How can you tell that $$ (448+217+34) imes 9 $$ is three times as large as $$ (448+217+34) imes 3 $$ 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.

Question

How can you tell that $$ (448+217+34) 	imes 9 $$ is three times as large as $$ (448+217+34) 	imes 3 $$ 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.

UpStudy AI · Accepted Answer

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.
The first expression $$ (448+217+34) \times 9 $$ is three times as large as the second expression $$ (448+217+34) \times 3 $$ because multiplying by 9 is the same as multiplying by 3 three times.

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