Sure! To find a fraction between [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex], we can use the method of averaging the two fractions. Here is the step-by-step solution:
1. Identify the given fractions:
- First fraction: [tex]\(\frac{1}{3}\)[/tex]
- Second fraction: [tex]\(\frac{3}{5}\)[/tex]
2. Add the two fractions:
To add [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex], we need a common denominator. The least common denominator (LCD) for 3 and 5 is 15.
Therefore:
[tex]\[
\frac{1}{3} = \frac{5}{15}
\][/tex]
[tex]\[
\frac{3}{5} = \frac{9}{15}
\][/tex]
Now, add the numerators of the equivalent fractions:
[tex]\[
\frac{5}{15} + \frac{9}{15} = \frac{5 + 9}{15} = \frac{14}{15}
\][/tex]
3. Find the average of the two fractions:
To find a fraction that lies between [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex], we calculate the average of [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]:
[tex]\[
\text{Average} = \frac{\left(\frac{1}{3} + \frac{3}{5}\right)}{2} = \frac{\frac{14}{15}}{2} = \frac{14}{15} \times \frac{1}{2} = \frac{14 \times 1}{15 \times 2} = \frac{14}{30}
\][/tex]
4. Simplify the resulting fraction:
Simplify [tex]\(\frac{14}{30}\)[/tex] by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2:
[tex]\[
\frac{14}{30} = \frac{14 \div 2}{30 \div 2} = \frac{7}{15}
\][/tex]
So, the fraction between [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] is given by [tex]\(\frac{7}{15}\)[/tex].
