To solve the problem, we need to determine the rate at which Heather and Joel complete problems and then include their initial number of completed problems.
First, let's find the rates for Heather and Joel:
- Heather completes 2 problems in 10 minutes. This is equivalent to:
\[
\text{Rate of Heather} = \frac{2 \text{ problems}}{10 \text{ minutes}} = \frac{2 \text{ problems}}{1/6 \text{ hours}} = 12 \text{ problems per hour}
\]
- Joel completes 3 problems in 15 minutes. This is equivalent to:
\[
\text{Rate of Joel} = \frac{3 \text{ problems}}{15 \text{ minutes}} = \frac{3 \text{ problems}}{1/4 \text{ hours}} = 12 \text{ problems per hour}
\]
Both Heather and Joel complete problems at the same rate of 12 problems per hour.
They also had some problems completed when they started:
- Heather had already completed 3 problems.
- Joel had already completed 2 problems.
The total initial number of completed problems is:
\[
3 \text{ (Heather)} + 2 \text{ (Joel)} = 5 \text{ problems}
\]
The combined rate at which they complete problems is:
\[
12 \text{ (Heather)} + 12 \text{ (Joel)} = 24 \text{ problems per hour}
\]
We want to find the time \( x \) (in hours) such that the total number of problems completed (including the problems completed initially) is between 30 and 50:
\[
30 \leq 5 + 24x \leq 50
\]
So, the correct inequality for \( x \), the number of hours that have passed, is:
\[
30 \leq 24x + 5 \leq 50
\]
The correct choice is:
A. 30 ≤ 24x + 5 ≤ 50
Heather can complete two problems in ten minutes, and when she started she had three problems done. Joel can complete three problems in fifteen minutes, and when he started he had two problems done. If you include the number of problems that were complete when they started, when will they have completed between 30 and 50 problems?
Which inequality would you use to solve this problem, if x stands for the number of hours that have passed?
Responses
A 30 ≤ 24x + 5 ≤ 50
B 30 ≤ 5x + 5 ≤ 50
C 30 < 24x + 5 < 50
D 30 < 10x + 5 < 50
3 answers
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Let's confirm the solution step-by-step.
1. **Determine the rates at which Heather and Joel complete problems:**
- Heather completes 2 problems in 10 minutes:
\[
\text{Rate of Heather} = \frac{2 \text{ problems}}{10 \text{ minutes}} = \frac{2 \text{ problems}}{1/6 \text{ hours}} = 12 \text{ problems per hour}
\]
- Joel completes 3 problems in 15 minutes:
\[
\text{Rate of Joel} = \frac{3 \text{ problems}}{15 \text{ minutes}} = \frac{3 \text{ problems}}{1/4 \text{ hours}} = 12 \text{ problems per hour}
\]
2. **Initial number of completed problems:**
- Heather had already completed 3 problems.
- Joel had already completed 2 problems.
Total initial number of completed problems:
\[
3 \text{ (Heather)} + 2 \text{ (Joel)} = 5 \text{ problems}
\]
3. **Combined problem-solving rate:**
\[
12 \text{ problems/hour (Heather)} + 12 \text{ problems/hour (Joel)} = 24 \text{ problems/hour}
\]
4. **Include initial problems and set up the inequality:**
- Let \( x \) be the number of hours that have passed.
- Total number of problems completed includes initial problems and problems completed over \( x \) hours:
\[
5 + 24x
\]
We want this total to be between 30 and 50:
\[
30 \leq 5 + 24x \leq 50
\]
Therefore, the correct inequality is:
\[
30 \leq 24x + 5 \leq 50
\]
So, the correct choice is indeed:
A. 30 ≤ 24x + 5 ≤ 50
1. **Determine the rates at which Heather and Joel complete problems:**
- Heather completes 2 problems in 10 minutes:
\[
\text{Rate of Heather} = \frac{2 \text{ problems}}{10 \text{ minutes}} = \frac{2 \text{ problems}}{1/6 \text{ hours}} = 12 \text{ problems per hour}
\]
- Joel completes 3 problems in 15 minutes:
\[
\text{Rate of Joel} = \frac{3 \text{ problems}}{15 \text{ minutes}} = \frac{3 \text{ problems}}{1/4 \text{ hours}} = 12 \text{ problems per hour}
\]
2. **Initial number of completed problems:**
- Heather had already completed 3 problems.
- Joel had already completed 2 problems.
Total initial number of completed problems:
\[
3 \text{ (Heather)} + 2 \text{ (Joel)} = 5 \text{ problems}
\]
3. **Combined problem-solving rate:**
\[
12 \text{ problems/hour (Heather)} + 12 \text{ problems/hour (Joel)} = 24 \text{ problems/hour}
\]
4. **Include initial problems and set up the inequality:**
- Let \( x \) be the number of hours that have passed.
- Total number of problems completed includes initial problems and problems completed over \( x \) hours:
\[
5 + 24x
\]
We want this total to be between 30 and 50:
\[
30 \leq 5 + 24x \leq 50
\]
Therefore, the correct inequality is:
\[
30 \leq 24x + 5 \leq 50
\]
So, the correct choice is indeed:
A. 30 ≤ 24x + 5 ≤ 50
1 answer