How can you tell that \( (448+217+34) \times 9 \) is three times as large as \( (448+217+34) \times 3 \) without doing complicated calculations? Compare the parts that are the same. The first expression shows a sum multiplied by a multiple of 3 . The second shows the same sum multiplied by a multiple of 3 . So, the two expressions are both 3 times as large as the sum. Compare the parts that are the same. The first expression shows a sum multiplied by a number. The second shows the same sum multiplied by a number. So, both expressions must be greater than the sum. Compare the parts that are different. The first expression shows a sum multiplied by 9 . The second shows the same sum multiplied by 3 . So, the first expression is three times as large as the second expression. Compare the parts that are different. The first expression shows a sum multiplied by 9 . The second shows the same sum multiplied by 3 . So, the first expression must be less than the second expression.

To determine how \( (448+217+34) \times 9 \) is three times as large as \( (448+217+34) \times 3 \) without performing complicated calculations, we can analyze the expressions step by step. 1. **Identify the common part**: Both expressions contain the same sum, which is \( 448 + 217 + 34 \). Let's denote this sum as \( S \). Therefore, we can rewrite the expressions as: - First expression: \( S \times 9 \) - Second expression: \( S \times 3 \) 2. **Compare the multipliers**: The first expression is multiplied by 9, while the second expression is multiplied by 3. 3. **Relate the multipliers**: Notice that \( 9 \) is three times \( 3 \): \[ 9 = 3 \times 3 \] 4. **Conclusion**: Since both expressions are based on the same sum \( S \), we can conclude that: \[ S \times 9 = S \times (3 \times 3) = (S \times 3) \times 3 \] This shows that the first expression is indeed three times as large as the second expression. Thus, the correct reasoning is: - Compare the parts that are different. The first expression shows a sum multiplied by 9, while the second shows the same sum multiplied by 3. Therefore, the first expression is three times as large as the second expression.