How to calculate this expression and get an integer number?

Question

Hello there I don't have idea how to calculate this:

$$\left[\frac {116690151}{427863887} \times \left(3+\frac 23\right)\right]^{-2} - \left[\frac{427863887}{116690151} \times \left(1-\frac 8{11}\right)\right]^{-2}$$

I have tried dividing these numbers but couldn't get anything, also I get a 11/3 when I add and subtract these in brackets, but don't know how to use that.

Answer 1

The two "small" fractions are $\frac {11}3$ and $\frac 3{11}$

Two methods of proceeding come to mind - the first is to see whether the factors $3$ and $11$ cancel (there are easy tests to show that both do), and see what happens.

The second would be to use the Euclidean Algorithm to see whether there are any common factors of the "large" fraction which can be cancelled out.

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This is not a full answer, but you have not at the moment asked a really full question. When you get $\frac {11}3$ and you are working with fractions, then immediately you should test whether these cancel.

If you develop any intuition at all about this kind of problem, it should be that if you are having to do it by hand, something is likely to cancel somewhere.

Answer 2

Using a simple calculator, the big fraction is

$$r=0.27272727\cdots$$

Then $100r-r=27.27272727\cdots-0.27272727\cdots=27$ and $$r=\frac3{11}.$$

Indeed $\dfrac{116690151}3=\dfrac{427863887}{11}=38896717$.

The rest is trivial.

Answer 3

Let's look at just the part inside the first set of brackets:

$$ \begin{eqnarray} \frac {116690151}{427863887} \times \left(3+\frac 23\right) &=& \frac {116690151}{427863887} \times \frac{11}{3} \\ &=& \frac {116690151 \times 11}{427863887 \times 3} \\ &=& \frac {1283591661}{1283591661}. \end{eqnarray} $$

I think you can take it from there. A similar method works for the other part.

It may seem a bit unobvious that one should multiply $116690151 \times 11,$ since we usually want to separate the factors of the numerator and denominator of a fraction we wish to simplify, but noticing that $11$ is not quite four times $3$ and that $427863887$ is not quite four times $116690151$, one might hope that $\frac{427863887}{116690151}$ and $\frac{11}{3}$ might be equal, and cross-multiplying is one way to confirm that they are. (As Mark Bennet remarked, if we're expected to work this out exactly then we should look for things to cancel out, and we certainly get very nice cancellation if the numbers we are multiplying are in fact exact inverses.)