The value of $x$ in the equation $$ \frac{113 \times 4 - x \times 2}{13 \times 9 - 5 \times 7} = 5 $$ is

### Concept & Strategy This is a linear equation featuring basic arithmetic operations in the numerator and denominator. The core strategy is to simplify the arithmetic expressions completely before attempting to isolate the variable $x$. The process follows the standard cross-multiplication principle: $$ \frac{A}{B} = C \implies A = B \times C $$ ### Step-by-Step Solution * **Step 1:** Simplify the multiplication terms in the numerator. Numerator $= (113 \times 4) - 2x = 452 - 2x$ * **Step 2:** Simplify the multiplication terms in the denominator. Denominator $= (13 \times 9) - (5 \times 7) = 117 - 35 = 82$ * **Step 3:** Substitute the simplified terms back into the equation. $$ \frac{452 - 2x}{82} = 5 $$ * **Step 4:** Cross-multiply to eliminate the fraction. $ 452 - 2x = 5 \times 82 $ $ 452 - 2x = 410 $ * **Step 5:** Isolate $x$. $ 2x = 452 - 410 $ $ 2x = 42 $ $ x = 21 $ ### Exam Strategy & Shortcut Use the **Unit Digit Method** to verify your arithmetic quickly. Numerator unit digit: $(3 \times 4) - 2x \implies 2 - 2x$. Denominator unit digit: $(3 \times 9) - (5 \times 7) \implies 7 - 5 = 2$. Equation: $\frac{2 - 2x}{2} = 5 \implies 2 - 2x = 0$ (unit digit of 10). So, $2x$ must end in 2. Looking at the options: $2 \times 21 = 42$ (ends in 2). $2 \times 27 = 54$ (ends in 4). Option A is the only one that fits the unit digit logic. ### Common Pitfall Students often make simple subtraction errors in the denominator (e.g., calculating $117 - 35$ as $72$ instead of $82$). A wrong denominator will lead to an incorrect multiplier for the right-hand side, ruining the rest of the calculation. Always double-check basic arithmetic. ### Final Answer **Therefore, the correct answer is 21.**