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 fraction where the numerator is a sum and difference of products involving mass, specific heat capacity, and temperature differences. 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. **Sum the numerator terms:** $$35950.2 - 27211.81 - 1447.2 = 35950.2 - 28659.01 = 5291.19$$ 5. **Calculate the denominator:** $$0.20 \times (27.7 - 21) = 0.20 \times 6.7 = 1.34$$ 6. **Divide numerator by denominator:** $$\frac{5291.19}{1.34}$$ 7. **Simplify the fraction:** $$\frac{\cancel{5291.19}}{\cancel{1.34}} = 3946.41$$ **Final answer:** $$3946.41$$ This value represents the result of the given expression.