Question

Select the option in which the numbers share the same relationship as that shared by the given pair of numbers.
(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.)
80 : 99

• A. 25 : 122
• B. 79 : 90
• C. 110 : 169
• D. 120 : 143

Hint:

When dealing with number analogies, if the numbers are close to well-known squares or cubes (like 80 is near 81, 99 is near 100), always check for patterns like \(n^2+1\), \(n^2-1\), \(n^3+1\), etc.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to find the logical pattern connecting the numbers 80 and 99 and then find an option pair that follows the same pattern.
Step 2: Key Formula or Approach:
The numbers are close to perfect squares. This suggests a pattern involving squares of integers. Let's analyze the given pair: - 80 can be written as \(81 - 1\), which is \(9^2 - 1\). - 99 can be written as \(100 - 1\), which is \(10^2 - 1\). The relationship is \(x^2 - 1 : (x+1)^2 - 1\), where \(x=9\).
Step 3: Detailed Explanation:
Now we will apply this pattern, \(x^2 - 1 : (x+1)^2 - 1\), to each of the options.
(A) 25 : 122 If \(x^2 - 1 = 25\), then \(x^2 = 26\). Here, x is not an integer. So, this option is incorrect.
(B) 79 : 90 If \(x^2 - 1 = 79\), then \(x^2 = 80\). Here, x is not an integer. So, this option is incorrect.
(C) 110 : 169 If \(x^2 - 1 = 110\), then \(x^2 = 111\). Here, x is not an integer. So, this option is incorrect.
(D) 120 : 143 Let's test the first number: \(x^2 - 1 = 120\). This gives \(x^2 = 121\), which means \(x = 11\). Now, let's check if the second number follows the pattern \((x+1)^2 - 1\). \((11+1)^2 - 1 = 12^2 - 1 = 144 - 1 = 143\). This matches the second number in the option. So, this pair follows the same relationship.
Step 4: Final Answer:
The pair 120 : 143 shares the same relationship as 80 : 99.