1. **Convert 6815.48 (base 9) to base 7** Step 1: Convert 6815.48 from base 9 to decimal. $$6815.48_9 = 6 \times 9^3 + 8 \times 9^2 + 1 \times 9^1 + 5 \times 9^0 + 4 \times 9^{-1} + 8 \times 9^{-2}$$ Calculate powers: $$9^3 = 729, \quad 9^2 = 81, \quad 9^1 = 9, \quad 9^0 = 1, \quad 9^{-1} = \frac{1}{9}, \quad 9^{-2} = \frac{1}{81}$$ Calculate each term: $$6 \times 729 = 4374$$ $$8 \times 81 = 648$$ $$1 \times 9 = 9$$ $$5 \times 1 = 5$$ $$4 \times \frac{1}{9} = \frac{4}{9} \approx 0.4444$$ $$8 \times \frac{1}{81} = \frac{8}{81} \approx 0.0988$$ Sum integer part: $$4374 + 648 + 9 + 5 = 5036$$ Sum fractional part: $$0.4444 + 0.0988 = 0.5432$$ Total decimal value: $$5036 + 0.5432 = 5036.5432$$ Step 2: Convert decimal 5036.5432 to base 7. Convert integer part 5036 to base 7: Divide 5036 by 7 repeatedly: $$5036 \div 7 = 719 \text{ remainder } 3$$ $$719 \div 7 = 102 \text{ remainder } 5$$ $$102 \div 7 = 14 \text{ remainder } 4$$ $$14 \div 7 = 2 \text{ remainder } 0$$ $$2 \div 7 = 0 \text{ remainder } 2$$ Reading remainders from last to first: 2 0 4 5 3 So integer part in base 7 is: $$20453_7$$ Convert fractional part 0.5432 to base 7: Multiply fractional part by 7 repeatedly and take integer parts: $$0.5432 \times 7 = 3.8024 \rightarrow 3$$ $$0.8024 \times 7 = 5.6168 \rightarrow 5$$ $$0.6168 \times 7 = 4.3176 \rightarrow 4$$ $$0.3176 \times 7 = 2.2232 \rightarrow 2$$ Stop here for 4 decimal places. Fractional part in base 7 is approximately: $$.3542_7$$ Final answer: $$6815.48_9 = 20453.3542_7$$ --- 2. **Evaluate 726 (base 8) divided by 56 (base 8)** Step 1: Convert both numbers to decimal. $$726_8 = 7 \times 8^2 + 2 \times 8^1 + 6 \times 8^0 = 7 \times 64 + 2 \times 8 + 6 = 448 + 16 + 6 = 470$$ $$56_8 = 5 \times 8^1 + 6 \times 8^0 = 5 \times 8 + 6 = 40 + 6 = 46$$ Step 2: Divide decimal values: $$\frac{470}{46} = 10.2173913...$$ Step 3: Convert result back to base 8 if needed. Integer part 10 in base 8: $$10 \div 8 = 1 \text{ remainder } 2$$ So integer part is 12 in base 8. Fractional part 0.2173913... multiply by 8: $$0.2173913 \times 8 = 1.7391304 \rightarrow 1$$ $$0.7391304 \times 8 = 5.9130432 \rightarrow 5$$ $$0.9130432 \times 8 = 7.3043456 \rightarrow 7$$ Stop here for 3 decimal places. Fractional part approx: .157 Final answer: $$10.2173913_{10} = 12.157_8$$ --- 3. **Find the product of 856.382 and 81.44 to 2 decimal places** Step 1: Multiply the numbers: $$856.382 \times 81.44 = 69744.34908$$ Step 2: Round to 2 decimal places: $$69744.35$$ Final answer: $$69744.35$$ --- 4. **If $23y = 1111_{2}$ find the value of $y$** Step 1: Convert $1111_2$ to decimal: $$1111_2 = 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 4 + 2 + 1 = 15$$ Step 2: Solve for $y$: $$23y = 15 \implies y = \frac{15}{23}$$ Final answer: $$y = \frac{15}{23}$$