Let's solve the given proportion step-by-step.
The given proportion is:
[tex]\[
\frac{\frac{1}{5}}{\frac{2}{3}}=\frac{\frac{1}{4}}{m}
\][/tex]
1. Calculate the left fraction \(\frac{\frac{1}{5}}{\frac{2}{3}}\):
First, let's remember the rule for dividing fractions: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\).
Applying this to our left fraction:
[tex]\[
\frac{\frac{1}{5}}{\frac{2}{3}} = \frac{1}{5} \times \frac{3}{2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10}
\][/tex]
So, the left fraction simplifies to \(\frac{3}{10}\).
2. Set up the proportion:
Now, substitute \(\frac{3}{10}\) back into the proportion:
[tex]\[
\frac{3}{10} = \frac{\frac{1}{4}}{m}
\][/tex]
3. Solve for \(m\):
To find \(m\), we can set up an equation by multiplying both sides of the proportion by \(m\):
[tex]\[
m \cdot \frac{3}{10} = \frac{1}{4}
\][/tex]
Next, isolate \(m\) by dividing both sides by \(\frac{3}{10}\):
[tex]\[
m = \frac{\frac{1}{4}}{\frac{3}{10}}
\][/tex]
Again, remember the rule for dividing fractions:
[tex]\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
\][/tex]
Applying this to our problem:
[tex]\[
m = \frac{1}{4} \times \frac{10}{3} = \frac{1 \times 10}{4 \times 3} = \frac{10}{12}
\][/tex]
4. Simplify the fraction \(\frac{10}{12}\):
We can simplify \(\frac{10}{12}\) by finding the greatest common divisor of the numerator and the denominator, which is 2:
[tex]\[
\frac{10}{12} = \frac{10 \div 2}{12 \div 2} = \frac{5}{6}
\][/tex]
Therefore, the value of \(m\) in the simplest form is:
[tex]\[
m = \frac{5}{6}
\][/tex]
So, the final answer is:
[tex]\[
m = \frac{5}{6}
\][/tex]
