A number when multiplied by $13$ is increased by $180$. The number is

### Concept & Logic When a number is multiplied by a factor, the "increase" represents the difference between the new product and the original number. We can set up a basic linear equation to find the unknown value. ### Step-by-step Solution **Given:** * Let the original number be $x$. * Multiplying it by $13$ gives $13x$. * This new value is $180$ more than the original number $x$. **Calculation:** Set up the algebraic equation: $$ 13x = x + 180 $$ Subtract $x$ from both sides to group the variables: $$ 13x - x = 180 $$ $$ 12x = 180 $$ Divide by $12$ to isolate $x$: $$ x = \frac{180}{12} $$ $$ x = 15 $$ ### Exam Strategy & Shortcut **Logical Deduction:** If multiplying a number by $13$ increases it by $180$, it means you added $12$ times the original number to itself (since $1x + 12x = 13x$). Therefore, $12$ times the number equals $180$. Number = $180 \div 12 = 15$. This skips writing the equation entirely. ### Common Pitfall A frequent mistake is interpreting "increased by 180" as multiplying by $180$ or setting the equation as $13x = 180$. Always remember that "increased by" denotes addition relative to the original base value. ### Final Answer **Therefore, the correct answer is 15.**