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?

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- May 28th 2007, 08:46 AMGodfatherNumerators and Denominators
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?

- May 28th 2007, 09:23 AMSoroban
Hello, Godfather!

Quote:

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?

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

For the largest value, we want the*smallest denominator.*

. . This occurs for: .$\displaystyle b = 2,\:d = 3$ .(or vice versa).

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

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

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

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

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