To work out the expression without using a calculator, we will simplify the numerator and the denominator separately. The expression is: $$ \frac{\frac{4}{5} + \frac{7}{8}}{1 + \left(\frac{1}{4} \times \frac{5}{18}\right)} $$ Step 1: Simplify the numerator. 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: $$ \text{Numerator} = \frac{32}{40} + \frac{35}{40} = \frac{32+35}{40} = \frac{67}{40} $$ Step 2: Simplify the term in the parenthesis in the denominator. $$ \frac{1}{4} \times \frac{5}{18} = \frac{1 \times 5}{4 \times 18} = \frac{5}{72} $$ Step 3: Simplify the denominator. Denominator $= 1 + \frac{5}{72}$ Convert 1 to a fraction with denominator 72: $$ 1 = \frac{72}{72} $$ Now, add the fractions: $$ \text{Denominator} = \frac{72}{72} + \frac{5}{72} = \frac{72+5}{72} = \frac{77}{72} $$ Step 4: Divide the simplified numerator by the simplified denominator. $$ \frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{67}{40}}{\frac{77}{72}} $$ 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 $5 \times 7 \times 11$. The final answer is $\boxed{\frac{603}{385}}$. That's 2 down. 3 left today — send the next one.