Introduction / Context:
We are given totals and proportions for two equal groups of trees (sandal and Ashoka). The task is to infer a statement guaranteed by the data, regardless of how flowering trees are distributed.
Given Data / Assumptions:
- Total trees = T.
- Sandal = T/2; Ashoka = T/2.
- Old trees = 3T/4.
- Flowering trees = T/2 (distribution unspecified).
Concept / Approach:
- Use bounds: find the minimum number of old Ashoka trees consistent with the totals.
- Maximize old sandal trees to minimize old Ashoka trees, then compute the remainder of old trees that must belong to Ashoka.
Step-by-Step Solution:
Let old sandal = at most T/2 (all sandal trees old).Old total = 3T/4, so old Ashoka = 3T/4 − T/2 = T/4.Since Ashoka = T/2, old Ashoka proportion ≥ (T/4) / (T/2) = 1/2.Thus, at least one-half of the Ashoka trees are old, even in the extreme case.
Verification / Alternative check:
Any reduction in old sandal increases old Ashoka beyond T/4, so the one-half lower bound is safe.
Why Other Options Are Wrong:
a,b: “All” flowering is not guaranteed—flowering trees are T/2 and may be split in any way. d: “One-half of sandal are flowering” is also not forced by the data.
Common Pitfalls:
Assuming specific splits of flowering trees without evidence.
Final Answer:
At least one-half of the Ashoka trees are old.
Discussion & Comments