To find the ratio of the volumes of a small cube to a big cube when the side length of the small cube is half the side length of the big cube, follow these steps:
1. Let the side length of the big cube be [tex]\( x \)[/tex].
2. Determine the side length of the small cube:
Given that the side length of the small cube is half of the side length of the big cube, the side length of the small cube is [tex]\( \frac{x}{2} \)[/tex].
3. Calculate the volume of the big cube:
The volume [tex]\( V \)[/tex] of a cube with side length [tex]\( x \)[/tex] is given by:
[tex]\[
\text{Volume of big cube} = x^3
\][/tex]
4. Calculate the volume of the small cube:
The volume [tex]\( V \)[/tex] of a cube with side length [tex]\( \frac{x}{2} \)[/tex] is given by:
[tex]\[
\text{Volume of small cube} = \left( \frac{x}{2} \right)^3
\][/tex]
This simplifies to:
[tex]\[
\text{Volume of small cube} = \frac{x^3}{8}
\][/tex]
5. Determine the ratio of the volumes:
To find the ratio of the volume of the small cube to the big cube, divide the volume of the small cube by the volume of the big cube:
[tex]\[
\text{Ratio} = \frac{\text{Volume of small cube}}{\text{Volume of big cube}} = \frac{\frac{x^3}{8}}{x^3}
\][/tex]
6. Simplify the ratio:
Simplifying the fraction:
[tex]\[
\frac{\frac{x^3}{8}}{x^3} = \frac{1}{8}
\][/tex]
Therefore, the ratio of the volumes of the small cube to the big cube is [tex]\( 1:8 \)[/tex].
