Reading a book

Algebra Level pending

Boris is reading a book. On the first day, he reads $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day, he reads $n+1$ pages in $t+1$ minutes. On each day thereafter, he reads one more page in one more minute, until he has completely read the 374 page book. It took him 319 minutes to do so. What is $n+t$ ?

The answer is 53.

1 solution

Brian Moehring
Mar 8, 2017

Suppose Boris reads the book during $d$ days. Then the given information becomes $dn + \frac{d(d-1)}{2} = \sum_{k=0}^{d-1} (n+k) = 374 \\ dt + \frac{d(d-1)}{2} = \sum_{k=0}^{d-1} (t+k) = 319$

By subtracting these equations, we find $d(n-t) = 374-319 = 55 \,\,\,\,\, \Rightarrow \,\,\,\,\, d \mbox{ divides } 55.$ In particular, this shows us that $d-1$ is even, so we can re-write the first equation above as $d\left(n + \frac{d-1}{2}\right) = 374 \,\,\,\,\, \Rightarrow \,\,\,\,\, d \mbox{ divides } 374.$ Combining these two, we find that $d$ must divide $\gcd(55,374) = 11$ , and since the problem references a "second day", we must have $d > 1$ , so $d=11$ .

Putting this value into the original pair of equations yields $11n + 55 = 374 \\ 11t + 55 = 319$ which we may solve to find $n=29, t=24$ , and $n+t = \boxed{53}$ .

Reading a book

The answer is 53.

1 solution

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