TV Sales Puzzle: Who Sold What?

Let's dive into this intriguing TV sales puzzle! We've got Ethan, Frank, Gina, and Gabriel, and some intriguing clues about their sales figures this quarter. Our goal is to figure out which statement must be true based on the information provided. This is a classic logic and math problem, perfect for sharpening those analytical skills! We'll break down each clue step-by-step, assign variables, and see if we can deduce the definitive truth.

Unpacking the Clues: A Step-by-Step Breakdown

To solve this, we need to translate the word problem into a series of mathematical equations. Let's assign variables to represent the number of TVs each person sold:

• Let E = the number of TVs Ethan sold.
• Let F = the number of TVs Frank sold.
• Let G = the number of TVs Gina sold.
• Let R = the number of TVs Gabriel sold. (Using 'R' for Gabriel to avoid confusion with 'G' for Gina).

Now, let's convert each statement into an equation or inequality:

1. "Ethan sold exactly eight TVs this quarter."

   • This is straightforward: E = 8.

2. "Frank sold exactly twice as many TVs as Gina."

   • This gives us: F = 2 * G.

3. "Gabriel sold more TVs than Ethan."

   • This means: R > E. Since we know E = 8, we can refine this to R > 8.

4. "Gina sold exactly three more TVs than Gabriel and Ethan combined."

   • "Gabriel and Ethan combined" means R + E.
   • "Three more TVs than" means we add 3 to that sum.
   • So, the equation is: G = (R + E) + 3.

Now, let's combine our known value for E into the equations where it appears:

• From R > 8, we know Gabriel sold at least 9 TVs (since we're dealing with whole TVs).
• From G = (R + E) + 3, we can substitute E=8: G = (R + 8) + 3, which simplifies to G = R + 11.

We now have a system of equations and inequalities:

• E = 8
• F = 2G
• R > 8
• G = R + 11

This system allows us to start making deductions. Since G = R + 11 and R must be greater than 8, G must be greater than 8 + 11, meaning G > 19. This tells us Gina sold at least 20 TVs.

Let's also consider Frank. Since F = 2G and G > 19, Frank must have sold more than 2 * 19, so F > 38. Frank sold at least 39 TVs.

We can also express Gina's sales (G) in terms of Gabriel's sales (R): G = R + 11. This is a crucial relationship. It means Gina always sells 11 more TVs than Gabriel, regardless of the exact number Gabriel sells, as long as Gabriel sells more than 8.

Analyzing the Statements: Which Must Be True?

Now, let's consider the options provided in the original question (A, B, C, etc. - assuming there were multiple-choice options that were omitted). Since the prompt asks "Which statement must be true?", we need to find a statement that is a logical consequence of our derived equations and inequalities, no matter what specific number of TVs Gabriel sold (as long as R > 8).

Let's re-evaluate our key findings:

• E = 8
• R > 8 (Gabriel sold more than 8)
• G = R + 11 (Gina sold 11 more than Gabriel)
• F = 2G (Frank sold double Gina's amount)

From G = R + 11, we can see a direct relationship between Gina's and Gabriel's sales. Because R > 8, we know G must be greater than 8 + 11, so G > 19.

Let's consider some possible scenarios for Gabriel's sales (R), keeping in mind R > 8:

• Scenario 1: If Gabriel sold 9 TVs (R=9).

  • Gina sold G = 9 + 11 = 20 TVs.
  • Ethan sold E = 8 TVs.
  • Frank sold F = 2 * 20 = 40 TVs.
  • Check: R > E (9 > 8) - True. G = (R+E)+3 (20 = (9+8)+3 = 17+3=20) - True.

• Scenario 2: If Gabriel sold 15 TVs (R=15).

  • Gina sold G = 15 + 11 = 26 TVs.
  • Ethan sold E = 8 TVs.
  • Frank sold F = 2 * 26 = 52 TVs.
  • Check: R > E (15 > 8) - True. G = (R+E)+3 (26 = (15+8)+3 = 23+3=26) - True.

• Scenario 3: If Gabriel sold 25 TVs (R=25).

  • Gina sold G = 25 + 11 = 36 TVs.
  • Ethan sold E = 8 TVs.
  • Frank sold F = 2 * 36 = 72 TVs.
  • Check: R > E (25 > 8) - True. G = (R+E)+3 (36 = (25+8)+3 = 33+3=36) - True.

