Can someone help with how many objects fit in a volume?

[Beowulf]

This keeps disappearing, so I've copied and pasted from the last time.

The question I am trying to answer, and keep getting wrong is, how many 1/4 cm cubes would fill a rectengular prism 1cm x 7/4cm x 2cm?

My reasoning is volume = L * W * H.
So I get 7/4 * 1/1 * 2/1, which gives me 14/4, so it takes fourteen 1/4cm cubes to fill that space.
But 14 appears to be the wrong answer.

It seems the correct answer is found by converting all of the fractions to quarters, so you get 7/4 * 8/4 * 4/4, and multiply the numerators only giving 224.

Can someone help me understand my mistake? I understand how 224 was determined, but that seems to break the rules for multiplying fractions, and if determining the volume for the shape and then breaking that volume into pieces doesn't work, then I have really misunderstood something basic.

Any help and insight into my error is greatly appreciated.

 

[mmm4444bot]

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… how many 1/4 cm cubes would fill a rectangular prism 1cm x 7/4cm x 2cm?

My reasoning is volume = L * W * H.
So I get 7/4 * 1/1 * 2/1, which gives me 14/4, so it takes fourteen 1/4cm cubes to fill that space.
But 14 appears to be the wrong answer … Can someone help me understand my mistake?

… if determining the volume for the shape and then breaking that volume into pieces doesn't work, then I have really misunderstood something basic.

This is a valid approach, but you broke the volume into wrong-sized pieces.

Good exercise! Your volume calculation for the prism is correct. Your mistake resulted from not considering units.

Yes, it's true that 14/4 can be viewed as 14 times 1/4th, but that 1/4th is a volume. In other words:

V = 14 * 1/4 cm^3

You then divided this volume by 1/4 cm, to get what you thought was 14, but you also need to consider the units:

14/4 cm^3 divided by 1/4 cm = 14 cm^2

Instead of dividing the prism's volume by the cube's side length, divide it by the cube's volume. You're trying to find the number of cube volumes that fit inside the prism's volume. :cool:

It seems the correct answer is found by converting all of the fractions to quarters, so you get 7/4 * 8/4 * 4/4, and multiply the numerators only giving 224.

I understand how 224 was determined, but that seems to break the rules for multiplying fractions …

This is a different approach, and, in this approach, we don't actually multiply the fractions.

The reason for converting each dimension to fourths is to view how many 1/4 cm "slices" or "cuts" we have in each dimension. Think of a Rubic's Cube (google an image, if you need to). The Rubic's Cube is divided into horizontal slices, and each slice is cut into smaller cubes. To count all of the smaller cubes, we would multiply the number of cuts through each dimension.

Since the smaller cubes in your exercise measure 1/4 cm on each dimension, the fractions below indicate the number of cuts along each dimension of the prism.

7/4 * 8/4 * 4/4

It's the numerators, which show the number of cuts in each dimension! To count all of the smaller cubes, we multiply the number of cuts through each dimension.

7 * 8 * 4

If you still have questions, feel free to ask.

 

[Beowulf]

Your mistake resulted from not considering units.
Instead of dividing the prism's volume by the cube's side length, divide it by the cube's volume.

THANK YOU! I had to re-read that three times to really understand it. But then it clicked and I worked out the volume of a single cube, divided it into the total volume and got an answer that matched the 2nd method.

Your explanation of the 2nd method was clear, I just need to play with all of that a bit more to internalize better.

 

[mmm4444bot]

I had to re-read that three times to really understand it. But then it clicked …

Excellent effort. You just grew your brain! :cool: