Numerators and Denominators

**Godfather** · May 28th 2007, 08:46 AM

What is the largest possible value for the sum of two fractions such that each of the four 1 digit prime numbers occurs as one of the numerators or denominators?

**Soroban** · May 28th 2007, 09:23 AM

Hello, Godfather!

What is the largest possible value for the sum of two fractions such that
each of {2, 3, 5, 7} occurs as one of the numerators or denominators?

Here's one approach . . .

The sum of two fractions is: . $S\:=\:\frac{a}{b} + \frac{c}{d} \:=\:\frac{ad + bc}{bd}$

For the largest value, we want the smallest denominator.
. . This occurs for: . $b = 2,\:d = 3$ .(or vice versa).

We have: . $S \:=\:\frac{3a + 2b}{6}$

Then we have two choices: . $\begin{array}{c}a = 5,\,b = 7 \\ a=7,\,b=5\end{array}$

If $a=5,\,b=5:\;S\:=\:\frac{3\!\cdot\!5+2\!\cdot\!7}{6 } \:=\:\frac{29}{6}$

If $a=7,\,b=5:\;S\:=\:\frac{3\!\cdot\!7 + 2\!\cdot\!5}{6}\:=\:\frac{31}{6}\quad\Leftarrow\:\ text{larger!}$

Therefore: . $\frac{7}{2} + \frac{5}{3}\:=\:\boxed{\frac{31}{6}}$ .is the largest sum.

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