Why does $0.75 \times 1.25$ equal $0.93$ and not $1$?

Solution 1:

The mistake you're making is that first taking away some percentage (in this case $25\%$) and then adding the same percentage (again $25\%$) does not give you back the number that you started with.

For instance, you start with the number $1$. You take away $25\%$, i.e. you obtain $0.75\cdot 1 = 0.75$. Then you add $25\%$, i.e. you obtain $1.25\cdot0.75 = 0.94 \neq 1$.

The point is that adding $25\%$ here amounts to adding $25\%$ of the reduced value, which is, of course, less than $25\%$ of the original value.

So indeed $0.75\cdot 1.25 \neq 1$, which is correct because $125\%$ of $75\%$ of $1$ is not $1$.

Solution 2:

You're correct in your first statement, but another way of thinking of it is to say that multiplying any number by $1.25$ is the same as adding on $\frac14$ to the existing number. So, $1.25 \times 100 = 100 + 25 = 125$

In the context of $0.75$, a quarter of $0.75$ is $0.1875$, so $125\%$ of $0.75$ is $0.75+0.1875 = 0.9375$.

A fairly simple/basic answer, but I hope it helps!

Solution 3:

All these answers show that the answer is 0.9375 by a series of manipulations, however I think it is helpful to think why this makes sense.

You are decreasing the first number by 25% to a produce a smaller number. Then this smaller number you are increasing by 25%. Even though you are taking an equal proportion of each number (25%), the initial number was larger so this means that the total size of this proportion is also larger.

So basically you are taking away a certain amount, then adding a slightly smaller amount.

After you become more experienced with maths these kinds of questions become intuitive and you take it for granted, a very good question though. Good luck with your future mathematical endeavours!

Solution 4:

You can also see this as the following: $$(1+0.25)(1-0.25)=1^2-0.25^2=0.9375$$

Solution 5:

The thing you are missing here is assuming that $25\%$ of $0.75$ is $0.25$, when in fact it's $33.\dot3\%$ of $0.75$ that is $0.25$.

$25\%$ of $1$ is $0.25$