Final answer:
The pattern established between the first and second number is to square the first number and multiply it by 5. Applying this to the second pair, we square 6 to get 36, then multiply by 5 to reach 180, which is not among the provided options.
Step-by-step explanation:
The question given is an analogy problem where you have to determine the relationship between the first pair of numbers to apply that same relationship to the second pair. Looking at the first pair, 4 is related to 80. To find this relationship, notice that 4 times 4 equals 16, and if you multiply that by 5, you get 80. Hence, the pattern is to square the first number and multiply the result by 5. Applying this pattern to the second pair, if the first number is 6, then squaring it gets us 36, and multiplying that by 5 gives us 180.
The relation looks like this: 4 : 4<sup>2</sup> × 5 :: 6 : 6<sup>2</sup> × 5. Therefore, following this pattern, the answer is 180, which is not listed in the provided options. So, it seems there might be a mistake in the question, or perhaps an alternative pattern was intended. However, based on the pattern we have deduced, 180 appears to be the proper solution in line with the relationship established between 4 and 80.
To find the relationship between the second and third numbers, we need to examine the relationship between the first and second numbers. In this case, the first number (4) is multiplied by 20 to get the second number (80). Therefore, we can deduce that the second number is obtained by multiplying the first number by 20. Applying the same logic, we can multiply the third number (6) by 20 to find the missing number:
6 * 20 = 120
So, the missing number is 120.