In the number analogy "7.25 : 7.75 :: 9 : ____", which value completes the pattern with the same increase?

Introduction / Context:
This numerical analogy deals with decimal numbers and a constant difference between the first and second element of each pair. The relationship "7.25 : 7.75" shows that some fixed amount has been added to 7.25 to obtain 7.75. You must identify this increment and apply it to 9 in order to choose the correct missing value from the options. Such questions help test basic arithmetic skills, especially with decimal fractions.

Given Data / Assumptions:

• First pair: 7.25 is related to 7.75.
• Second pair: 9 is related to an unknown value.
• Options: 9.5, 7, 10, 8.5, 8.25.
• We assume a simple addition pattern rather than complex operations.

Concept / Approach:
First check the difference between 7.25 and 7.75. Subtract the smaller from the larger. The difference is 7.75 minus 7.25, which equals 0.50. Therefore, the rule in the first pair is: add 0.50 to the first number to obtain the second number. To maintain the analogy, we must add the same amount, 0.50, to 9. This yields 9.50, which can be written as 9.5. That value must be selected from the options.

Step-by-Step Solution:

Compute the difference in the first pair: 7.75 - 7.25 = 0.50. Recognise that the rule is "add 0.50" or "add half a unit". Apply the rule to the number 9: 9 + 0.50 = 9.50. Rewrite 9.50 in standard decimal form as 9.5. Locate 9.5 in the options and select it as the correct answer.

Verification / Alternative check:
Check alternative options briefly. If you add 1 instead of 0.5, you would get 10, which does not match the pattern of the first pair. Adding 0.25 would give 9.25, which is not listed. Subtracting 0.5 would give 8.5, which also does not correspond to the increase seen in the original numbers. The only consistent rule is addition of 0.5, and this leads uniquely to 9.5.

Why Other Options Are Wrong:
The values 7, 10, 8.5 and 8.25 do not result from adding 0.5 to 9. Choosing any of them would mean applying a different increment than the one used between 7.25 and 7.75. Since analogies require the same operation to be used in both pairs, these options are incorrect.

Common Pitfalls:
One common error is to try to match whole numbers only and ignore the decimal fraction, which may lead candidates to choose 10 or 8. However, careful subtraction of the decimal values reveals the true pattern. Another pitfall is confusion between adding and subtracting, which can be avoided by explicitly performing the calculation rather than guessing.

Final Answer:
Using the rule "add 0.5", 7.25 maps to 7.75 and 9 maps to 9.5. So the completed analogy is "7.25 : 7.75 :: 9 : 9.5".