Question

David notices this pattern.

19= 1 x 9 + 1 + 9

29= 2 x 9 + 2+ 9

39= 3 x 9 + 3+ 9

Based on this pattern, David concludes that any
two-digit number ending in 9.is equal to n x 9 + n + 9, where n is the tens digit of the
number.

Part a. Is this an example of inductive or deductive
reasoning? Explain.

Part b. Is David's conclusion correct? Support your
answer.

Answer 1

Part a: David's pattern is an example of inductive reasoning.

Part b: Yes, David's conclusion is correct.

Explanation:

• Inductive reasoning is a type of reasoning that draws a general conclusion from specific observations.
  It is based on the assumption that if something is true in a few cases, it is likely to be true in all cases.
• Deductive reasoning is a type of reasoning that moves from general principles to a specific conclusion.
  It is based on the assumption that if the premises are true, then the conclusion must also be true.

Part a:

Inductive reasoning is a type of reasoning that draws a general conclusion from specific observations.

In this case,

David observed a pattern in the three examples he provided and then generalized that any two-digit number ending in 9 can be expressed as:

`\sf n * 9 + n + 9`

where n is the tens of the number.

So,

David's pattern is an example of inductive reasoning.

`\hrulefill`

Part b:

We can verify this by testing it with a few examples.

For instance, let's consider the two-digit number 59.

The tens digit of 59 is 5.

Therefore, according to David's formula,

We should have:

`\sf 59 = 5 * 9 + 5 + 9 = 45+5 + 9 =59`

This is indeed true.

We can also test this formula with other two-digit numbers ending in 9 and we will find that it holds true for all such numbers.

Therefore, we can conclude that David's formula is correct for any two-digit number ending in 9.