Question - Solving a Fraction Problem
Solution:
Let's denote the numerator of the original fraction as $$ n $$ and the denominator as $$ d $$. According to the problem, the denominator is four more than the numerator, so:$$ d = n + 4 $$The problem also states that if both the numerator and denominator are increased by one, the resulting fraction is simplified to $$ \frac{5}{7} $$:$$ \frac{n + 1}{d + 1} = \frac{5}{7} $$Now we have a system of two equations to solve for the values of $$ n $$ and $$ d $$:1) $$ d = n + 4 $$2) $$ \frac{n + 1}{n + 5} = \frac{5}{7} $$ (since $$ d = n + 4 $$, we replaced $$ d $$ with $$ n + 4 $$ in the second equation)Next, we can cross-multiply in the second equation to solve for $$ n $$:$$ 7(n + 1) = 5(n + 5) $$Expand both sides:$$ 7n + 7 = 5n + 25 $$Subtract $$ 5n $$ from both sides:$$ 2n + 7 = 25 $$Subtract 7 from both sides:$$ 2n = 18 $$Divide by 2:$$ n = 9 $$Now that we have $$ n $$, we can find $$ d $$ using either of the original equations. Let's use the first one:$$ d = n + 4 $$Replace $$ n $$ with 9:$$ d = 9 + 4 $$Thus:$$ d = 13 $$The original fraction is $$ \frac{n}{d} = \frac{9}{13} $$.
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