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

Introduction / Context:
This is an algebra problem involving fractions of an unknown number. You are told that if 50 is subtracted from two thirds of a number, the result is equal to the sum of 40 and one fourth of the same number. The task is to find that number. Such questions are common in quantitative aptitude tests and help you practice forming and solving linear equations involving fractional coefficients.

Given Data / Assumptions:

• Let the number be N.
• Two-thirds of the number is (2 / 3) * N.
• One-fourth of the number is (1 / 4) * N.
• (2 / 3) * N − 50 = 40 + (1 / 4) * N.
• We must solve for N.

Concept / Approach:
We translate the word statement into an algebraic equation. Because the equation has fractions with different denominators (3 and 4), it is convenient to multiply through by the least common multiple of the denominators to clear the fractions. This gives a simpler linear equation with integer coefficients, which we can solve using basic algebra. After finding N, we should substitute it back into the original equation to verify that it satisfies the condition.

Step-by-Step Solution:
Step 1: Let the number be N. Step 2: According to the problem, two-thirds of the number minus 50 equals 40 plus one-fourth of the number: (2 / 3) * N − 50 = 40 + (1 / 4) * N. Step 3: To eliminate fractions, multiply the entire equation by 12, the least common multiple of 3 and 4. Step 4: Multiply term by term: 12 * (2 / 3) * N = 8N, 12 * (−50) = −600, 12 * 40 = 480, and 12 * (1 / 4) * N = 3N. Step 5: The new equation is 8N − 600 = 480 + 3N. Step 6: Subtract 3N from both sides to get 5N − 600 = 480. Step 7: Add 600 to both sides: 5N = 480 + 600 = 1,080. Step 8: Divide both sides by 5 to find N: N = 1,080 / 5 = 216.

Verification / Alternative check:
Substitute N = 216 into the original statement. Two-thirds of 216 is (2 / 3) * 216 = 144. Now compute (2 / 3) * 216 − 50 = 144 − 50 = 94. One-fourth of 216 is (1 / 4) * 216 = 54. The sum of 40 and one-fourth of the number is 40 + 54 = 94. Since both sides are equal, N = 216 satisfies the equation and is therefore correct.

Why Other Options Are Wrong:
Option 156, 179, 198 and 144: Substituting any of these values into the original equation results in different values on the two sides, so they do not satisfy the condition (2 / 3) * N − 50 = 40 + (1 / 4) * N.

Common Pitfalls:
A common mistake is to misinterpret the words and form the wrong equation, for example by subtracting 50 from the whole number instead of from two-thirds of the number. Another error is to handle the fractions incorrectly when clearing denominators, which leads to wrong coefficients. Carefully multiplying every term by the least common multiple and checking the final equation prevents such mistakes. Always verify your solution by substituting it back into the original statement.

Final Answer:
The number that satisfies the given condition is 216.