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

Strategy:

We need to find the logical pattern connecting the numbers 80 and 99 and then find an option pair that follows the same pattern.

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 \(x=9\).

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.

The pair 120 : 143 shares the same relationship as 80 : 99.