Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is \(2\) greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator of the previous fraction by \(1\). He then multiplies all his fractions together. He has \(3\) fractions, and their product equals \(10\). What is the value of the first fraction he wrote?
+2666 We have the equation: \(\large{{{x +2} \over x} \times {x+3 \over x+1} \times {x+4 \over x+2} = 10}\)
Because there is the term \(x+2\) in both the numerator and the denominator, we can cancel them out.
This gives us: \(\large{{x+3 \over x+1} \times {x+4 \over x} = 10 }\)
Simplifying the left-hand side gives us: \(\large{{{x^2 + 7x + 12} \over {x^2 + x}} = 10 }\).
From the equation, we know that the numerator (\(x^2 + 7x + 12 \)) must be 10 times the denominator (\(x^2 + x \))
Thus, we have: \(10(x^2 +x) = x^2 + 7x + 12 \)
Now, we have to solve the equation, and subsitute our values into the first fraction (\(\large{x+2 \over x}\))
Can you take it from here?
