Solving Math Puzzles: Pages with a Digit Sum of 9

Mathematics can be both a challenge and a joy, especially when solving puzzles that require a bit of logical thinking and mathematical reasoning. One such intriguing problem involves finding two consecutive pages in a math book where the sum of the digits of one page is 9, and their combined page numbers total 53. Let's break down this puzzle step-by-step.

Understanding the Puzzle

When you open a math book, two pages face you. For simplicity, let's call one page Page A and the other corresponding page Page B. In many book formats, the page numbers are typically consecutive, differing by just one, with Page B being the next page number after Page A.

Our goal is to find the page numbers of these two facing pages, given the following clues:

The sum of the digits of Page A is 9. The sum of the page numbers of Page A and Page B is 53.

Let's denote the page numbers of Page A and Page B as 10a b and 10a b 1, where a and b are digits.

Setting Up the Equations

First, we need to express the digits sum condition, which is that the sum of the digits of Page A is 9. This can be written as:

a b 9

Next, we know that the combined sum of the page numbers of Page A and Page B is 53:

(10a b) (10a b 1) 53

Simplifying the equation, we get:

20a 2b 1 53

Subtracting 1 from both sides, we have:

20a 2b 52

Dividing the entire equation by 2:

10a b 26

Solving the Equations

Now we have two equations:

a b 9 10a b 26

Subtract the first equation from the second:

(10a b) - (a b) 26 - 9

This simplifies to:

9a 17

Solving for a:

a 3

Substituting a 3 back into the first equation:

3 b 9

Solving for b:

b 6

Therefore, the page numbers are:

Page A 10a b 10 * 3 6 36 Page B Page A 1 36 1 37

However, these page numbers do not satisfy the condition that the sum of the digits of Page A is 9. Therefore, let's try another approach by checking the next consecutive pair of numbers that might fit the criteria.

Checking Other Possibilities

Let's try the next consecutive pair:

Page A 26

Comparing the digits of 26: 2 6 8 (not 9; let's check the next pair)

Page A 27

Comparing the digits of 27: 2 7 9 (yes, this fits the digit sum condition)

Page B is then:

Page B 27 1 28

Checking the sum of Page A and Page B:

26 27 53

This fits the total sum condition.

Conclusion

The two pages that meet the given conditions are Page 26 and Page 27.

By breaking down the problem, setting up the equations, and solving step-by-step, we can find the solution to the intriguing math puzzle. If you have any more math puzzles or need help with math problems, feel free to reach out!

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