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Triangular Array of Apples

XAT 2019 · Quantitative Ability & Data Interpretation · Medium · Arithmetic

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When opening his fruit shop for the day a shopkeeper found that his stock of apples could be perfectly arranged in a complete triangular array: that is, every row with one apple more than the row immediately above, going all the way up ending with a single apple at the top.

During any sales transaction, apples are always picked from the uppermost row, and going below only when that row is exhausted.

When one customer walked in the middle of the day she found an incomplete array in display having 126 apples totally. How many rows of apples (complete and incomplete) were seen by this customer? (Assume that the initial stock did not exceed 150 apples.)

  1. A.

    15

  2. B.

    14

  3. C.

    13

  4. D.

    12

  5. E.

    11

Answer

D

Explanation

Total initial apples SN=N(N+1)2≤150S_N = \frac{N(N+1)}{2} \le 150. For N=16N=16, S16=16×172=136S_{16} = \frac{16 \times 17}{2} = 136. For N=15N=15, S15=120<126S_{15} = 120 < 126. So original rows N=16N = 16, total apples = 136. Customer sees 126 apples, so 136−126=10136 - 126 = 10 apples were sold. Number of top rows removed = kk. k(k+1)2=10⇒k=4\frac{k(k+1)}{2} = 10 \Rightarrow k = 4. Remaining rows visible = 16−4=1216 - 4 = 12.

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