Question : The numerator of a fraction is a multiple of two numbers. One of the numbers is greater than the other by 2. The greater number is smaller than the denominator by 4. If the denominator $7 + c (c > –7)$ is a constant, then the minimum value of the fraction is:
Option 1: 5
Option 2: $\frac{1}{5}$
Option 3: –5
Option 4: $-\frac{1}{5}$
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Correct Answer: $-\frac{1}{5}$
Solution : Let the first number be $x$. Then, the second number would be $(x+2)$. According to the question, ⇒ $x + 2+ 4 = 7 + c$ ⇒ $x + 6 = 7 + c$ ⇒ $x = 1 + c$ Fraction is $\frac{{x}({x}+2)}{7+{c}}= \frac{(1+c)(3+c)}{7+c}$ For the minimum value, $–3 < c < –1$ ⇒ $c = -2$ The required value of the fraction is $\frac{(1-2)(3-2)}{7-2}=-\frac{1}{5}$ Hence, the correct answer is $-\frac{1}{5}$.
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