To write the ratio \( 6 k^2 : 4 h k \) in the form \( 1 : n \), we need to simplify the given ratio step-by-step. Here’s how we can do it:
1. Start with the ratio \( \frac{6 k^2}{4 h k} \).
2. Simplify the ratio by dividing both the numerator and the denominator by any common factors. Notice that both the numerator and the denominator share a factor of \( 2k \).
[tex]\[
\frac{6 k^2}{4 h k} = \frac{6 k^2 \div 2k}{4 h k \div 2k}
\][/tex]
3. Perform the division:
[tex]\[
\frac{6 k^2 \div 2k}{4 h k \div 2k} = \frac{3 k}{2 h}
\][/tex]
4. The simplified form of the ratio \( 6 k^2 : 4 h k \) is \( \frac{3 k}{2 h} \).
Now, to express the ratio in the form \( 1 : n \), we set \( \frac{3 k}{2 h} \) equal to \( \frac{1}{n} \):
5. Solve for \( n \):
[tex]\[
\frac{3 k}{2 h} = \frac{1}{n}
\][/tex]
6. To isolate \( n \), take the reciprocal of both sides:
[tex]\[
n = \frac{2 h}{3 k}
\][/tex]
Therefore, the expression for \( n \), in terms of \( h \) and \( k \), is:
[tex]\[
n = \frac{2 h}{3 k}
\][/tex]
So the ratio \( 6 k^2: 4 h k \) in the form \( 1 : n \) simplifies to:
[tex]\[
n = \frac{2 h}{3 k}
\][/tex]
