Find the rational number having the following decimal expansions:… - Vidyadip

Question

Find the rational number having the following decimal expansions:$3.\overline{52}$

✓

Answer

$3.\overline{52}=3+0.52222\ \dots$$=3+0.5+0.02+0.002+0.0002+\ \dots$
$=3.5+\frac{2}{10^2}+\frac{2}{10^3}+\frac{2}{10^4}+\ \dots$
$=3.5+\frac{2}{10^2}\Big(1+\frac{1}{10}+\frac{1}{10^2}+\ \dots\Big)$
$=\frac{35}{10}+\frac{2}{100}\Bigg(\frac{1}{1-\frac{1}{10}}\Bigg)$
$=\frac{35}{10}+\frac{2}{100}\times\Big(\frac{10}{9}\Big)$
$=\frac{35}{10}+\frac{2}{90}$
$=\frac{315+2}{90}$
$3.\overline{52}=\frac{317}{90}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Geometric Progressions→Geometric Progressions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Solve the following equations:
$4\sin\text{x}\cos\text{x}+2\sin\text{x}+2\cos\text{x}+1=0$

Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:
$5\text{x}^2+4\text{y}^2=1$

Differentiate the functions with respect to 'x'.
$\frac{\text{ax}+\text{b}}{\text{cx}+\text{d}}$

The mean and standard deviation of a group of $100$ observation were found to be $20$ and $3$ respectively. Later on it was found that three observations were incorrect, which were recorded as $21, 21$ and $18$. Find the mean and standard deviation if the incorrect observations are omitted.

A wheel makes 360 revolutions per minute. Through how many radians does it turn in 1 second?

While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the mean and variance as 45 and 16 respectively. Find the correct mean and the variance.

If a, b, c, d are in G.P., prove that:
$\frac{\text{ab}-\text{cd}}{\text{b}^2-\text{c}^2}=\frac{\text{a}+\text{c}}{\text{b}}$

Use the Principle of Mathematical Induction in the following Exercis.
Prove that $\frac{1}{\text{n}+1}+\frac{1}{\text{n}+2}+\ .....\ +\frac{1}{2\text{n}}>\frac{13}{24},$ for all natural numbers n > 1.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{2}-\sqrt{1+\cos\text{x}}}{\sin^2\text{x}}$

show that the solution set of the following system of linear inequalities is an unbounded region $2\text{x}+\text{y}\geq8,\text{x}+2\text{y}\geq10,\text{x}\geq0,\text{y}\geq0.$