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Each piglet in a litter is fed exactly \(\frac{1}{2}\) pound of a mixture of oats and barley. The ratio of the amount of barley to that of oats varies from piglet to piglet, but each piglet is fed some of both grains. how many piglets are there in the litter?
We need to find the exact number of piglets in the litter.
- Each piglet gets exactly \(\frac{1}{2}\) pound of feed (mixture of oats and barley)
- The ratio of barley to oats varies from piglet to piglet
- Each piglet gets some of both grains (no piglet gets only one type)
For sufficiency, we need information that leads to one specific number of piglets—not a range or multiple possibilities.
Since each piglet gets exactly \(\frac{1}{2}\) pound, if we can determine the total amount of feed used, we can find the number of piglets using:
\(\mathrm{Number\ of\ piglets} = \mathrm{Total\ feed} \div 0.5\)
Statement 1: Piglet A received exactly \(\frac{1}{4}\) of all the oats in the litter.
This tells us what fraction of the total oats goes to one specific piglet. However, we don't know:
- The total amount of oats
- How much barley Piglet A received
- How the remaining oats and all the barley are distributed
Without these crucial pieces, we cannot determine the total feed amount or the number of piglets.
Example: If there were 2 pounds of oats total, Piglet A would get 0.5 pounds of oats. But Piglet A also needs some barley (since each piglet gets both grains). Without knowing the barley amounts, we can't determine how many piglets there are.
Statement 1 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: Piglet A received exactly \(\frac{1}{6}\) of all the barley in the litter.
Similar to Statement 1, this gives us a fraction of one type of grain going to one piglet. We still don't know:
- The total amount of barley
- How much oats Piglet A received
- How the remaining barley and all the oats are distributed
Again, without these pieces of information, we cannot determine the total feed amount or the number of piglets.
Statement 2 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice B.
Now we know that Piglet A gets:
- \(\frac{1}{4}\) of all oats
- \(\frac{1}{6}\) of all barley
- These must total to \(\frac{1}{2}\) pound (since every piglet gets exactly \(\frac{1}{2}\) pound)
Here's where the magic happens. The fact that specific fractions (\(\frac{1}{4}\) and \(\frac{1}{6}\)) of the total oats and barley must sum to exactly \(\frac{1}{2}\) pound creates a unique mathematical constraint.
Why this constraint is so powerful:
Think of it this way—if \(\frac{1}{4}\) of the total oats plus \(\frac{1}{6}\) of the total barley equals \(\frac{1}{2}\) pound, there's only one specific relationship between the total oats and total barley that works.
To visualize: Let's say there are O pounds of oats and B pounds of barley in total.
- Piglet A gets: \(\left(\frac{1}{4}\right)\mathrm{O} + \left(\frac{1}{6}\right)\mathrm{B} = \frac{1}{2}\) pound
This single equation, combined with the fact that all feed must be distributed evenly in \(\frac{1}{2}\) pound portions, uniquely determines:
1. The relationship between total oats and total barley
2. The total amount of feed \((\mathrm{O} + \mathrm{B})\)
3. Since each piglet gets exactly \(\frac{1}{2}\) pound, this gives us exactly one possible number of piglets
Both statements together are sufficient.
[STOP - Sufficient!] This eliminates choice E.
The specific fractions of total grains that Piglet A receives (\(\frac{1}{4}\) of oats and \(\frac{1}{6}\) of barley), combined with the constraint that each piglet gets exactly \(\frac{1}{2}\) pound, uniquely determines the number of piglets in the litter.
Answer Choice C: Both statements together are sufficient, but neither statement alone is sufficient.