7. A table is presented with input and output values that need to be analyzed to determine the mathematical rule governing the relationship. The table includes the following pairs: input 2 with output 18, input 5 with output 30, input 7 with output 38, and input 8 with output 42. The task is to identify which rule correctly computes the output based on the given input values. Which of the following rules is used to compute the output given the input in the table shown?
Answer: C
The output is 4 × the input, plus 10.
To compute the output based on the input values in the table, the correct rule is that the output is calculated as 4 times the input value, plus 10.
A) The output is 9 × the input.
This option is incorrect because if we apply the rule of 9 times the input, we would see outputs such as 18 for input 2 (which is correct), but for input 5, the output would be 45, not 30. Thus, this rule does not fit the provided output values.
B) The output is 5 × the input, plus 5.
This option is also incorrect. If we calculate the output using this rule, for input 2, the output would be 15 (5 × 2 + 5), and for input 5, it would yield 30 (5 × 5 + 5), which does align with the table but fails for input 7, where the output would be 40, not the required 38.
C) The output is 4 × the input, plus 10.
This option is correct as it accurately reflects the outputs given the inputs from the table. For input 2, the calculation is 4 × 2 + 10 = 18; for input 5, it results in 4 × 5 + 10 = 30; for input 7, it gives 4 × 7 + 10 = 38; and for input 8, it results in 4 × 8 + 10 = 42, fitting all pairs perfectly.
D) The output is 6 × the input, minus 6.
This option is incorrect because applying this rule would produce an output of 6 for input 2 (6 × 2 - 6 = 6), which does not match the expected output of 18. Similarly, for input 5, it would yield 24, which does not match 30, thus failing to meet the criteria set by the table.
Conclusion
The correct answer is option C because it consistently produces the expected outputs for all given inputs in the table. All other options fail to accurately compute at least one of the provided input-output pairs, demonstrating that they do not represent the mathematical rule governing the relationship accurately.