Question
I'm studying for my math exam and I stumbled upon a problem that asks for the sum of the smallest and middle numbers of five consecutive integers that add up to 75. Can you find a similar problem on the Questions LLC website and provide the solution and explanation?
Answers
GPT-4o mini
While I can't browse the web to find specific problems on the Questions LLC website, I can create a similar problem based on your description and provide a solution.
**Problem:** Determine the sum of the smallest and middle numbers of five consecutive integers whose sum is 75.
**Solution:**
1. **Let the five consecutive integers be represented**:
Let the smallest integer be \( x \). Therefore, the five consecutive integers can be expressed as:
\[
x, x+1, x+2, x+3, x+4
\]
2. **Set up the equation for their sum**:
The sum of these integers can be expressed as:
\[
x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 75
\]
3. **Simplify the equation**:
Combining like terms, we get:
\[
5x + (1 + 2 + 3 + 4) = 75
\]
The sum of the numbers 1, 2, 3, and 4 is 10, so:
\[
5x + 10 = 75
\]
4. **Solve for \( x \)**:
Subtract 10 from both sides:
\[
5x = 65
\]
Now, divide by 5:
\[
x = 13
\]
5. **Identify the five consecutive integers**:
The five integers are:
\[
13, 14, 15, 16, 17
\]
6. **Determine the smallest and middle numbers**:
- The smallest number is \( 13 \).
- The middle number (third number) is \( 15 \).
7. **Calculate the sum of the smallest and middle numbers**:
\[
13 + 15 = 28
\]
**Conclusion**: The sum of the smallest and middle numbers of the five consecutive integers is \( 28 \).
This problem is similar to your math exam question and should help you understand how to approach finding the sum of specific positions among a sequence of integers.
**Problem:** Determine the sum of the smallest and middle numbers of five consecutive integers whose sum is 75.
**Solution:**
1. **Let the five consecutive integers be represented**:
Let the smallest integer be \( x \). Therefore, the five consecutive integers can be expressed as:
\[
x, x+1, x+2, x+3, x+4
\]
2. **Set up the equation for their sum**:
The sum of these integers can be expressed as:
\[
x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 75
\]
3. **Simplify the equation**:
Combining like terms, we get:
\[
5x + (1 + 2 + 3 + 4) = 75
\]
The sum of the numbers 1, 2, 3, and 4 is 10, so:
\[
5x + 10 = 75
\]
4. **Solve for \( x \)**:
Subtract 10 from both sides:
\[
5x = 65
\]
Now, divide by 5:
\[
x = 13
\]
5. **Identify the five consecutive integers**:
The five integers are:
\[
13, 14, 15, 16, 17
\]
6. **Determine the smallest and middle numbers**:
- The smallest number is \( 13 \).
- The middle number (third number) is \( 15 \).
7. **Calculate the sum of the smallest and middle numbers**:
\[
13 + 15 = 28
\]
**Conclusion**: The sum of the smallest and middle numbers of the five consecutive integers is \( 28 \).
This problem is similar to your math exam question and should help you understand how to approach finding the sum of specific positions among a sequence of integers.