A is greater than B by 25% then by what percentage B is less than A?

Question

Q. $A$ is greater than $B$ by $25\text{%}$ then by what percentage $B$ is less than $A$ ?

my approach:

$A$ is greater than $B$ by $25$% then

$${A-B\over B}\times 100=25$$

$$A=B+0.25B=1.25B$$

Now, $B$ is less than $A$ by %

$$\frac{A-B}{B}\times 100=\frac{1.25B-B}{B}\times100=25\text{%}$$

Is my answer correct? I don't know where I am wrong.

Help me solve this question. Thanks.

Answer 1

Your answer is not correct. You must take $\text{%}$ of B w.r.t. A as follows $$\frac{A-B}{A}\times 100=\frac{1.25B-B}{1.25B}\times 100=\frac{1}{5}\times 100=20\text{%}$$

Answer 2

In such questions is most important to clearing which number we are taking as 100%.

When we say $A$ is greater than $B$ by 25% of $B$ then $A=B+B\cdot 0.25$

When we say $A$ is greater than $B$ by 25% of $A$ then $A=B+A\cdot 0.25$

Same for reverse.

Answer 3

There's a problem in your question. In Q you say

$A$ is greater than $B$ by $20\%$

but then in your attempt you say

$A$ is greater than $B$ by $25\%$.

I will assume the second statement is correct.

I think the best way to think of percentage change is as multiplication by a factor. Then "$A$ is greater than $B$ by $25\%$" means $$ A = 1.25B . $$ So (solving for $B$)

$$ B= \frac{A}{1.25}= \frac{1}{1.25}A = 0.8A = (1-0.2)A $$ so $B$ is $20\%$ less than $A$.

For a slightly more abstract argument, see How to get the reverse percentage (not amount of reverse percentage)?

Answer 4

Working with percentages is always, always, always a multiplication, never, never, never and addition or a subtraction, like in this case:

A = 1.25 * B
B = 1/1.25 * A
  = 0.8 * A

And 0.8*... means, subtract 20%.