Answer :

We start by converting each mixed number to an improper fraction.

1. For
[tex]$$30 \frac{3}{5},$$[/tex]
we have
[tex]$$30 \frac{3}{5} = \frac{30 \times 5 + 3}{5} = \frac{150+3}{5} = \frac{153}{5}.$$[/tex]

2. For
[tex]$$22 \frac{14}{15},$$[/tex]
we have
[tex]$$22 \frac{14}{15} = \frac{22 \times 15 + 14}{15} = \frac{330+14}{15} = \frac{344}{15}.$$[/tex]

Next, we need a common denominator to subtract the fractions. The least common denominator for [tex]$5$[/tex] and [tex]$15$[/tex] is [tex]$15$[/tex]. We convert [tex]$\frac{153}{5}$[/tex] to an equivalent fraction with denominator [tex]$15$[/tex] by multiplying the numerator and denominator by [tex]$3$[/tex]:

[tex]$$\frac{153}{5} = \frac{153 \times 3}{5 \times 3} = \frac{459}{15}.$$[/tex]

Now, subtract the two fractions:

[tex]$$\frac{459}{15} - \frac{344}{15} = \frac{459 - 344}{15} = \frac{115}{15}.$$[/tex]

Finally, we simplify the fraction [tex]$\frac{115}{15}$[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is [tex]$5$[/tex]:

[tex]$$\frac{115}{15} = \frac{115\div 5}{15\div 5} = \frac{23}{3}.$$[/tex]

Thus, the simplified result is

[tex]$$\boxed{\frac{23}{3}}.$$[/tex]

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