If 50 is subtracted from two-third of a number, the result is equal to sum of 40 and one-fourth of that number. What is the number?

### Concept & Formula This problem translates a word sentence directly into a linear equation. We set the expression for subtracting 50 from a fraction of the number equal to adding 40 to another fraction of the number. $$ \frac{2}{3}x - 50 = \frac{1}{4}x + 40 $$ ### Step-by-Step Solution * Let the unknown number be $x$. * "50 is subtracted from two-third of a number" translates to: $$ \frac{2}{3}x - 50 $$ * "sum of 40 and one-fourth of that number" translates to: $$ \frac{1}{4}x + 40 $$ * Equate both expressions as stated in the problem: $$ \frac{2}{3}x - 50 = \frac{1}{4}x + 40 $$ * Group the variable terms ($x$) on one side and constants on the other by adding 50 and subtracting $\frac{1}{4}x$ from both sides: $$ \frac{2}{3}x - \frac{1}{4}x = 40 + 50 $$ $$ \frac{2}{3}x - \frac{1}{4}x = 90 $$ * Find a common denominator (12) for the fractions: $$ \frac{8x - 3x}{12} = 90 $$ $$ \frac{5x}{12} = 90 $$ * Isolate $x$: $$ 5x = 90 \times 12 $$ $$ x = \frac{1080}{5} $$ $$ x = 216 $$ ### Exam Strategy & Shortcut To avoid dealing with fractions, find the Least Common Multiple (LCM) of the denominators 3 and 4, which is 12. Assume the number is $12k$. Two-thirds of $12k = 8k$. One-fourth of $12k = 3k$. The equation becomes: $8k - 50 = 3k + 40$. $5k = 90$, which means $k = 18$. Since the number is $12k$, calculate $12 \times 18 = 216$. This LCM method heavily reduces calculation time and errors. ### Common Pitfall A standard mistake is subtracting the fraction from 50 (i.e., $50 - \frac{2}{3}x$) instead of subtracting 50 from the fraction ($\frac{2}{3}x - 50$). Always respect the "subtracted from" wording. ### Final Answer **Therefore, the correct answer is 216.**