To work out the expression without using a calculator, we will simplify the numerator and the denominator separately.

Numerator $= \frac{4}{5} + \frac{7}{8}$
To add these fractions, find a common denominator, which is 40.

$$\frac{4}{5} = \frac{4 \times 8}{5 \times 8} = \frac{32}{40}$$ $$\frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40}$$

Now, add the fractions:

$$\text{Numerator} = \frac{32}{40} + \frac{35}{40} = \frac{32+35}{40} = \frac{67}{40}$$

Denominator $= 1 + \frac{5}{72}$
Convert 1 to a fraction with denominator 72:

$$1 = \frac{72}{72}$$

Now, add the fractions:

$$\text{Denominator} = \frac{72}{72} + \frac{5}{72} = \frac{72+5}{72} = \frac{77}{72}$$

$$\frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{67}{40}}{\frac{77}{72}}$$

To divide by a fraction, multiply by its reciprocal:

$$\frac{67}{40} \times \frac{72}{77}$$

We can simplify by dividing 40 and 72 by their greatest common divisor, which is 8:

$$40 \div 8 = 5$$ $$72 \div 8 = 9$$

So the expression becomes:

$$\frac{67}{5} \times \frac{9}{77}$$

Now, multiply the numerators and the denominators:

$$\text{Result} = \frac{67 \times 9}{5 \times 77}$$ $$67 \times 9 = 603$$ $$5 \times 77 = 385$$

Therefore, the final answer is:

$$\frac{603}{385}$$

This fraction cannot be simplified further as the prime factors of 603 are $3^2 \times 67$ and the prime factors of 385 are $5 \times 7 \times 11$.