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Problem 509

Algebra Difficulty 2.4 Multiple choice CEMC Fermat · Canada · 2023

Points $A$ , $B$ , $C$ , and $D$ lie along a line, in that order. If
$AB:AC = 1:5$ , and $BC:CD = 2:1$ , then the ratio $AB:CD$ is equal to

Pick one

Official solution

Suppose that $AB = x$ for some
$\text{x gt; 0}$ .

Since $AB:AC = 1:5$ , then $AC = 5x$ .

This means that $BC = AC - AB = 5x - x = 4x$ .

Since $BC:CD = 2:1$ and $BC = 4x$ , then $CD = 2x$ .

Therefore, $AB:CD = x:2x = 1:2$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.