Question

In the figure, the number in any cell is obtained by adding the two numbers in the cells directly below it. For example, the 9 marked in the figure is obtained by adding the 4 and 5 directly below it.

The value of X - Y is

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

Write the row above the base first using the addition rule, then match the cell the figure labels "Y + 29" with the same cell built from the base row using X.

Answer 1

The Correct Option is C

Step 1: Set up the base row and the rule.
Every cell in this pyramid is the sum of the two cells sitting directly below it. The base row, read left to right, is Y, 4, 5, 2, X. We climb the pyramid one row at a time using this rule.

Step 2: Build the row just above the base.
Add each pair of neighbours in the base row. The first pair gives Y + 4. The second pair gives 4 + 5 = 9, and this is exactly the 9 marked on the figure, so we know we are reading the rows correctly. The third pair gives 5 + 2 = 7. The fourth pair gives 2 + X. So this row reads: Y + 4, 9, 7, X + 2.

Step 3: Build the next row (three cells).
Add neighbouring pairs from the row in Step 2. (Y + 4) + 9 = Y + 13. Next, 9 + 7 = 16. Next, 7 + (X + 2) = X + 9. So this row reads: Y + 13, 16, X + 9.

Step 4: Build the row of two cells and use the given label.
Adding the last pair from Step 3 gives 16 + (X + 9) = X + 25. The figure already marks this exact cell as Y + 29. Since two ways of writing the same cell must agree:
\[ X + 25 = Y + 29 \]

Step 5: Solve for X - Y.
Move the Y-terms and constants across the equals sign:
\[ X - Y = 29 - 25 = 4 \]

Step 6: Check the answer against the apex.
The other cell in that same row is (Y + 13) + 16 = Y + 29 as well, so both cells of this row equal Y + 29, which also equals X + 25. Call this shared value k. The apex is the sum of these two cells, so \(2k = 68\), giving \(k = 34\). Then \(Y + 29 = 34\) gives \(Y = 5\), and \(X + 25 = 34\) gives \(X = 9\). Rebuilding the whole pyramid from the base row 5, 4, 5, 2, 9 gives exactly 68 at the apex, so these values are consistent.

Final Answer:
X - Y = 4, which is option (CC).\[ \boxed{X-Y=4} \]