The value of $$ \left( \frac{0.896 \times 0.763 + 0.896 \times 0.237}{0.7 \times 0.064 + 0.7 \times 0.936} \right) $$ is

### Concept & Strategy Similar to the previous problems, this complex-looking fraction breaks down instantly when you use the **Distributive Property**. The question is designed with "complementary decimals"—numbers inside the brackets that perfectly sum up to $1.000$, neutralizing the complex multiplication. ### Step-by-Step Solution **Given:** $$ \frac{0.896 \times 0.763 + 0.896 \times 0.237}{0.7 \times 0.064 + 0.7 \times 0.936} $$ **Calculation:** * **Simplify the Numerator:** Factor out the common multiplier, $0.896$. $$ 0.896 \times (0.763 + 0.237) $$ Add the decimals inside the parenthesis. Notice they sum to exactly $1$. $$ 0.763 + 0.237 = 1.000 $$ Numerator becomes: $0.896 \times 1 = 0.896$. * **Simplify the Denominator:** Factor out the common multiplier, $0.7$. $$ 0.7 \times (0.064 + 0.936) $$ Add the decimals inside the parenthesis. They also sum to exactly $1$. $$ 0.064 + 0.936 = 1.000 $$ Denominator becomes: $0.7 \times 1 = 0.7$. * **Evaluate the Simplified Fraction:** $$ \frac{0.896}{0.7} $$ * **Clear the Decimals:** Multiply the numerator and the denominator by $1000$ to work with whole numbers. $$ \frac{896}{700} $$ * **Divide:** Divide both by $7$: $896 / 7 = 128$ $700 / 7 = 100$ $$ \frac{128}{100} = 1.28 $$ ### Exam Strategy & Shortcut **Complementary Scanning:** In fraction equations with multiple decimal terms, always check the non-repeated decimals first. $0.763 + 0.237 = 1$ and $0.064 + 0.936 = 1$. The moment you spot this, the entire equation collapses in your head to just $0.896 / 0.7$. To solve $0.896 / 0.7$ mentally, shift the decimal one place to get $8.96 / 7$. Since $7 \times 1 = 7$, the answer must start with $1$. The only option starting with $1$ and having the correct decimal magnitude is $1.28$. ### Common Pitfall Failing to recognize that $0.064 + 0.936 = 1.000$. Students sometimes misalign the decimals when adding and think it sums to $0.1$ or $10$, which throws off the final division scale by a factor of 10. Always align decimals vertically: 0.936 + 0.064 ------- 1.000 ### Final Answer **Therefore, the correct answer is 1.28.**