If $$ A = \frac{58^2 - 25^2}{46^2 - 37^2} $$ and $$ B = \frac{26^2 - 15^2}{56^2 - 15^2}, $$ then find the value of $$ \frac{1}{B} - \frac{20}{A}. $$

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**Solution:**

1. Simplify expression for *A* using difference of squares formula:
   $$
   A = \frac{58^2 - 25^2}{46^2 - 37^2} = \frac{(58 - 25)(58 + 25)}{(46 - 37)(46 + 37)}
   $$
   $$
   A = \frac{33 \times 83}{9 \times 83} = \frac{33}{9} = \frac{11}{3}
   $$

2. Simplify expression for *B* similarly:
   $$
   B = \frac{26^2 - 15^2}{56^2 - 15^2} = \frac{(26 - 15)(26 + 15)}{(56 - 15)(56 + 15)}
   $$
   $$
   B = \frac{11 \times 41}{41 \times 71} = \frac{11}{71}
   $$

3. Calculate $ \frac{1}{B} - \frac{20}{A} $:
   $$
   \frac{1}{B} - \frac{20}{A} = \frac{1}{\frac{11}{71}} - \frac{20}{\frac{11}{3}} = \frac{71}{11} - \frac{20 \times 3}{11} = \frac{71 - 60}{11} = \frac{11}{11} = 1
   $$

**Answer:** The value of $ \frac{1}{B} - \frac{20}{A} $ is **1**.

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