Tom and Jerry are working together to weed the garden. Working together it takes them 2 hours and 24 minutes. If it takes Tom two hours longer to finish the job alone than it takes Jerry to finish the job alone, how long would it take each of them to weed the garden if they worked alone?
1/J + 1/120 min = 144 min
they each do a fraction of the job
the fractions add to one
the whole job
(2.2 / T) + (2.2 / J) = 1
substituting
[2.2 / (J + 2)] + (2.2 / J) = 1
solve for J, add 2 to find T
Well, let's see if we can solve this puzzle on our green thumbs!
If we let T represent the time it takes Tom to finish the job alone (in minutes), and J represent the time it takes Jerry to finish the job alone (in minutes), we can set up some equations.
We know that when they work together, it takes them 2 hours and 24 minutes, which is 144 minutes. So we can set up the equation:
1/J + 1/T = 1/144
Now we also know that it takes Tom two hours longer to finish the job alone than it takes Jerry. So we can set up another equation:
T = J + 120
Now we have to solve these two equations simultaneously to find the values of T and J.
But hey, let's take a moment to appreciate the irony here. Tom and Jerry, those legendary cartoon characters, are finally working together peacefully to weed the garden. It's a real-life redemption story!
Okay, back to the math. Solving these equations may take more brainpower than it takes them to pull out weeds, but we can do it! Let's substitute the second equation into the first:
1/J + 1/(J + 120) = 1/144
Now let's multiply every term by 144*J*(J+120) to get rid of those pesky fractions:
144*(J+120) + 144*J = J*(J+120)
Expand everything:
144J + 17280 + 144J = J^2 + 120J
Combine like terms:
288J + 17280 = J^2 + 120J
And simplify:
J^2 - 168J - 17280 = 0
Now we have a quadratic equation, but no fear, my horticultural friend! We can solve it using the quadratic formula:
J = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values for a, b, and c:
J = (-(-168) ± √((-168)^2 - 4(1)(-17280)))/(2*1)
J = (168 ± √(28224 + 69120))/2
J = (168 ± √97344)/2
J = (168 ± 312)/2
Now we have two possible solutions for J:
J = (168 + 312)/2 = 240
J = (168 - 312)/2 = -72
Hmmm, that second solution doesn't make sense in the context of this problem. We can't have negative time! So, we'll ignore that one.
Therefore, it would take Jerry 240 minutes to weed the garden alone.
Since we know that Tom takes two hours longer, or 120 minutes longer, we can add that to Jerry's time:
T = J + 120
T = 240 + 120
T = 360
So, it would take Tom 360 minutes to weed the garden alone.
And there you have it! Tom and Jerry, the dynamic gardening duo, would take 240 minutes and 360 minutes, respectively, to weed the garden on their own. Now that's teamwork!
To solve this problem, we can set up an equation based on the given information.
Let's say Jerry takes x minutes to finish the job alone. According to the problem, Tom takes 2 hours longer than Jerry to finish the job alone, which can be represented as (x + 2*60) minutes.
When they work together, their combined work rate is equal to 1 job completed in 2 hours and 24 minutes, or 144 minutes.
We can express their combined work rate as the sum of their individual work rates:
1/x + 1/(x + 2*60) = 1/144
To solve this equation, we can first simplify it by finding a common denominator:
(1/x*(x + 2*60)) + (1/(x + 2*60)*x) = 1/144
Simplifying further:
((x + 2*60) + x) / (x*(x + 2*60)) = 1/144
Combine like terms:
(2x + 2*60) / (x*(x + 2*60)) = 1/144
Cross multiply:
(2x + 2*60)*144 = x*(x + 2*60)
Simplify:
(2x + 120)*144 = x^2 + 120x
Expand and rearrange:
288x + 17280 = x^2 + 120x
Rearrange the equation:
x^2 - 168x - 17280 = 0
At this point, we can solve this quadratic equation to find the value of x. Factoring or using the quadratic formula would be the suitable methods to find the solutions.
To solve this problem, we can set up an equation using the concept of work rates. Let's suppose that it takes Jerry x minutes to finish the job alone.
Since Tom takes 2 hours longer to finish the job alone than Jerry, we can say that it takes Tom (x + 120) minutes to finish the job alone. Here, we convert 2 hours to minutes by multiplying it by 60.
Now, let's consider the work rates of Jerry and Tom. The work rate is defined as the fraction of the job done per unit of time.
Jerry's work rate is 1/x (since he finishes the entire job in x minutes).
Tom's work rate is 1/(x + 120) (since he finishes the entire job in x + 120 minutes).
When they work together, their total work rate is 1/(2 hours and 24 minutes) = 1/144 minutes.
According to the concept of work rates, the total work rate when two people work together is the sum of their individual work rates. Therefore, we can set up the following equation:
1/x + 1/(x + 120) = 1/144
Now, let's solve this equation to find the values of x and (x + 120):
To simplify the equation, we can multiply through by the least common denominator, which is x(x + 120)(144):
144(x + 120) + 144x = x(x + 120)
Simplify the equation:
144x + 144(120) + 144x = x^2 + 120x
288x + 17280 = x^2 + 120x
Rearrange the equation:
x^2 + 120x - 288x - 17280 = 0
x^2 - 168x - 17280 = 0
Now, we can use the quadratic formula to solve for x:
x = [-(-168) ± sqrt((-168)^2 - 4(1)(-17280)))] / (2(1))
x = [168 ± sqrt(28224 + 69120)] / 2
x = [168 ± sqrt(97344)] / 2
x = [168 ± 312] / 2
Now, let's consider both the positive and negative values of x:
When x = (168 + 312) / 2 = 480 / 2 = 240, Tom's time would be (240 + 120) = 360 minutes.
When x = (168 - 312) / 2 = -144 / 2 = -72, the negative value is not applicable in this situation.
Therefore, it would take Jerry 240 minutes and Tom 360 minutes to weed the garden if they worked alone.