20/(16+m) + 12/m = 1
First, we'll simplify the equation by finding a common denominator:
(20m + 12(16+m)) / (m(16+m)) = 1
Now, distribute and combine like terms in the numerator:
(20m + 192 + 12m) / (m(16+m)) = 1
Combine the m terms in the numerator:
(32m + 192) / (m(16+m)) = 1
Multiply both sides by m(16+m) to clear the fraction:
32m + 192 = m(16+m)
Expand the right side:
32m + 192 = 16m + m^2
Rearrange the equation to bring all terms to one side and set equal to zero:
m^2 - 20m - 192 = 0
Now we can factor the quadratic equation:
(m - 24)(m + 8) = 0
This gives us two possible values for m:
m - 24 = 0 or m + 8 = 0
Solving each equation separately:
m = 24 or m = -8
These are the two solutions to the equation. However, we must check if they satisfy the original equation. Plugging in m = 24:
20/(16+24) + 12/24 = 1
20/40 + 1/2 = 1
1/2 + 1/2 = 1
1 = 1
The value of m = 24 satisfies the original equation. For m = -8:
20/(16-8) + 12/-8 = 1
20/8 - 1.5 = 1
2.5 - 1.5 = 1
1 = 1
The value of m = -8 also satisfies the original equation. Therefore, both m = 24 and m = -8 are valid solutions.
