Problem Analysis
This problem requires us to find the number of pencils Anushka has, given the counts of erasers, the relationship between pencils and sharpeners, and the total number of items.
- Number of erasers (E) = 4.
- Number of sharpeners (S) is one-fourth the number of pencils (P): $S = \frac{1}{4}P$.
- Total number of items (erasers + pencils + sharpeners) = 49.
Mathematical Setup
We can express the total number of items as an equation:
$E + P + S = 49$Substitute the known value of E:
$4 + P + S = 49$Isolate the terms involving pencils and sharpeners:
$P + S = 49 - 4$ $P + S = 45$Solving for Pencils
Now, substitute the relationship $S = \frac{1}{4}P$ into the equation $P + S = 45$:
$P + \frac{1}{4}P = 45$To solve for P, combine the terms with P:
$\frac{4}{4}P + \frac{1}{4}P = 45$ $\frac{5}{4}P = 45$Isolate P:
$P = 45 \times \frac{4}{5}$ $P = \frac{180}{5}$ $P = 36$Verification
If the number of pencils (P) is 36:
- Number of sharpeners (S) = $\frac{1}{4} \times 36 = 9$.
- Total items = Erasers + Pencils + Sharpeners = $4 + 36 + 9 = 49$.
The calculated total matches the given total, confirming our answer.
Anushka has 36 pencils.
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