The sum of the digits of a two-digit number is 9 less than the number itself. Which digit must appear in the units place of this two-digit number?

Introduction / Context:
This question is about forming an equation for a two-digit number using its digits and a condition relating the sum of the digits to the number itself. It tests understanding of place value representation and basic algebra. We are asked to determine whether a specific units digit can be identified from the given information or whether the data are insufficient to determine a unique digit.

Given Data / Assumptions:

• Let the two-digit number be 10a + b, where a is the tens digit and b is the units digit.
• The sum of the digits is a + b.
• The sum of the digits is 9 less than the number: a + b = (10a + b) − 9.
• We must decide which digit is in the units place, or whether the data are inadequate.

Concept / Approach:
We represent the number using its tens and units digits, then use the given relation to form an equation. Simplifying the equation will reveal a relationship between a and b. If this relationship determines b uniquely, we can choose a specific digit. If the relationship leaves b free to take multiple values, then the information is insufficient to determine the units digit, and the correct conclusion is that the data are inadequate.

Step-by-Step Solution:
Step 1: Let the two-digit number be 10a + b. Step 2: The sum of its digits is a + b. Step 3: According to the problem, this sum is 9 less than the number itself: a + b = (10a + b) − 9. Step 4: Simplify the right side: (10a + b) − 9 = 10a + b − 9. Step 5: Set up the equation: a + b = 10a + b − 9. Step 6: Subtract a + b from both sides: 0 = 9a − 9. Step 7: This simplifies to 9a − 9 = 0, so 9a = 9. Step 8: Divide by 9 to get a = 1. Step 9: The tens digit is therefore fixed at 1, so the number is 10 + b. Step 10: However, there is no restriction on b from this equation alone. Any b from 0 to 9 gives a number whose sum of digits is 9 less than the number.

Verification / Alternative check:
Take examples. If b = 0, the number is 10. Sum of digits is 1 + 0 = 1, and 10 − 9 = 1, so it works. If b = 5, the number is 15. Sum of digits is 1 + 5 = 6, and 15 − 9 = 6, which also works. Clearly several choices of b satisfy the condition. Therefore, there is no unique digit that must occupy the units place, confirming that the data are inadequate to identify a single units digit.

Why Other Options Are Wrong:
Options (a) 1, (b) 2, and (c) 3 each suggest a specific units digit. However, as shown, the units digit can be 0, 1, 2, 3 and so on, as long as the tens digit is 1. Therefore, none of these specific digits is forced by the conditions, so these options are incorrect.

Common Pitfalls:
A frequent mistake is to assume that because the tens digit is fixed as 1, the units digit must also be determined, or to try to guess a particular b based on a single example. Another error is mishandling the algebraic simplification, leading to expressions that seem to involve b but actually do not. Recognizing that multiple valid numbers satisfy the condition is key to concluding that the data are inadequate.

Final Answer:
No unique units digit can be determined, so the correct conclusion is data inadequate.