SOLUTION: I cannot figure this problem out! I cannot figure out how to set up an equation and rationalize the denominators. To setup the equation, you need to make a table, which I do not kn

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Question 856547: I cannot figure this problem out! I cannot figure out how to set up an equation and rationalize the denominators. To setup the equation, you need to make a table, which I do not know how to do. I would really appreciate it if you could completely solve this problem for me. I will be sure to thank you for your help.
By splitting up the problems, Noah and Bella can do a homework assignment in 30 minutes. Bella alone can do the same assignment in 25 minutes less time than Noah. Find the time it takes each of them to do the assignment alone.
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Uniform rates problem, task or job completion, or rate of work. Do you absolutely NEED to make a table to do this one? (Also the problem has no need to "rationalize" any denominators. The only irrational component occurs in a numerator.)

Use work rates as number of JOBS per MINUTE. That will keep your thinking clear so R*T=J Rate, Time, Job.

Noah's rate, we do not know; call this 1/m jobs per minute.
Bella's rate, 1/(m-25) jobs per minute;
Noah And Bella Together, 1/30 jobs per minute.
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Their jobs are simply added for when they are working simultaneously. This will mean, if Noah's rate is r, then the rate sum equation should be:
highlight_green%281%2Fm%2B1%2F%28m-25%29=1%2F30%29
If this description and equation make sense to you then you are ready to just solve for m, and that will be Noah's work rate for when he works alone.

LCD is 30m(m-25); multiply both members by it.---(CORRECTED SOLUTION STEPS FOLLOW HERE)----(*Previous mistaken solution at bottom)
%2830m%28m-25%29%29%2Fm%2B%2830m%28m-25%29%29%2F%28m-25%29=30m%28m-25%29%2F30
30%28m-25%29%2B30m=m%28m-25%29
30m-750%2B30m=m%5E2-25m
m%5E2-25m-30m%2B750-30m=0
highlight_green%28m%5E2-85m%2B750=0%29
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Discriminant is 85%5E2-4%2A750=4225=169%2A25
Discriminant refactored is %2813%5E2%29%285%5E2%29=highlight_green%2865%5E2%29
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Continuing with general solution for a quadratic equation even though the equation obtained is factorable, the solution for m is:
m=%2885%2B-+sqrt%2865%5E2%29%29%2F2
m=%2885%2B65%29%2F2, using only the positive square root form which prevents a nonsense answer as one of the solutions.
m=150%2F2
highlight%28m=75%29
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This means, 75 minutes for Noah alone, and therefore 75-25=50 minutes for Bella alone.


* ---------------------------------Previous Mistaken Solution Steps----------------------
30%28m-25%29%2B30m=m%28m-25%29
30m-30%2A25%2B30m=m%5E2-25m
m%5E2-25m-30m-30%2A25=0
m%5E2-55m-750=0
Not sure if that is factorable,
disciminant is 55^2-4(-750)=3025+3000=6025=5*5*241;
m=%2855%2Bsqrt%286025%29%29%2F2
highlight%28m=%2855%2B5%2Asqrt%28241%29%29%2F2%29
highlight%28m=66.31%29 approximately, minutes.
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Noah's rate is 1%2F66.31 jobs per minute, approximately.
Noah could do the assignment alone in 66%261%2F2 minutes.