Math Competition Problems - Problem of the Week 6 - MOEMS Counting Problem

How many 2-digit odd numbers are greater than 30?

Source: MOEMS (Math Olympiad) Grades 4-6

MOEMS Problem of the Week - Counting Problem

Counting Problems like this appear frequently in MOEMS, Noetic, CML and other math competitions. Instead of performing lengthy calculations, students are expected to recognize patterns and count efficiently.

Math Competition Problems - Problem of the Week 6

This can be easily solved if we make an ordered list. And yes, this is one of the easier problems in MOEMS (Math Olympiad for Elementary and Middle Schools)

So let’s start from the 30’s and make a list

31, 33, 35, 37, 39

41, 43, 45, 47, 49

51, …..

61, …..

71, …..

81, …..

91, 93, 95, 97, 99.

99 is the last 2-digit number and the last 2-digit odd number

So each row has exactly 5 numbers, and there are 7 rows.

Hence, there are 35 2-digit odd numbers greater than 30.

Note: I still wrote the whole of the last row to make sure that there are 5 numbers even in the last row (and nothing is missing). Always make sure that you list out the first and the last row to ensure that it follows the same pattern.

Solution 2 - Multiplication Principle

The tens digit has 7 options (3-9).

For each choice of the tens digit, the ones digit has 5 options (1, 3, 5, 7 or 9).

Hence total number of options = 7*5=35.

Common Mistake: Some students incorrectly assume that the tens digit must be greater than 3, so they count only 4, 5, 6, 7, 8, and 9 as possibilities. This gives only 6 choices for the tens digit.

However, numbers such as 31, 33, 35, 37, and 39 are all greater than 30, so the tens digit can also be 3. Therefore, the tens digit has 7 choices: 3, 4, 5, 6, 7, 8, and 9.

When to use the Multiplication Principle?

The multiplication principle is often faster than making a list, especially when the numbers become larger.

Try it with the challenge problem

Now try a similar problem on your own:

How many 2-digit even numbers are greater than 30?

Want a challenge?

How many 3-digit odd numbers are greater than 500?

Post the answer to the problem in the comments section below.

About the Author

Vasudha is the founder of V's Online Math Tutoring. She specializes in Beast Academy, AoPS, math competitions, and enrichment programs, helping students become confident and independent problem solvers.

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