When a book is opened and the page numbers multiplied the product is 2,970. What was the numbers of the pages?

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samhouston's profile pic

samhouston | Middle School Teacher | (Level 1) Associate Educator

Posted on

The best way to solve this problem is by using a system of equations.

Since the pages are consecutive, one of the equations would be:

x + 1 = y, where x is the page on the left and y is the page on the right.

Since the product of the page numbers equals 2970, the other equation would be x * y = 2970.

So here is your system:

  • x + 1 = y
  • x * y = 2970

Solve the system by substituting (x + 1) in for y in the second equation.

x * (x + 1) = 2970

x^2 + x = 2970

x^2 + x - 2970 = 0

(x + 54)(x - 54) = 0

Since the page number cannot be negative, x = 54.  This means that y = 55.  The book is opened to pages 54 and 55.

rugbykats's profile pic

rugbykats | High School Teacher | (Level 2) Adjunct Educator

Posted on

If you are looking at facing pages, the numbers will be consecutive. A simple way to solve the problem is

n (the number of the first page) x (n+1) (the number of the second page) = 2,970

n^2 + n = 2,970

n^2 + n - 2,970 = 0

Factoring: (n + 55) (n - 54) = 0 (Tip: Find the square root of 2,970, and try the numbers just above and below that number.)

That gives -55 and +54 as your roots. You cannot have a negative page number, so throw out -55, and check +54.

54 x (54 + 1) = 2,970

54 X 55 = 2,970, which is correct. So your answers are pages 54 and 55.

Hope this helps, and good luck!

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

When you open a book and multiply the two page numbers you see, the page numbers are consecutive integers.

The numbers can be denoted by n and n+1. The product of the two numbers is 2970.

=> n(n+1) = 2970

=> n^2 + n = 2970

=> n^2 + n - 2970 = 0

=> n^2 + 55n - 54n - 2970 = 0

=> n(n + 55) - 54(n + 55) = 0

=> (n - 54)(n + 55) = 0

=> n = 54 and n = -55

We can ignore the negative root.

The page numbers are 54 and 55

shaznl1's profile pic

shaznl1 | High School Teacher | (Level 1) Salutatorian

Posted on

well if you open a book, you will find that the two pages in front of you will be consecutive

Given that the two pages are consecutive

let the first page be page X

then the second page is page x+1

By definition of multiplication

X(X+1)=2970

X^2+X-2970=0

well this needs some factoring

first we see there is a 5 in a 2970 (obvious)

2970/5=594

i see a two in the number

594/2=297

2+9+7= 18 thats a multiple of 3

297/3=99 which is still a multiple of 3

99/3= 33 which is still a multiple of 3

33/3=11 Youa re left with a prime number

so it is 5,2,3,3,3,11

you need to form two numbers that the difference is one

by calculation, we find out

5*11 = 55

2*3*3*3= 54

so the factoring is (X+55)(X-54)=0

I dont know a book that has negative page numbers

X= 54

so the two pages are 54 and 55

 

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