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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

Ask by Brooks Frazier. in the United States
Mar 16,2025

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(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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Beyond the Answer

Carmen’s algorithm cleverly utilizes the properties of addition, specifically the associative property. By rearranging numbers and grouping them in a way that makes the math easier, she effectively showcases how flexible addition can be in solving subtraction problems. However, her final answer of 38 for \(54 - 16\) is incorrect; the correct answer is actually 38—but Carmen's steps are a bit convoluted! Now, let’s apply Carmen’s method to solve \(72 - 38\). Following a similar approach, we can reframe it: 1. Start with \(72 + 2 = 74\). 2. Then consider what you need to subtract: \(38 + 2 = 40\) means you are essentially calculating \(74 - 40\). 3. Which gives us \(74 - 40 = 34\). Traditional checking: \(72 - 38 = 34\), confirming that unsurprisingly, Carmen's method works!

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