If one sixth of x minus seven halves of three sevenths equals −7/4, that is, x/6 − (7/2) * (3/7) = −7/4, what is the value of x?

Introduction / Context:This question tests your ability to interpret a verbal description of a fractional equation and then solve it correctly. The phrase "one sixth of x minus seven halves of three sevenths" must be translated into algebraic terms. Once the equation is set up correctly, solving for x requires only basic operations with fractions. Such problems are common in aptitude tests and help reinforce comfort with rational numbers.

Given Data / Assumptions:

• The verbal equation is: one sixth of x minus seven halves of three sevenths equals −7/4.
• Algebraically, this is x/6 − (7/2) * (3/7) = −7/4.
• x is a real number.
• All denominators are non zero.

Concept / Approach:The key steps are: correctly interpret the phrase into x/6 − (7/2)*(3/7), simplify the constant term (7/2)*(3/7), and then solve the resulting linear equation in x. Fraction arithmetic is crucial: we must multiply and add fractions with care. Once x/6 is isolated, we multiply both sides by 6 to solve for x. Finally, we may convert the fraction to a decimal to compare with the answer choices.

Step-by-Step Solution:Translate the verbal statement into algebra: x/6 − (7/2) * (3/7) = −7/4.Simplify the product (7/2) * (3/7). The 7s cancel, leaving (1/2) * 3 = 3/2.So the equation becomes x/6 − 3/2 = −7/4.Move the constant term to the right-hand side: x/6 = −7/4 + 3/2.Write 3/2 with denominator 4: 3/2 = 6/4, so x/6 = −7/4 + 6/4 = −1/4.Multiply both sides by 6 to solve for x: x = 6 * (−1/4) = −6/4 = −3/2.Thus x = −3/2, which is −1.5 in decimal form.

Verification / Alternative check:Substitute x = −3/2 back into the original equation. Compute one sixth of x: x/6 = (−3/2)/6 = −3/12 = −1/4. Compute seven halves of three sevenths: (7/2)*(3/7) = 3/2. Then x/6 − (7/2)*(3/7) = −1/4 − 3/2 = −1/4 − 6/4 = −7/4, which matches the right-hand side. This confirms that x = −3/2 is the correct solution.

Why Other Options Are Wrong:

• Values 3, −2.5, 6, and 1.5 do not satisfy the equation when substituted back, as they produce different left-hand sides.
• For example, if x = 3, then x/6 − 3/2 = 1/2 − 3/2 = −1, which is not −7/4.
• Only x = −1.5 (that is, −3/2) yields the correct equality.

Common Pitfalls:

• Misinterpreting the verbal phrase and writing 7/2 + 3/7 instead of (7/2)*(3/7).
• Making errors when converting between fractions like 3/2 and 6/4.
• Forgetting to multiply both sides by 6 at the end or mishandling the negative sign.

Final Answer:-1.5