3. The sum, difference, and quotient of two numbers are in the ratio $20: 4$ : 1. Determine these numbers.
First method:

Let the first number be $x$, and the second number be $y$. Then we have: $(x+y):(x-y): \frac{x}{y}=20: 4: 1$.

Therefore, there exists a rational number $k$ such that:

$x+y=20 k, x-y=4 k \text{ and } \frac{x}{y}=k$.

1 POINT

If we add the equations $x+y=20 k$ and $x-y=4 k$, we get:

$2 x=24 k$, or $x=12 k$.

1 POINT

Now we have $12 k+y=20 k$, so $y=8 k$.

1 POINT

From $\frac{x}{y}=k$ it follows that $\frac{12 k}{8 k}=k$, or $k=1.5$.

1 POINT

Now $x=12 \cdot 1.5=18$,

1 POINT

and $y=8 \cdot 1.5=12$.

1 POINT

The required numbers are 18 and 12.

TOTAL 6 POINTS

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