Equation for "$a$ is greater than $b$ by $40\%$"

Question

Most people answer $$ a = b + \frac{40}{100}b $$

but I think this should also be right $$ a - \frac{40}{100}a = b $$ Is one of them wrong or both are correct?
Or the statement is lacking details of 40% of who (b or a).

Help me solve this question. Thanks.

Answer 1

The phrasing is not clear. Let me rephrase:

$$\textrm{"a is greater than b by 40%"}$$

Let start with the untrue equality $a=b\quad(1) $.

At the moment the LHS is greater than the RHS by $40\%$. To obtain an equality we add $40\%$ on the right side.

$$a=b+0.4\cdot b\Rightarrow \boxed{a=1.4b}$$

Next we can set up an equation for $\textrm{"b is smaller than a by 40%"}$. If we look at $(1)$ we have to subtract $40\%$ at the left side to obtain a true equality.

$$a-0.4a=b\Rightarrow \boxed{0.6a=b}$$ It is worth to notice that this two equations are not equivalent (equal).

Answer 2

This phrase always means that $a$ is $100$% of $b$ plus $40$% of $b$. Thus, your first thought is correct.