Depth · one concept from first principles · step 2 of 4

Equal side products cancel the shared corner

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Number systemExpressions

Your answer

How to solve

Strategy Convert to Algebra — We only need the ratio, not x and y themselves. The left and right sides share the top corner -3/2, so set their products equal and divide that shared factor away: what is left is a direct link between x and y.

1STEP 1

Set the two slanted sides equal

Both slanted sides contain the top number -3/2.

(-3/2) × 4 × x = (-3/2) × 3.25 × y

2STEP 2

Cancel the shared corner

Divide both sides by -3/2, which is not 0.

4 × x = 3.25 × y

3STEP 3

Read off the ratio

Divide both sides by 4 × y and write 3.25 as 13/4.

x/y = 13/4 ÷ 4 = 13/16

Answer

13/16

Find the actual values. Setting the left side equal to the bottom side and dividing by x gives -6 = -2/3 × y, so y = 9 and then x = 13/16 × 9 = 117/16. Each side's product is then -351/8: for example (-3/2) × 4 × 117/16 = -351/8 and 117/16 × (-2/3) × 9 = -351/8.

Takeaway

When two products share a factor, cancel it first and the rest falls out.

Set the two slanted sides equal
Cancel the shared corner
Read off the ratio

Where next?

Level upsuggested

Turn the condition into x+y in terms of xy Rewrite numerators using the given total

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▶ Practice — 12 problems