a, b, c, d are positive integers $\frac{a}{b} < \frac{c}{d}$. What is bigger $\frac{a + c}{b + d}$ or $\frac{c}{d}$

Question

The answer is $\frac{c}{d}$ is bigger. Why? What is the intuition behind? How to prove it? *I took this question from GRE preparation, the resource just says plug in numbers and no intuition or proof or proper explanation.

Thanks

Answer 1

$$\frac{a}{b} < \frac{c}{d}$$ $$ad< bc $$ $$ad+cd< bc + cd$$

$$(a+c)d < c(b+d)$$

$$\frac{a+c}{b+d}<\frac{c}{d}$$