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In the following number pairs, the second number is obtained by applying certain mathematical operations to the first number. Which numbers should replace X and Y so that the pattern followed by the two numbers on the left side of:: is the same as that on the right side of::? (NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 - Operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
X = 7, Y = 538
X = 4, Y = 512
X = 5, Y = 546
X = 6, Y = 488
X = 6, Y = 488
Look at the relation between the numbers: 8 cubed is 512. Check if the pattern is n cubed + n squared or n cubed - n. Here, 8 cubed (512) + 8 squared (64) = 576, not 8:Y. Wait, let's test cube minus something or cube plus something.
, let X = 5, then 5 cubed + 5 squared + 5 = 125 + 25 + 5 = 155. Let's check the official option D: X=6, Y=488. If X=6: 6 cubed (216) +... no. Let's test n cubed + n squared or n times n squared. If X=6, 6 cubed + 6 = 216 + 6 = 222. Wait! For Option D: X = 6, Y = 488. Let's check 6 and 488: 8 cubed = 512, 512 - 24 = 488 (8 $\times 3).$For X=6: 6 cubed = 216, 216 - 18 = 198 (6 $\times 3)$! Perfect! Rule: First number n gives second number n cubed minus 3n.
Look at the relation between the numbers: 8 cubed is 512. Check if the pattern is n cubed + n squared or n cubed - n. Here, 8 cubed (512) + 8 squared (64) = 576, not 8:Y. Wait, let's test cube minus something or cube plus something.
, let X = 5, then 5 cubed + 5 squared + 5 = 125 + 25 + 5 = 155. Let's check the official option D: X=6, Y=488. If X=6: 6 cubed (216) +... no. Let's test n cubed + n squared or n times n squared. If X=6, 6 cubed + 6 = 216 + 6 = 222. Wait! For Option D: X = 6, Y = 488. Let's check 6 and 488: 8 cubed = 512, 512 - 24 = 488 (8 $\times 3).$For X=6: 6 cubed = 216, 216 - 18 = 198 (6 $\times 3)$! Perfect! Rule: First number n gives second number n cubed minus 3n.
Step 1: Analyze the Right Pair (8 : Y)
Given the first number is 8. Let us apply common mathematical relationships involving cubes and squares. For 8, 8 cubed is 512. Subtracting 3 times 8 (which is 24) from 512 gives 512 - 24 = 488. This matches Y = 488 in Option D.
Step 2: Verify the Rule with X : 198
Test the rule f(n) = n cubed - 3n on X = 6. We get 6 cubed minus 3 times 6 = 216 - 18 = 198. This perfectly matches the second number 198 on the left side.
Step 3: Confirm Option D
Since both X = 6 and Y = 488 satisfy the exact same relation n cubed minus 3n, Option D is the correct answer.
A: X=7 and Y=538 does not satisfy n cubed minus 3n for either pair. B: X=4 and Y=512 gives 64 - 12 = 52 for X, not 198. C: X=5 and Y=546 gives 125 - 15 = 110 for X, not 198.
D is correct because for any number n, applying the formula n cubed minus 3n yields the second number, giving X = 6 and Y = 488.
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