Could be a rule for this input-output pair

Question

Could not be a rule for this input-output pair

Sign

Double the input, then add 15.

Multiply by 3, then add 5.

If even, write 19. If odd, subtract 3.

Multiply by 2, then subtract 18.

Multiply by 9, then add 1.

Add 10.

Answer 1

Let's analyze the input-output pairs to identify the potential rule based on the given operations. For each operation:

1. Could be a rule for this input-output pair:

   • "Sign": This might be referring to something higher up on an abstraction where it can be an indication. However, it may likely indicate a categorization based on the positivity or negativity of input.
   • "Double the input, then add 15": This is a clear mathematical operation that could be applicable depending on what the input is.
   • "Multiply by 3, then add 5": Again, this describes a straightforward operation that could yield various outputs based on different inputs.
   • "If even, write 19. If odd, subtract 3": This rule distinguishes between even and odd numbers and provides a different outcome based on that parity.
   • "Multiply by 2, then subtract 18": Another clear operation where the outcome can be easily determined from the input.
   • "Multiply by 9, then add 1": Once more, a mathematical operation would define the output based on a straightforward input.
   • "Add 10": This could straightforwardly apply to any numeric input.

2. Could not be a rule for this input-output pair:

   • The term "Sign" may not have a definitive mathematical operation and could be more of a categorization rather than a rule that produces a predictable output.

The core differentiator here is whether the suggested "rule" provides a consistent or predictable output for all possible inputs within the defined constraints.

In conclusion:

• The operations involving specific mathematical steps (doubling, multiplying, adding) can function as rules for input-output pairs.
• "Sign" appears to be less straightforward and cannot clearly be mapped to an input-output relationship, making it an unlikely candidate for a rule.

Thus, while the specific operations mentioned are viable rules for calculation, "Sign" does not fit well into this structure, making it the likely choice for having not being a rule for this input-output pair.

Answer 2

Could be a rule for this input-output pair

Could not be a rule for this input-output pair

Double the input, then add 15.

Multiply by 3, then add 5.

If even, write 19. If odd, subtract 3.

Multiply by 2, then subtract 18.

Multiply by 9, then add 1.

Add 10.

What ones could or couldn't be a rule

Answer 3

To determine which operations could or could not be a rule for a given input-output pair, we need to evaluate them based on whether they produce a consistent and predictable output for all possible inputs. Let's examine each operation:

Could be a rule:

1. Double the input, then add 15:

   • This is a clear numerical transformation that can be applied to any input.
   • For example, with an input \( x \): \[ \text{Output} = 2x + 15 \]

2. Multiply by 3, then add 5:

   • Similar to the first, this is a well-defined operation.
   • Example: \[ \text{Output} = 3x + 5 \]

3. Multiply by 9, then add 1:

   • This also defines a consistent relationship.
   • Example: \[ \text{Output} = 9x + 1 \]

4. Add 10:

   • A simple operation that can be universally applied.
   • Example: \[ \text{Output} = x + 10 \]

5. Multiply by 2, then subtract 18:

   • This operation consistently transforms the input.
   • Example: \[ \text{Output} = 2x - 18 \]

Could not be a rule:

1. If even, write 19. If odd, subtract 3:
   • This operation is conditional based on the parity (even or odd) of the input, meaning it produces different outputs for different classes of inputs. It does not consistently apply the same mathematical operation across all inputs, therefore it cannot be a universal rule for every input-output pair.

Summary:

Rules that could be applied to an input-output pair:

• Double the input, then add 15.
• Multiply by 3, then add 5.
• Multiply by 9, then add 1.
• Multiply by 2, then subtract 18.
• Add 10.

Rule that would not work for an input-output pair:

• If even, write 19. If odd, subtract 3.

The conditional nature of the last rule makes it inconsistent as a general rule for any arbitrary input.