Carmen has leamed an algorithm for subtraction. She uses it to find $$ 54-16 $$ as follows. $$ \begin{array}{l}54+10=64 \ -(16+10)=-26\end{array} $$ so $$ 54+10=50+14 $$ So her answer is $$ 30+8=38 $$. (a) What properties does Carmen's algorithm use? Is her method correct? (b) Now solve another subtraction problem, $$ 72-38 $$, with Carmen's algorithm. Use the traditional algorithm to check your work. (a) Which properties of addition does Carmen's algorithm use? Select all that apply. $$ \square $$ A. Transitivity property $$ \square $$ B. Associative property $$ \square $$ C. Closure property $$ \square $$ D. None of the above

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Let's break down the problem step by step.

### Part (a)

1. **Carmen's Algorithm Analysis:**
   - Carmen starts with $$ 54 - 16 $$.
   - She rewrites it as $$ 54 + 10 - (16 + 10) $$.
   - This can be interpreted as adjusting both numbers by adding 10, which keeps the difference the same.

2. **Properties Used:**
   - **Associative Property:** This property allows the grouping of numbers to be changed without affecting the sum. Carmen groups the numbers differently.
   - **Closure Property:** This property states that the sum of any two real numbers is also a real number. Carmen is still working within the realm of real numbers.
   - **Transitivity Property:** This property is not directly applicable here since it relates to equality rather than addition.

3. **Correctness of Method:**
   - Carmen's method is incorrect because her final calculation $$ 30 + 8 = 38 $$ does not yield the correct result for $$ 54 - 16 $$. The correct answer is $$ 38 $$.

### Answer for Part (a):
- The properties of addition that Carmen's algorithm uses are:
  - $$ \square $$ A. Transitivity property
  - $$ \square $$ B. Associative property
  - $$ \square $$ C. Closure property
  - $$ \checkmark $$ D. None of the above (since the transitivity property is not applicable).

### Part (b)

Now, let's solve the subtraction problem $$ 72 - 38 $$ using Carmen's algorithm.

1. **Using Carmen's Algorithm:**
   - Start with $$ 72 - 38 $$.
   - Add 10 to both numbers:
     $$
     72 + 10 - (38 + 10) = 82 - 48
     $$
   - Now, calculate $$ 82 - 48 $$:
     - Break it down further:
     $$
     82 - 48 = (80 - 40) + (2 - 8) = 40 + (-6) = 34
     $$

2. **Traditional Algorithm Check:**
   - Now, let's check using the traditional method:
   $$
   72 - 38 = 34
   $$

### Conclusion for Part (b):
- Using Carmen's algorithm, we found $$ 72 - 38 = 34 $$, which matches the traditional method.

Thus, the final answers are:
- (a) Properties used: $$ \checkmark $$ D. None of the above.
- (b) The result of $$ 72 - 38 $$ using Carmen's algorithm is $$ 34 $$, which is confirmed by the traditional method.
(a) Carmen's algorithm uses the associative and closure properties of addition. Her method is incorrect because it doesn't yield the right answer for $$54 - 16$$. (b) Using Carmen's algorithm, $$72 - 38 = 34$$, which matches the traditional method. So, the answers are: - (a) Properties used: Associative and Closure properties. - (b) $$72 - 38 = 34$$.

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