Directions: What approximate value should come in place of question mark($x$) in the following questions? (NOTE: You are not expected to calculate the exact value). $421 \div 35 \times 299.97 \div 25.05 = x^2$

### Concept & Strategy This is an **Approximation** problem involving a chain of multiplication and division. Because multiplication and division have the same precedence in BODMAS (evaluated left-to-right), rounding the numbers to highly compatible multiples ensures smooth cancellation of fractions without needing to compute large intermediary decimals. ### Step-by-Step Solution **Calculation:** Step 1: Approximate the values to nearest compatible integers. * $421 \approx 420$ (Since it is easily divisible by 35) * $35$ remains $35$ * $299.97 \approx 300$ * $25.05 \approx 25$ Step 2: Rewrite the equation with approximations. $$ 420 \div 35 \times 300 \div 25 = x^2 $$ Step 3: Convert the chain of operations into a single fraction for easier visual cancellation. $$ \left(\frac{420}{35}\right) \times \left(\frac{300}{25}\right) = x^2 $$ Step 4: Simplify the individual parts. $$ \frac{420}{35} = 12 $$ $$ \frac{300}{25} = 12 $$ Step 5: Multiply the simplified results. $$ 12 \times 12 = x^2 $$ $$ 144 = x^2 $$ Step 6: Solve for $x$. $$ x = 12 $$ ### Exam Strategy & Shortcut **Fraction Scaling:** If $\frac{420}{35}$ isn't instantly obvious, scale it up or down. Divide both by 7 to get $\frac{60}{5} = 12$. Similarly for $\frac{300}{25}$, think of 100 having four 25s. Therefore, 300 has $3 \times 4 = 12$ twenty-fives. Realizing both blocks equal 12 means you are looking at $12 \times 12 = x^2$, which immediately gives $x=12$ without even calculating 144. ### Common Pitfall A severe mistake here is evaluating strictly left to right without converting to fractions, like trying to do $(420 \div 35) = 12$, then $12 \times 300 = 3600$, and then $3600 \div 25$. While it gives the same answer, dealing with larger numbers like 3600 increases the mental load and risk of simple errors compared to keeping the numbers small and modular. ### Final Answer **Therefore, the correct answer is 12.**