https://file.aiquickdraw.com/mathbot/file/1730821472934q2g9j5ms.jpg

Let's go through each part of the question step-by-step:

### Given Data:
1. **All Real Bar (60g):** 
   - Protein: 20g
   - Sugar: 12g
   - Calories: 240

2. **NutraGo Bar (40g):**
   - Protein: 10g
   - Sugar: 16g
   - Calories: 200

### Part (a) Absolute Percentage Difference in Protein
To find the absolute percentage difference, use the formula:

\[
\text{Absolute Percentage Difference} = \left|\frac{\text{All Real Protein} - \text{NutraGo Protein}}{\text{Average of Protein Values}}\right| \times 100
\]

1. All Real Protein: 20g
2. NutraGo Protein: 10g
3. Average Protein = \(\frac{20 + 10}{2} = 15\)

\[
\text{Absolute Percentage Difference} = \left|\frac{20 - 10}{15}\right| \times 100 = \frac{10}{15} \times 100 = 66.67\%
\]

### Part (b) Absolute Percentage Difference in Sugar
1. All Real Sugar: 12g
2. NutraGo Sugar: 16g
3. Average Sugar = \(\frac{12 + 16}{2} = 14\)

\[
\text{Absolute Percentage Difference} = \left|\frac{12 - 16}{14}\right| \times 100 = \frac{4}{14} \times 100 \approx 28.57\%
\]

### Part (c) Relative Percentage Difference in Protein (NutraGo to All Real)
The relative percentage difference is calculated by comparing NutraGo to All Real, using the formula:

\[
\text{Relative Percentage Difference} = \left(\frac{\text{All Real Protein} - \text{NutraGo Protein}}{\text{NutraGo Protein}}\right) \times 100
\]

\[
= \left(\frac{20 - 10}{10}\right) \times 100 = 1 \times 100 = 100\%
\]

### Part (d) Relative Percentage Difference in Sugar (NutraGo to All Real)
\[
\text{Relative Percentage Difference} = \left(\frac{\text{All Real Sugar} - \text{NutraGo Sugar}}{\text{NutraGo Sugar}}\right) \times 100
\]

\[
= \left(\frac{12 - 16}{16}\right) \times 100 = -0.25 \times 100 = -25\%
\]

### Part (e) Ratio of Calories per Gram for All Real : NutraGo
1. **All Real Calories per Gram** = \(\frac{240}{60} = 4\) calories per gram
2. **NutraGo Calories per Gram** = \(\frac{200}{40} = 5\) calories per gram

**Ratio (All Real : NutraGo)** = \(4 : 5\)

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### Summary of Answers
1. **Absolute Percentage Difference (Protein):** 66.67%
2. **Absolute Percentage Difference (Sugar):** 28.57%
3. **Relative Percentage Difference (Protein):** 100%
4. **Relative Percentage Difference (Sugar):** -25%
5. **Ratio of Calories per Gram (All Real : NutraGo):** 4:5

Would you like further explanations on any part of these calculations, or have any other questions?

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Here are five related questions for further exploration:

1. How would the differences change if the NutraGo bar size was adjusted to match the All Real bar size?
2. What would the calorie ratio be if the calorie content of both bars were proportionally increased?
3. Can you calculate the absolute and relative percentage differences if we consider fat content (if data was available)?
4. How does the sugar-to-calorie ratio compare between the two bars?
5. What is the protein density per gram for each bar, and how does it compare?

**Tip:** When calculating percentage differences, be mindful of whether you are asked for an absolute or relative difference, as the formulas vary slightly but can lead to different interpretations.

Comparing NutraGo to All Real, determine the absolute percentage difference between the two brands for: a) Protein b) Sugar. Comparing NutraGo to All Real, determine relative percentage difference of: c) Protein d) Sugar. Write a ratio denoting Calories per gram for: e) All Real : NutraGo

Learn how to find the absolute and relative percentage differences in protein and sugar content between NutraGo and All Real protein bars. This problem also includes calculating the ratio of calories per gram for both bars, using basic percentage and ratio formulas. Suitable for students in Grades 8-10.

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