The product of the page numbers on two facing pages of a book is 132. Find the page numbers.
Select the correct choice below and fill in the answer box(es) to complete your choice.
A. There is only one solution. The consecutive page numbers are ____.
B. There are two solutions. The smaller consecutive page numbers are ____, and the larger consecutive page numbers are ____.
Answer (2)
The consecutive page numbers in the book whose product is 132 are 11 and 12. There is only one solution to this problem. Hence, the answer is option A.
;
Let n be the first page number, so the next page is n + 1 .
Write the equation n ( n + 1 ) = 132 .
Solve the quadratic equation n 2 + n − 132 = 0 , which factors to ( n + 12 ) ( n − 11 ) = 0 .
The positive solution is n = 11 , so the page numbers are 11 and 12. The final answer is 11 , 12 .
Explanation
Understanding the Problem We are given that the product of two facing page numbers in a book is 132. Our goal is to find these page numbers. Since facing pages are consecutive, we can set up an equation to represent this situation. Let's call the first page number n . Then the next page number will be n + 1 .
Setting up the Equation We know that the product of these two numbers is 132, so we can write the equation: n ( n + 1 ) = 132
Expanding the Equation Expanding the left side of the equation, we get: n 2 + n = 132
Rearranging to Quadratic Form To solve for n , we need to rearrange the equation into a standard quadratic form by subtracting 132 from both sides: n 2 + n − 132 = 0
Factoring the Quadratic Now we need to factor the quadratic equation. We are looking for two numbers that multiply to -132 and add to 1. These numbers are 12 and -11. So we can factor the equation as: ( n + 12 ) ( n − 11 ) = 0
Solving for n Setting each factor equal to zero gives us two possible solutions for n :
n + 12 = 0 ⇒ n = − 12 n − 11 = 0 ⇒ n = 11
Choosing the Correct Solution Since page numbers must be positive, we discard the negative solution n = − 12 . Therefore, the first page number is n = 11 . The next page number is n + 1 = 11 + 1 = 12 .
Checking the Solution The two consecutive page numbers are 11 and 12. We can check our answer by multiplying these numbers: 11 × 12 = 132 , which confirms our solution.
Final Answer Therefore, the two consecutive page numbers are 11 and 12.
Examples
This type of problem can be used in real life when trying to determine dimensions of rectangular objects. For example, if you know the area of a rectangular garden is 132 square feet and the length is one foot longer than the width, you can use a similar approach to find the dimensions of the garden. This involves setting up a quadratic equation and solving for the unknown variable, which in this case would be the width of the garden. Understanding how to solve these types of problems can help with various practical applications involving areas and dimensions.
