The numbers x, y, and z are proportional to 2, 3, and 5 respectively, and x + y + z = 80. If z is given by the relation z = a x - 8, what is the value of a?

Introduction / Context:
This algebra question involves proportional relationships and a linear equation relating two variables. You are told that three numbers x, y, and z are in proportion 2:3:5 and that their sum is 80. Additionally, z can be expressed in terms of x using the linear expression z = a x - 8. The task is to determine the value of the coefficient a. This type of question is common in problems on ratios, proportions, and linear equations.

Given Data / Assumptions:
- x : y : z = 2 : 3 : 5.
- x + y + z = 80.
- z can be written as z = a x - 8 for some constant a.
- We must find the value of a that satisfies all these conditions.

Concept / Approach:
First, we use the proportional relationship to express x, y, and z in terms of a single scaling factor k. This will allow us to use the given sum to find specific numeric values for x, y, and z. Once we know x and z explicitly, we substitute them into z = a x - 8 and solve for a. This two step process moves from general ratios to concrete numbers and then to the linear relation between x and z.

Step-by-Step Solution:
Step 1: Since x, y, and z are proportional to 2, 3, and 5, let x = 2k, y = 3k, and z = 5k for some positive real number k.Step 2: Use the sum condition: x + y + z = 80.Step 3: Substitute the expressions in terms of k: 2k + 3k + 5k = 80.Step 4: Simplify the left side: 2k + 3k + 5k = 10k.Step 5: So 10k = 80, and dividing both sides by 10 gives k = 8.Step 6: Now compute the actual values of x, y, and z. We have x = 2k = 2 * 8 = 16.Step 7: y = 3k = 3 * 8 = 24, and z = 5k = 5 * 8 = 40.Step 8: Check that these values sum to 80: 16 + 24 + 40 = 80, which is correct.Step 9: Use the relation z = a x - 8. Substitute x = 16 and z = 40 into this equation.Step 10: This gives 40 = a * 16 - 8.Step 11: Add 8 to both sides: 40 + 8 = 16a, so 48 = 16a.Step 12: Divide both sides by 16: a = 48 / 16 = 3.

Verification / Alternative check:
Once we have a = 3, we can check the relation z = a x - 8 directly. With x = 16, a x - 8 = 3 * 16 - 8 = 48 - 8 = 40, which matches z = 40 that we obtained from the proportional relationship. The sum x + y + z remains 16 + 24 + 40 = 80, and the ratio 16 : 24 : 40 simplifies by dividing by 8 to 2 : 3 : 5. This consistency check confirms that our value of a is correct.

Why Other Options Are Wrong:
If a were 6, then z would be 6 * 16 - 8 = 96 - 8 = 88, which does not match z = 40 and would violate the sum condition. If a were 3/2, then z would be (3/2) * 16 - 8 = 24 - 8 = 16, again not equal to 40. Similar contradictions arise for a = 5/2 or a = 4. These values fail to satisfy both the ratio and the linear equation for the same x and z.

Common Pitfalls:
A common mistake is to misapply the ratio and write x = 2, y = 3, and z = 5 directly without using a scaling factor, which conflicts with the sum of 80. Another error is to forget to apply the sum condition and attempt to solve for a using only the ratio relationship. Always introduce a common factor for ratios, use the sum to find the exact values, and then apply any additional relations among the variables.

Final Answer:
The value of a is 3.