Step by step
The question
Question: two similar solids have matching lengths 3 and 7.5, and the small volume is 16: find the large volume.
1 Length scale factor
Similar means the same shape: one is a bigger copy.
The scale factor k says how many times longer: the large one over the small one, 7.5/3, that is 5/2.
Divide top and bottom by 1.5.
2 Volume factor
Volume grows in three directions: k times k times k.
On top, 5 times 5 times 5 is 125.
At the bottom, 2 times 2 times 2 is 8.
That is 125/8.
Why?
For example, a cube 1 by 1 by 1, made 2 times longer every way, holds 8 little cubes, not 2.
3 Use it
Going to the large one, we multiply: 16 times 125/8.
Times 125/8 means times 125, then divide by 8.
We can do them in any order.
Keep the numbers small: 16 divided by 8 is 2.
Then 2 times 125 is 250.
So the large volume is 250.
Careful!
The volume does not grow like the lengths.
It grows by k cubed.
4 Check
Let's check: the large volume divided by the small one is 125/8, which is k cubed.
So it is right!
Let's look back.
First: length scale factor.
Then: volume factor.
Then: use it.
Last: check.