Question

A store owner is packing small radios into larger boxes that measure 25 in. by 42 in. by 60 in. If the measurement of each radio is 7 in. by 6 in. by 5 in., then how many radios can be placed in the box?

• A. 300
• B. 325
• C. 400
• D. 420
• E. 480

Hint:

For packing problems, simply dividing the volumes is a quick first step. However, always double-check by aligning the dimensions. If the larger dimensions are not integer multiples of the smaller dimensions, space will be wasted, and the volume division method will give an incorrect (overestimated) answer.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem asks for the number of small boxes (radios) that can fit inside a larger box. This can be solved by comparing the volumes, but it's more accurate to check how the dimensions align to ensure there's no wasted space.
Step 2: Key Formula or Approach:
The number of small items that can fit in a large container is found by dividing the volume of the large container by the volume of one small item, provided the dimensions align perfectly.
\[ \text{Number of radios} = \frac{\text{Volume of large box}}{\text{Volume of one radio}} \] Alternatively, and more reliably, divide the corresponding dimensions:
\[ \text{Number of radios} = \left(\frac{\text{Length}_{\text{box}}}{\text{Length}_{\text{radio}}}\right) \times \left(\frac{\text{Width}_{\text{box}}}{\text{Width}_{\text{radio}}}\right) \times \left(\frac{\text{Height}_{\text{box}}}{\text{Height}_{\text{radio}}}\right) \] Step 3: Detailed Explanation:
Method 1: Using Volumes
First, calculate the volume of the large box.
\[ V_{\text{box}} = 25 \text{ in} \times 42 \text{ in} \times 60 \text{ in} = 63000 \text{ in}^3 \] Next, calculate the volume of one radio.
\[ V_{\text{radio}} = 7 \text{ in} \times 6 \text{ in} \times 5 \text{ in} = 210 \text{ in}^3 \] Now, divide the volume of the box by the volume of a radio.
\[ \text{Number of radios} = \frac{63000}{210} = \frac{6300}{21} = 300 \] This method works because the dimensions of the box are perfect multiples of the radio's dimensions. Method 2: Aligning Dimensions
Let's check if the dimensions fit perfectly. We need to match the radio's dimensions (5, 6, 7) to the box's dimensions (25, 42, 60).
- Along the 25 in. side of the box, we can fit the 5 in. side of the radios: \( \frac{25}{5} = 5 \) radios.
- Along the 42 in. side of the box, we can fit the 6 in. or 7 in. side of the radios. Let's try the 7 in. side: \( \frac{42}{7} = 6 \) radios.
- Along the 60 in. side of the box, we can fit the remaining 6 in. side of the radios: \( \frac{60}{6} = 10 \) radios.
Since each division results in a whole number, there is no wasted space with this orientation.
Total number of radios = \( 5 \times 6 \times 10 = 300 \).
Step 4: Final Answer
A total of 300 radios can be placed in the box.