Translating a mixed-fraction condition: The sum of one-half and one-fifth of a number exceeds one-third of that number by 7 1/3. What is the number?
-
A15
-
B18
-
C20
-
D30
Answer
Correct Answer: 20
Explanation
Introduction / Context:Convert words into an algebraic equation involving fractions, then solve for the unknown number. Mixed fractions should be converted to improper fractions to simplify arithmetic.Given Data / Assumptions:
- (1/2 + 1/5)x − (1/3)x = 7 1/3.
- 7 1/3 = 22/3.
Concept / Approach:Combine like terms on the left: (1/2 + 1/5 − 1/3)x. Compute the common denominator and simplify to get a single coefficient times x. Then isolate x.
Step-by-Step Solution:
1/2 + 1/5 = 7/10.7/10 − 1/3 = 21/30 − 10/30 = 11/30.(11/30)x = 22/3 ⇒ x = (22/3) * (30/11) = (22*30)/(33) = 660/33 = 20.Verification / Alternative check:Check: (1/2 + 1/5) of 20 = (10 + 4) = 14; one-third of 20 = 20/3 ≈ 6.666…; difference 14 − 6.666… = 7.333… = 7 1/3, correct.
Why Other Options Are Wrong:
- 15, 18, 30: Do not satisfy the exact fractional relation; plugging them in fails the difference test.
Common Pitfalls:Misconverting 7 1/3 to an improper fraction or adding fractions without a common denominator.
Final Answer:
20