Why is the answer of $\frac{ab}{a+b}$ always smaller than the smallest number substituted?

If $a,b>0$ then $a<a+b$ and $\dfrac a{a+b}<1$ so $\dfrac {ab}{a+b}<b$.

A similar argument shows that $\dfrac{ab}{a+b}<a$.


Another way to think about it: Assuming $0<a\leq b$, divide the top and bottom of your fraction by $b$ to get

$$\frac{a}{\frac{a}{b}+1}.$$

$a$ is the smaller number and you're dividing it by a number greater than one, so the result is smaller than $a$.

Tags:

Inequality

Algebra Precalculus

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