A rectangular auditorium seats 1862 people. The number of seats in each row exceeds the number of rows by 11. Find the number of seats in each row.
3 answers
Let the number of rows be $r$. Since the number of seats in each row exceeds the number of rows by 11, there must be $r+11$ seats in each row. Thus, we have $r(r+11)=1862$. Factoring 1862, we quickly find the prime factorization 1862=$2 \cdot 19 \cdot 49$. $49=7^2$, so $r=7$. Our answer is $r+11=\boxed{18}$.
A rectangular auditorium seats 1862 people. The number of seats in each row exceeds the number of rows by 11. Find the number of seats in each row.
Let $r$ be the number of rows and $s$ be the number of seats in each row. We know that $s = r + 11$ and that $rs = 1862$. Substituting the first equation into the second equation, we have $(r+11)r = 1862$. Expanding the left side, we get $r^2 + 11r = 1862$. Rearranging this equation, we have $r^2 + 11r - 1862 = 0$. We can factor this into $(r + 34)(r - 23) = 0$. This gives us two possible values for $r$: $r = -34$ and $r = 23$. Since the number of rows cannot be negative, we have $r = 23$. Plugging this into the first equation, we have $s = 23 + 11 = 34$. Therefore, there are $\boxed{34}$ seats in each row.