Question 80·Medium·Linear Equations in One Variable
A library added 1,200 identical copies of a popular book to its collection. Each month afterward, an average of 35 of these copies are damaged and removed from the collection, and no additional copies are added. After how many months will exactly 815 copies of the book remain in the collection?
For word problems about a quantity that starts at some value and increases or decreases at a constant rate, define a variable for time, write a linear expression like starting amount ± (rate × time), and set it equal to the target amount given in the question. Then solve the resulting one-variable linear equation carefully, paying attention to negative signs and checking each answer option by plugging back into the original context if you have time.
Hints
Identify the starting amount and the change per month
What number represents the initial number of copies in the collection? How many copies are removed each month?
Write an expression for remaining copies
After months, how can you express the number of remaining copies using the starting amount 1,200 and the monthly decrease of 35?
Set up the equation to match the target amount
Set your expression for the remaining copies equal to 815, then solve that linear equation for .
Carefully solve the linear equation
After you move the constants to one side, you should get a negative coefficient on . Be careful with the negative signs when you divide.
Desmos Guide
Enter the equation representing the remaining books
In Desmos, type the equation 1200 - 35x = 815. Desmos will automatically graph the line and the horizontal line representing 815.
Find the solution from the graph
Look for the point where the graph of intersects the line , or use the Desmos solver to find the value of that makes the equation true. The -value of that intersection is the number of months.
Step-by-step Explanation
Define the variable and write an expression
Let represent the number of months after the library added the books.
Each month, 35 copies are removed. Starting from 1,200 copies, the number of copies left after months is:
We are told this should equal 815 when we reach the desired time.
Set up the equation
Set the expression for the remaining books equal to 815:
Now we will solve this equation for .
Isolate the term with the variable
First, subtract 1200 from both sides (or think of moving 1200 to the right):
which simplifies to
Solve for the number of months
Now divide both sides of the equation by to solve for :
Compute this quotient:
So, it will take 11 months for exactly 815 copies of the book to remain in the collection.