for every 20 math books a school buys, they they get 7 aditional books for free. The school needs 297 math books. What is the total number of math books the school must buy in order to receive 297 math books in all?
297/27 = 11
11 * 20 = ?
To find the total number of math books the school must buy in order to receive 297 math books in all, we can set up a proportion based on the given scenario.
Let's define the following variables:
B = Number of math books that the school needs to buy
F = Number of additional free math books
According to the information given, for every 20 math books the school buys, they receive 7 additional books for free. This means the ratio of bought books to free books is 20:7, or simplified, it's 20/7.
We can set up the proportion:
B/20 = 297/(20 + F)
We know that the school needs a total of 297 math books in all, so B + F = 297.
We can substitute the value of F in terms of B into the proportion:
B/20 = 297/(20 + B - 297)
B/20 = 297/(B - 277)
Cross-multiplying the proportion gives:
B(B - 277) = 20 * 297
Expanding and simplifying the equation:
B^2 - 277B = 5940
Rearranging the equation to isolate B:
B^2 - 277B - 5940 = 0
Now, we can solve this quadratic equation to find the value of B. The equation can be factored as:
(B - 297)(B + 20) = 0
Setting each factor equal to zero gives:
B - 297 = 0 or B + 20 = 0
Solving each equation gives:
B = 297 or B = -20
Since the number of books cannot be negative, the solution is B = 297.
Therefore, the school must buy a total of 297 math books in order to receive 297 math books in all.