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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