The answer is 72 days.
The rate that the Math Club raises money can be represented by F(x) = 31.25x, where x represents the number of days and F(x) is the amount raised. We know that F(24) = 750, since the Math Club raises $750 in 24 days.
The National Forensic League, when working with the Math Club can raise $750 in 18 days. The function to represent this is G(18) = 31.25(18) + 18f, where f represents the money the National Forensic League raises per day.
31.25 multiplied by 18 is $562.50. Subtracting this from $750 leaves you with $187.50. This, when put back into the equation results in the expression 187.50 = 18f. The next step is to divide both steps of the equation by 18, to solve for f. The answer is 10.4166...67, which simplifies to the fraction 125/12.
Using this rate, we can create a new function, H(x) = 125/12x. If H(x) = 750, then we have to divide it by 125/12 to solve for x, to see how many days it will take the National Forensic Club to raise $750 on their own.
[tex] \frac{750}{( \frac{125}{12} )} =72[/tex]
This means that the National Forensic League will take 72 days to raise $750 on their own.
You can check this answer by adding the two rates of the clubs together and seeing if G(18) = 750. The National Forensic League raises $(125/12 ) per day and the Math Club raises $31.25 a day, which is equal to 375/12. 375/12 + 125/12 = 500/12, which is the amount the two clubs raise together in a day. Our old function, G(x) = 31.25x + 18f now becomes G(x) = 500/12x.
Since we want G(x) = 750, when x = 18, we plug in those values to create the equation 750 = 500/12(18). 500/12 * 18/1 = 9000/12, which simplifies to 750. 750 = 750, meaning that the National Forensic Club's daily earning rate is correct, therefore validating the answer.
