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, 2014

Algebra Difficulty 1.0 Junior Prove it Canada

What is the value of $\dfrac{\sqrt{16} + \sqrt{9}}{\sqrt{16 + 9}}$ ?
In the diagram, the angles of $\triangle ABC$ are shown in terms of $x$ .

What is the value of $x$ ?

Lisa earns two times as much per hour as Bart. Lisa works 6 hours and Bart works 4 hours. They earn $200 in total. How much does Lisa earn per hour?

Solution

Evaluating, $\dfrac{\sqrt{16} + \sqrt{9}}{\sqrt{16 + 9}} = \dfrac{4+3}{\sqrt{25}}=\dfrac{7}{5}$ .
Since the sum of the angles in a triangle is $180^\circ$ , then $(x-10)^\circ + (x+10)^\circ + x^\circ = 180^\circ$ or $(x-10)+(x+10)+x=180$ .

Thus, $3x=180$ and so $x=60$ .
Suppose Bart earns x $per hour. In 4 hours, he earns$ 4 $\times$ \$x = \$4x$.

Then Lisa earns \$2x per hour. In 6 hours, she earns 6 $\times$ \$2x = \$12x$.

Since they earn $200 in total, then$ 4x+12x=200 $or$ 16x = 200$.

Therefore, $x = 12.5$ .

Finally, since $2x=25$ , then Lisa earns \$25$ per hour.

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