By how much is $\frac{3}{4}$ th of $568$ lesser than $\frac{7}{8}$ th of $1008$?

### Concept & Formula This problem requires basic fractional multiplication and subtraction. We need to evaluate two independent expressions and find the absolute difference between them. ### Step-by-step Solution Let's calculate the two parts separately. **Step 1: Calculate $\frac{3}{4}$ of $568$** $$ \text{Value 1} = \frac{3}{4} \times 568 $$ $$ \text{Value 1} = 3 \times 142 $$ $$ \text{Value 1} = 426 $$ **Step 2: Calculate $\frac{7}{8}$ of $1008$** $$ \text{Value 2} = \frac{7}{8} \times 1008 $$ $$ \text{Value 2} = 7 \times 126 $$ $$ \text{Value 2} = 882 $$ **Step 3: Find the difference** $$ \text{Difference} = \text{Value 2} - \text{Value 1} $$ $$ \text{Difference} = 882 - 426 $$ $$ \text{Difference} = 456 $$ ### Exam Strategy & Shortcut Instead of full mental calculation, you can use the **unit digit method** for the final subtraction step to save a few seconds: * Unit digit of first part: $3 \times 2 = 6$ * Unit digit of second part: $7 \times 6 = 2$ (which effectively becomes $12$ for subtraction) * Difference of unit digits: $12 - 6 = 6$ Look at the options: only (d) $456$ ends in $6$. You can safely mark it without doing the full $882 - 426$ subtraction. ### Common Pitfall Students often make calculation errors when dividing large numbers like $1008$ by $8$. A good tip is to divide by $2$ three times sequentially (e.g., $1008 \rightarrow 504 \rightarrow 252 \rightarrow 126$) if you aren't comfortable with direct division. ### Final Answer **Therefore, the correct answer is 456.**