The ratio [tex]$6k^2 : 4hk[tex]$[/tex] can be written in the form [tex]$[/tex]1 : n$[/tex].

Write an expression for [tex]$n[tex]$[/tex], in terms of [tex]$[/tex]h[tex]$[/tex] and [tex]$[/tex]k$[/tex], in its simplest form.



Answer :

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]