# I always exaggerate everything I say by 25%. By how much do I truly exaggerate?

'I always exaggerate everything I say by 25%. By how much do I truly exaggerate?'

Hi, I asked myself this question and it is starting to give me a headache. Does anyone have a correct solution? Thanks

• Do we have to consider that you also exaggerate not all the time, but 75% of the time by x amount? – Ryan Oct 22 '19 at 6:56
• Nice, hadn't thought of this. If 'always' is treated as 100% then yes. – zoidstar Oct 22 '19 at 11:00
• In fact 'everything' could also be treated as a percentage. Rephrased this would be: '100% of the time I exaggerate 100% of what I say by 25%. By how much do I truly exaggerate?' – zoidstar Oct 22 '19 at 12:17

$$20.7\%$$ approximately.

If you exaggerate by $$x\%$$ then you are multiplying by $$1 + \frac{x}{100}$$.

We know that $$x \times (1 + \frac{x}{100}) = 25$$.

Solving for $$x$$ using the quadartic formula leads to the correct result.

Nice question. :)

Let $$x$$ be the fraction you exaggerate everything by.

Then $$25\%$$ is equal to $$x$$ exaggerated by $$x$$, so $$x(1+x) = 25\%$$.

Solving this quadratic equation yields $$x = \frac{\pm\sqrt{2} - 1}{2}$$.

So either you exaggerate everything by $$20.71067811865\%$$, or you exaggerate everything by $$-120.71067811865\%$$.