In all these valid scenarios, certain relationships hold:

• Ethan always sold 8 TVs.
• Gabriel always sold more than 8 TVs.
• Gina always sold exactly 11 more TVs than Gabriel (G = R + 11).
• Frank always sold twice as many TVs as Gina.

Let's consider the statement provided in the prompt's discussion category: "A. Gabriel and Gina combined sold". This appears to be an incomplete statement. However, if we assume it's leading into a comparison or a specific number, we can analyze the sum of Gabriel's and Gina's sales.

Gabriel + Gina = R + G Since G = R + 11, we can substitute: R + (R + 11) = 2R + 11.

This means the combined sales of Gabriel and Gina will always be 2R + 11. Since R > 8, the minimum combined sale would be 2*(9) + 11 = 18 + 11 = 29. So, Gabriel and Gina combined sold at least 29 TVs.

Let's re-examine the original question format, which implies specific choices. Suppose the options were:

A. Gabriel and Gina combined sold more than 30 TVs. B. Frank sold exactly 40 TVs. C. Gina sold exactly 20 TVs. D. Gabriel sold exactly 9 TVs.

Let's test these hypothetical options:

A. Gabriel and Gina combined sold more than 30 TVs. (2R + 11 > 30). Since R > 8, the smallest R can be is 9. If R=9, 2(9)+11 = 18+11 = 29. If R=10, 2(10)+11 = 20+11 = 31. So, it's possible they sold more than 30, but not guaranteed if R=9.

B. Frank sold exactly 40 TVs. (F = 40). We know F = 2G. So, 2G = 40, which means G = 20. If G = 20, and G = R + 11, then 20 = R + 11, meaning R = 9. This is a possible scenario, but not the only one. If R=10, G=21, F=42. So, Frank didn't necessarily sell 40.

C. Gina sold exactly 20 TVs. (G = 20). If G = 20, and G = R + 11, then 20 = R + 11, so R = 9. Again, this is possible but not guaranteed. Gabriel could have sold more than 9.

D. Gabriel sold exactly 9 TVs. (R = 9). This is also possible, but Gabriel could have sold 10, 11, or any number greater than 8.

It seems the original prompt may be missing the actual statements to choose from. However, based on the relationship G = R + 11, the statement that Gina sold exactly 11 more TVs than Gabriel must be true.

If the original prompt was structured to have options derived from the variables, let's analyze the strongest relationships:

• Ethan sold 8 TVs. (E=8)
• Gabriel sold > 8 TVs. (R > 8)
• Gina sold R + 11 TVs. (G = R+11)
• Frank sold 2(R+11) TVs.* (F = 2G)

From these, we can definitively say:

1. Gina sold more TVs than Gabriel. (Since G = R + 11, and 11 > 0, G is always greater than R).
2. Gina sold more TVs than Ethan. (Since G = R + 11 and R > 8, G > 8 + 11 = 19. So G is always greater than 8).
3. Gina sold at least 20 TVs. (Because R must be at least 9, G must be at least 9 + 11 = 20).
4. Frank sold more TVs than Gina. (Since F = 2G and G > 0, F is always greater than G).
5. Frank sold at least 40 TVs. (Because G must be at least 20, F must be at least 2 * 20 = 40).

Let's assume the missing part of statement A was something like "...at least 29 TVs."

A. Gabriel and Gina combined sold at least 29 TVs. (R + G = R + (R+11) = 2R + 11. Since R >= 9, 2R + 11 >= 2(9) + 11 = 18 + 11 = 29). This statement must be true.

Without the specific options, we rely on the derived relationships. The core deduction is G = R + 11, which dictates a fixed difference between Gina's and Gabriel's sales. Another key deduction is that all calculated sales figures (except Ethan's fixed 8) depend on Gabriel's sales (R), which can vary as long as R > 8. Therefore, statements that claim specific exact numbers for Gina, Frank, or Gabriel are unlikely to be must be true statements, unless they represent the minimum possible value.

If the question implies that the answer is one of the conditions derived, the statement "Gina sold exactly three more TVs than Gabriel and Ethan combined" is directly given and thus true by definition. However, the phrasing "Which statement must be true?" usually implies a derived conclusion. The derived conclusion that must be true is the relationship G = R + 11.

If we must pick a statement that is provably true from the options provided in the prompt's structure, and considering the common format of these puzzles, it's likely asking for a derived inequality or a minimum value. The statement "Gabriel and Gina combined sold at least 29 TVs" is a strong candidate for a statement that must be true.

Another possibility for a