Given the growth rate of one component, how much is the total growth rate?

Question

$x = a + b + c + d$

If I increase $a$ by a $5\%$, Can I know by how much $x$ will increase keeping all constant?

Answer 1

Assuming that "keeping all constant" means $b$, $c$ and $d$ don't change, the new value of $x$ is $$ x' = 1.05a + b + c + d = 0.05a + x $$ so $$ \frac{x'}{x} = \frac{0.05a}{x} + 1 $$ so the relative increase in $x$ is $\frac{0.05a}{x}$, which you can write as a percent.

Intuitively, the change in $x$ caused by a change in $a$ is reduced by the fraction $a$ contributes to $x$.

Answer 2

It cannot be known. You have that $x=a+b+c+d$ and wonder about how much bigger $x^\prime=1.05 a + b+c+d$ is. The change in $x$ is $\Delta x=x^\prime-x=0.05a$, but the percentage increase in $x$ is

$$ \frac{\Delta x}{x}=\frac{0.05a}{a+b+c+d},$$

which depends on $a+b+c+d$.