Which correctly compares \( 3 \times(54+21) \) and \( 6 \times(54+21) \) without doing any calculations? \( 3 \times(54+21)=6 \times(54+21) \) \( 3 \times(54+21)>6 \times(54+21) \) \( 3 \times(54+21)<6 \times(54+21) \) \( 6 \times(54+21)<3 \times(54+21) \)

To compare \( 3 \times(54+21) \) and \( 6 \times(54+21) \) without calculating the values, we can observe the coefficients in front of the expression \( (54+21) \). Since \( 6 \) is greater than \( 3 \), multiplying the same quantity \( (54+21) \) by a larger coefficient \( 6 \) will definitely yield a larger result than multiplying by \( 3 \). Thus, it follows that \( 3 \times(54+21)<6 \times(54+21) \). Additionally, understanding how distribution works can be insightful. The distributive property states that multiplying a number by a sum is the same as multiplying the number by each term of the sum and then adding the results. Thus, it emphasizes that the comparison of \( 3 \) and \( 6 \) being multiplied by the same value ensures that the inequality follows naturally, reinforcing the idea that a larger multiplier leads to a larger product.