Question
there are 8 studets on the school chess team. they need to raise at least $760 for field trip. their advisor says they still need to raise 85% of the funds, and she wants each of them to raise an equal share of the money still needed. diego wrote and solved the inequality 8m+0.85(760) ≥ 760 to fid the least amont of money,m,each student on the chess team still needs to raise. he says the solution is m ≥ 14.25 describe and correct diego error
Answers
GPT-4o mini
Let's break down Diego's approach to solving the problem.
1. **Calculate the Total Funds Needed**: The total amount needed for the trip is $760.
2. **Calculate the Amount Already Raised**: The advisor says they still need to raise 85% of the funds, which means they have already raised 15% of the funds.
\[
\text{Amount raised} = 0.15 \times 760 = 114
\]
3. **Calculate the Amount Still Needed**: Therefore, the amount still needed is:
\[
\text{Amount still needed} = 760 - 114 = 646
\]
4. **Equal Share for Each Student**: Since there are 8 students raising this amount equally, we find out how much each student needs to raise by dividing the total amount still needed by the number of students:
\[
m = \frac{646}{8} = 80.75
\]
Now, let’s examine Diego's inequality:
- He wrote the inequality \(8m + 0.85(760) \geq 760\).
- The term \(0.85(760)\) calculates the amount they still need (the correctly needed funds), but as we’ve demonstrated, it should lead to calculating the funds they still need to raise, which is misinterpreted.
5. **Diego's Error**: The core of Diego’s mistake lies in misunderstanding what needs to be calculated. He included \(0.85(760)\) as part of the funds that are already raised instead of calculating it as part of the amount they still need. The correct interpretation is that they need the total of $760 minus the amount they have raised (15% of 760).
6. **Correcting the Inequality**: The correct way to frame this situation would be:
\[
8m = 646
\]
Which leads to:
\[
m = \frac{646}{8} = 80.75
\]
Therefore, the actual amount each student needs to raise is $80.75, not $14.25.
### Conclusion
Diego’s solution was incorrect because he miscalculated the percentage of funds already raised and mistakenly set up the inequality. The correct amount each student needs to raise is $80.75.
1. **Calculate the Total Funds Needed**: The total amount needed for the trip is $760.
2. **Calculate the Amount Already Raised**: The advisor says they still need to raise 85% of the funds, which means they have already raised 15% of the funds.
\[
\text{Amount raised} = 0.15 \times 760 = 114
\]
3. **Calculate the Amount Still Needed**: Therefore, the amount still needed is:
\[
\text{Amount still needed} = 760 - 114 = 646
\]
4. **Equal Share for Each Student**: Since there are 8 students raising this amount equally, we find out how much each student needs to raise by dividing the total amount still needed by the number of students:
\[
m = \frac{646}{8} = 80.75
\]
Now, let’s examine Diego's inequality:
- He wrote the inequality \(8m + 0.85(760) \geq 760\).
- The term \(0.85(760)\) calculates the amount they still need (the correctly needed funds), but as we’ve demonstrated, it should lead to calculating the funds they still need to raise, which is misinterpreted.
5. **Diego's Error**: The core of Diego’s mistake lies in misunderstanding what needs to be calculated. He included \(0.85(760)\) as part of the funds that are already raised instead of calculating it as part of the amount they still need. The correct interpretation is that they need the total of $760 minus the amount they have raised (15% of 760).
6. **Correcting the Inequality**: The correct way to frame this situation would be:
\[
8m = 646
\]
Which leads to:
\[
m = \frac{646}{8} = 80.75
\]
Therefore, the actual amount each student needs to raise is $80.75, not $14.25.
### Conclusion
Diego’s solution was incorrect because he miscalculated the percentage of funds already raised and mistakenly set up the inequality. The correct amount each student needs to raise is $80.75.