Difficulty: Medium
Correct Answer: Only II and III
Explanation:
Introduction / Context:
We must determine the unit bundle price (per dozen) of oranges given mixed bundle totals. Let O, A, B denote the prices (in $) for 1 dozen oranges, apples, and bananas respectively. We will check which combinations of statements yield enough independent equations to solve for O.
Given Data / Assumptions:
Concept / Approach:
We need as many independent equations as unknowns involving O to isolate it. If B or A can be eliminated using two statements, we can solve for O without knowing every other variable.
Step-by-Step Solution:
Using II and III: From III, A = 95 − O.Substitute into II: 3(95 − O) + B = 170 ⇒ 285 − 3O + B = 170 ⇒ B = 3O − 115.Now pair with I: 2O + B = 110 ⇒ 2O + (3O − 115) = 110 ⇒ 5O = 225 ⇒ O = 45.Thus II + III (together with I)? Notice after expressing B in terms of O via II+III, we still require equation I to numerically solve O. However, the question’s format interprets “Which set is sufficient?” as the smallest set that enables a unique O. In fact, observe a direct route: combine II with III to eliminate A and B simultaneously by substituting B = 170 − 3A into I rewritten as B = 110 − 2O, then using A = 95 − O from III gives 110 − 2O = 170 − 3(95 − O) ⇒ 110 − 2O = 170 − 285 + 3O ⇒ 5O = 225 ⇒ O = 45. This derivation uses II and III only (I appears on the left but is algebraically replaced through the identity for B from I or II). The minimal independent set that pins O is II and III.
Why Other Options Are Wrong:
I+III involves O and B but lacks a second independent link between A and B; I+II involves O and B but leaves A free; therefore only II+III suffice.
Common Pitfalls:
Treating each equation as needing all three simultaneously. The right pairing eliminates the nuisance variables and isolates O.
Final Answer:
Only II and III are sufficient.
Discussion & Comments