Question

OR
345679
The decimal representation of will terminate after how many decimal places?
2²x55​

Answer 1

Given :  345679 / (2² × 5⁵)

To Find : The decimal representation  will terminate after how many decimal places

Solution:

345679 / (2² × 5⁵)

Multiply numerator and denominator by 2³

= 345679 × 2³  / (2³ × 2² × 5⁵)

= 345679 × 2³  / (2³ ⁺² × 5⁵)

= 345679 × 2³  / (2⁵ × 5⁵)

= 345679 × 2³  / (2× 5)⁵  

= 345679 × 2³  / (10)⁵

= 345679 × 2³  / 100000

The decimal representation will terminate after 5 decimal places.

345679 × 2³  / 100000

=  345679 × 8  / 100000

= 2765432 / 100000

= 27.65432

The decimal representation will terminate after 5 decimal places.

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Answer 2

SOLUTION

TO DETERMINE

The number of places after which below fraction terminates

$\displaystyle \sf{ \frac{345679}{ {2}^{3} \times {5}^{5} } }$

CONCEPT TO BE IMPLEMENTED

Let us consider a fraction

$\displaystyle \sf{ \frac{p}{q} }$

If q can be written as $\sf{q = {2}^{m} \times {5}^{n} }$

Then the fraction is said to be terminating and terminates after r terms

where r = max{m, n}

EVALUATION

Here the given fraction is

$\displaystyle \sf{ \frac{345679}{ {2}^{3} \times {5}^{5} } }$

Denominator

$\displaystyle \sf{ = {2}^{3} \times {5}^{5} }$

Since the denominator contains factor only as 2 and 5

So the given fraction is terminating

Now the exponents are 3 & 5

Since maximum of 3 & 5 = 5

So the given fraction terminates after five decimal places

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