Question

How many integers from 3 to 30, inclusive, are odd?

• A. 13
• B. 14
• C. 15
• D. 16
• E. 17

Hint:

For counting problems with inclusive ranges, the formula \(\text{Last} - \text{First} + 1\) gives the total number of integers. In the range 3 to 30, there are \(30-3+1 = 28\) integers. Since the range starts and ends with an odd and an even number, exactly half of them will be odd and half will be even. So, the number of odd integers is \(28 / 2 = 14\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a counting problem. We need to find the number of odd integers within a given inclusive range.
Step 2: Key Formula or Approach:
There are two main ways to solve this: by listing and counting, or by using a formula for arithmetic progressions.
The formula for the number of terms in a sequence is:
\[ \text{Number of terms} = \left(\frac{\text{Last term} - \text{First term}}{\text{Common difference}}\right) + 1 \] Step 3: Detailed Explanation:
Method 1: Listing and Counting
The range is from 3 to 30, inclusive. The odd integers in this range are:
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.
Counting these numbers, we find there are 14 odd integers.
Method 2: Using the Formula
This is an arithmetic sequence of odd numbers.
- The first term is 3.
- The last term is 29.
- The common difference between consecutive odd integers is 2.
Using the formula:
\[ \text{Number of terms} = \left(\frac{29 - 3}{2}\right) + 1 \] \[ \text{Number of terms} = \left(\frac{26}{2}\right) + 1 \] \[ \text{Number of terms} = 13 + 1 = 14 \] Step 4: Final Answer:
There are 14 odd integers between 3 and 30, inclusive.