What is $394$ times $113$?

### Concept & Strategy This is a direct multiplication problem. However, instead of traditional vertical multiplication, applying the **distributive property** allows you to break the numbers into easily calculable chunks. $$ A \times (B + C) = (A \times B) + (A \times C) $$ ### Step-by-Step Solution * **Given:** The expression $394 \times 113$. * **Step 1:** Break down $113$ into its expanded base-10 form to make multiplication simpler. $113 = 100 + 10 + 3$ * **Step 2:** Apply the distributive property. $394 \times (100 + 10 + 3) = (394 \times 100) + (394 \times 10) + (394 \times 3)$ * **Step 3:** Calculate the individual parts. $394 \times 100 = 39400$ $394 \times 10 = 3940$ $394 \times 3 = 1182$ * **Step 4:** Sum the calculated parts. $39400 + 3940 + 1182 = 44522$ ### Exam Strategy & Shortcut Use the **Digital Sum (Digital Root)** method to verify the answer instantly. 1. Find the digital sum of $394$: $3 + 9 + 4 = 16 \rightarrow 1 + 6 = 7$. 2. Find the digital sum of $113$: $1 + 1 + 3 = 5$. 3. Multiply their digital sums: $7 \times 5 = 35 \rightarrow 3 + 5 = 8$. The correct option must have a digital sum of $8$. Let's check the options: (a) $44402 \rightarrow 4+4+4+0+2 = 14 \rightarrow 5$ (b) $44522 \rightarrow 4+4+5+2+2 = 17 \rightarrow 8$ (Match!) (c) $44632 \rightarrow 4+4+6+3+2 = 19 \rightarrow 1$ (d) $44802 \rightarrow 4+4+8+0+2 = 18 \rightarrow 9$ Option (b) is the only valid answer. ### Common Pitfall Writing out the traditional three-tier multiplication stack increases the likelihood of addition misalignment or carrying errors. Mental math using distribution or digital sums is far safer and faster. ### Final Answer Therefore, the correct answer is **44522**.