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A mathematician works for t hours per day and solves p problems per hour, where t and p are positive integers. One day, the mathematician drinks some coffee and discovers that he can now solve 4p +2 problems per hour. In fact, he only works for t-11 hours that day, but he still solves twice as many problems as he would in a normal day. How many problems does he solve the day he drinks coffee?

 May 19, 2024
 #1
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Let's solve this step-by-step to find the number of problems the mathematician solves the day he drinks coffee.

 

Normal Day:

 

Problems solved in a normal day = t (hours) * p (problems/hour) = pt problems

 

Coffee Day:

 

Hours worked with coffee = t - 11 hours

 

Problems solved per hour with coffee = 4p + 2 problems/hour

 

Let x be the number of problems solved on the coffee day. Then, we have x = (t - 11) * (4p + 2)

 

Twice as many problems:

 

The number of problems solved with coffee (x) is twice the number of problems solved on a normal day (pt).

 

This translates to the equation: x = 2 * pt

 

Solving for x:

 

Now we can substitute the expression for x from step 2 into the equation from step 3: (t - 11) * (4p + 2) = 2 * pt

 

Expanding the left side: 4pt + 2t - 22p - 22 = 2pt

 

Combining terms with p and t: 2pt + 2t - 22p - 2pt = -22 2t - 22p = -22 t - 11p = -11 (dividing both sides by 2)

 

Special Condition:

 

We are given that t and p are positive integers. From the equation t - 11p = -11, we can see that t must be a multiple of 11 (because it's equal to -11p plus another integer, -11).

 

Finding possible values of t and p:

 

We can try different integer values for p that are multiples of 7 (since 4p is always even and adding 2 makes it even as well).

 

For example:

 

If p = 7, then t = (11 * 7) - 11 = 70 (valid integer).

 

Other values of p might not result in an integer value for t.

 

Coffee Day Problems:

 

Plugging p = 7 and t = 70 (from our example) into the expression for problems solved with coffee (x = (t - 11) * (4p + 2)): x = (70 - 11) * (4 * 7 + 2) = 59 * 30 = 1770 problems

 

Therefore, on the day the mathematician drinks coffee, he solves 1770 problems (given p = 7 and t = 70).

 May 19, 2024

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