}{9}+\frac{43}{90}$
$=\frac{54+70+43}{90}$
$=\frac{167}{90}$

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