AMC 10A 2021 Problem 22 | System of Equations

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

Problem-Solving Strategies by Arthur Engel

AMC 10A 2021 Problem 22

13

Let us assume that the roommate took sheets $a$ through $b$.
So, try to think what will be the changes in the page number?

So, page numbers $2 a-1$ through $2 b$. Because there are $(2 b-2 a+2)$ numbers.

Now apply the condition given there.

So we get, $\frac{(2 a-1+2 b)(2 b-2 a+2)}{2}$+$19(50-(2 b-2 a+2))$=$\frac{50 \cdot 51}{2}$

Now simplify this expression.

So , $2 a+2 b-39=25, b-a+1=13$

Now solve for $a, b$.

Find the number of pages using the values.

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