In a two digit number, the digit in the tens place is 8 and the digit in the units place is z. What is the correct algebraic expression for this two digit number?

Introduction / Context:
This question checks your understanding of how to represent a two digit number using place value and algebraic notation. Many aptitude and algebra problems use a variable for one digit and ask you to express the entire number. Understanding that the tens place contributes a multiple of ten and the units place contributes a single unit is the key concept here.

Given Data / Assumptions:

• The number is a two digit number.
• The tens digit is 8.
• The units digit is represented by the variable z.
• We are asked to write the numerical value of the number as an algebraic expression in terms of z.

Concept / Approach:
Any two digit number can be written as 10 multiplied by the tens digit plus the units digit. For example, 37 is 3 * 10 + 7. Here, the tens digit is fixed at 8, and the units digit is the variable z. So the number must be of the form 10 * 8 plus z. We simply need to express that clearly and avoid confusing digit multiplication with place value.

Step-by-Step Solution:
Step 1: Recognize that a general two digit number with tens digit a and units digit b can be written as 10 * a + b. Step 2: In this problem, the tens digit is 8, so a = 8. Step 3: The units digit is represented by z, so b = z. Step 4: Substitute these into the generic form to obtain 10 * 8 + z. Step 5: Compute 10 * 8 = 80, so the number is 80 + z. Step 6: Hence, the correct algebraic expression for the number is 80 + z.

Verification / Alternative check:
To verify, choose a simple value for z, for example z = 3. Then the actual number should be 83, since the tens digit is 8 and the units digit is 3. If we use the expression 80 + z, we get 80 + 3 = 83, which matches. If we try an incorrect expression such as 8z + 8 with z = 3, we get 8 * 3 + 8 = 32, which does not have tens digit 8 and units digit 3, so it is wrong. This quick substitute check helps confirm that 80 + z is the only correct representation.

Why Other Options Are Wrong:
The option 8z + 8 treats the tens digit as a coefficient multiplied by z, which misrepresents place value. The expression 80z + z factors the number as a multiple of z, which makes the tens digit depend on z as well, which is not true. The expression 80z + 8 is far too large for reasonable values of z and again confuses digit position with multiplication. The option 8 + z would only represent a one or two digit number formed by adding the digits, not arranging them in tens and units places. None of these incorrect forms match the place value rule.

Common Pitfalls:
Students sometimes mix up the idea of multiplying by ten with the use of variables and write forms like 8z or z8 instead of 10 * 8 + z. Others may try to directly concatenate symbols and think of 8z as a two digit number rather than a product. Remember that algebraic expressions follow arithmetic rules and digits in decimal notation correspond to powers of ten, not to coefficients on a variable unless clearly defined that way. Carefully separating place value from variable notation avoids these misunderstandings.

Final Answer:
The correct expression for the two digit number is 80 + z.