Emily collected standard 4″ by 6″ postcards during a road trip and wants to display them without overlapping in 12″ by 15″ picture frames. How many postcards can she display in one frame?
A. 6 postcards.
B. 7 postcards.
C. 8 postcards.
D. 9 postcards.
The correct answer and explanation is:
To determine how many 4″ by 6″ postcards can fit in a 12″ by 15″ picture frame without overlapping, the area of the frame and the area of each postcard needs to be considered.
The first step is to calculate the area of the frame. The frame is 12″ by 15″, so: Area of the frame=12 in×15 in=180 square inches\text{Area of the frame} = 12 \, \text{in} \times 15 \, \text{in} = 180 \, \text{square inches}
Next, calculate the area of one postcard. Each postcard is 4″ by 6″, so: Area of one postcard=4 in×6 in=24 square inches\text{Area of one postcard} = 4 \, \text{in} \times 6 \, \text{in} = 24 \, \text{square inches}
Now, divide the total area of the frame by the area of one postcard to find how many postcards can fit in the frame: Area of the frameArea of one postcard=18024=7.5\frac{\text{Area of the frame}}{\text{Area of one postcard}} = \frac{180}{24} = 7.5
Since 7.5 is not a whole number, it means that 7 full postcards can fit, but not 8. Therefore, the maximum number of postcards that can fit in the frame without overlapping is 7 postcards.
However, this is only an area-based calculation. To further verify, we can think about how the postcards might be arranged within the frame. The frame dimensions are 12″ by 15″, and the postcards are 4″ by 6″. The postcards can be arranged in rows and columns.
- Along the 12″ side of the frame, you can fit 124=3\frac{12}{4} = 3 postcards.
- Along the 15″ side of the frame, you can fit 156=2.5\frac{15}{6} = 2.5 postcards, but since it’s not possible to have half a postcard, you can fit only 2 postcards along this side.
Thus, in total, you can fit 3×2=63 \times 2 = 6 postcards in the frame. Therefore, the correct answer is 6 postcards.