1. **State the problem:** Calculate the value of the expression: $$\frac{0.429 \times 4190 \times (47.7 - 27.7) - 0.97 \times 4190 \times (27.7 - 21) + 0.54 \times 400 \times (27.7 - 21)}{0.20 \times (27.7 - 21)}$$ 2. **Identify the formula and rules:** This is a calculation involving multiplication and subtraction of terms with units of energy (Joules) and temperature differences (°C). The denominator is a product of mass and temperature difference. 3. **Calculate each term in the numerator:** - First term: $$0.429 \times 4190 \times (47.7 - 27.7) = 0.429 \times 4190 \times 20 = 0.429 \times 83800 = 35950.2$$ - Second term: $$0.97 \times 4190 \times (27.7 - 21) = 0.97 \times 4190 \times 6.7 = 0.97 \times 28073 = 27211.81$$ - Third term: $$0.54 \times 400 \times (27.7 - 21) = 0.54 \times 400 \times 6.7 = 0.54 \times 2680 = 1447.2$$ 4. **Combine the numerator terms:** $$35950.2 - 27211.81 + 1447.2 = (35950.2 - 27211.81) + 1447.2 = 8738.39 + 1447.2 = 10185.59$$ 5. **Calculate the denominator:** $$0.20 \times (27.7 - 21) = 0.20 \times 6.7 = 1.34$$ 6. **Divide numerator by denominator:** $$\frac{10185.59}{1.34}$$ 7. **Simplify the fraction:** $$\frac{\cancel{10185.59}}{\cancel{1.34}} = 7599.7$$ **Final answer:** $$7599.7$$