Explanation of calculator percentage key

Question

I would really appreciate some help with this question. I have asked few people but nobody could explain.

When you use calculator and enter these values you get different results. Why?

Example: 100 - 25% = 75 - It's correct. 100 * 1.25 = 125 - It's correct. 100 * 0.25 = 25 - It's correct. 100 / 1.25 = 80 - Is it correct??? 100 / 0.25 = 400 - what is this?

Can you explain why calculator is doing this? I have tried this on different calculators.

Answer 1

If you divide $100$ by $4$, you are asking how many portions of sugar you can get from a 100g bag of sugar, if each portion weighs 4g. The answer is $25$ portions.

If you divide $100$ by $0.25$, you are asking how many portions of sugar you can get from a 100g bag of sugar, if each portion weighs 0.25g. The answer is $400$ portions.

If you divide $100$ by $1.25$, you are asking how many portions of sugar you can get from a 100g bag of sugar, if each portion weighs 1.25g. The answer is $80$ portions.

That's what the calculator told you.

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Perhaps you were expecting $100 \div 1.25$ to give you the same answer as $100 - 25\%$. But those do not mean the same thing, and that is why you got different answers. $100 - x\%$ means $100 \times (1 - \frac{x}{100})$. You can calculate $100 - 25\%$ by doing $100 \times (1 - 0.25) = 100 \times 0.75 = 75$. This is not the same as $100 \div 1.25$.

Answer 2

$\boxed{100 \div 1.25} = 100 \div(1\frac14) = 100 \div \frac54 = 100 \times \frac45 = (100\times 4)\div 5 = 400\div 5 = \boxed{80}$

and

$\boxed{100\div 0.25} = 100 \div \frac14 = 100 \times \frac41 = (100\times 4)\div 1 = 400\div 1 = \boxed{400}$

Answer 3

$$100\% - 25\% = 75\%$$ represents the idea that a whole pie, minus a quarter of a pie, is equal to three quarters of a pie.

$$\frac{100\%}{1.25} = 80\%$$ is equivalent to writing $$100\% = 80\% \times 125\%$$ which in turn represents the idea that if you have five quarter-pieces of pie ($125\%$), four of those five pieces ($80\%$) will make a whole pie ($100\%$).

$$\frac{100\%}{0.25} = 400\%$$ is the same as $$100\% = 25\% \times 400\%$$ which represents the idea that if you have four whole pies ($400\%$), then one quarter ($25\%$) of those four pies is one whole pie ($100\%$).