Question

Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2,$ the second sheet contains pages $3$ and $4,$ and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19.$ How many sheets were borrowed?

$10$

$13$

$15$

$17$

$20$

Answer: B

Concepts:mean Diophantine Equation factoring

Difficulty rating: 1820

Video solution:

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Written solution:

Suppose the borrowed sheets are sheets $a$ through $b,$ and let $s=b-a+1.$ The borrowed pages run from $2a-1$ through $2b,$ so there are $2s$ borrowed pages and their sum is $s(2a+2b-1).$

The total sum of all page numbers is $50\cdot51/2=1275.$ If the remaining pages have mean $19,$ then $\begin{aligned} 1275&-s(2a+2b-1)\\ &=19(50-2s),\\ s(2a+2b-39)&=325. \end{aligned}$

Because $b-a+1$ is a positive divisor of $325$ and is at most $25,$ its only possibilities are $1,5,13,25.$ The first two would force $b>25,$ and $25$ would remove every sheet. Thus the only valid possibility is

$\begin{aligned} 2a+2b-39 &=25, \\ b-a+1 &=13. \end{aligned}$

Thus $a+b=32$ and $b-a=12,$ so $a=10$ and $b=22.$ Therefore $13$ sheets were borrowed.

Thus, B is the correct answer.

2021 AMC 10A Spring Problem 22

Answer: B

Difficulty rating: 1820

Video solution:

Written solution:

Problem 22 in Other Years