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Input Output Tables

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An input/output table gives pairs of numbers that follow a certain pattern or rule.

In this input/output table, the rule is that 2 has been added to each input.

Input
Output
5
7
10
12
1
3
6
8

To determine the rule for a table, simply look at one pair of input/output numbers and figure out what number has been added, subtracted, multiplied, or divided by it.

Then, make sure that pattern matches the rest of the numbers.

Once you know the rule for a table, you can finish filling in the rest of it.

Look at the input and apply the rule.

For example, if the rule is "Subtract 5" and the input is 9 then you subtract 5 from 9 to get 4, and 4 is the output.

Let's take a look at some examples:

Example 1

 

What is the rule for this table?

Input
Output
5
15
4
12
2
6
7
21

Looking at the first pair (5 and 15), we can tell that either 10 was added to 5 or 5 was multiplied by 3.

Let's see which pattern works for the rest of the table.

For the second pair (4 and 12) we can see that 10 was NOT added (only 8 was added.) But, 4 times 3 is 12,

So, it appears the rule is "multiply by 3."

Let's make sure this rule works for the rest of the table.

2 x 3 = 6 and 7 x 3 = 21, so yes, the rule is "multiply by 3."

If you needed to write this rule using a variable, you would say "3 times x" which can be written as 3x.

Remember that a number next to a variable means multiplication.

That's because x stands for the input. So the output is always 3 times the input (x).

 

Example 2

 

Find the rule for the table and finish filling it in.

Input
Output
10
6
8
4
11
7
4
0
5
-
20
-

First we must figure out the rule.

What's happened with the first pair of numbers (10 and 6)?

It looks like 4 has been subtracted by 6.

Let's see if this rule works for the rest of the table.

8 - 4 = 4, 11 - 4 = 7, and 4 - 4 = 0.

So, yes, the rule is "subtract 4."

So, yes, the rule is "subtract 4."

What would this rule be written with a variable?

It would be x - 4.

(because you start with x - the input - then subtract 4.)

Now we can fill in the rest of the table.

The next input is 5, and the rule is to subtract 4.

So what is 5 - 4?

1, of course.

This goes in the output spot. Now our table looks like this:

Input
Output
10
6
8
4
11
7
4
0
5
1
20
-

The last input is 20, and the rule is still to subtract 4.

20 - 4 = 16, so that goes in the output spot.

The final answer is that the rule is "subtract 4" and the filled-in table is

Input
Output
10
6
8
4
11
7
4
0
5
1
20
16
 

Example 3

 

Complete this table. The rule is "divide by 2 then add 1" (or, written as a variable: x ÷ 2 + 1)

Input
Output
20
-
6
-
10
-
18
-

In this case, we are told the rule and have to fill in the table.

Notice that the rule has two steps:

divide by 2 and then add 1.

Let's start with the first input (20).

We have to first divide by 2 then add 1.

20 ÷ 2 = 10 and then 10 + 1 = 11.

So 11 goes in the output spot.

Now the table looks like this:

Input
Output
20
11
6
-
10
-
18
-

For the next input (6), we follow the same rule (divide by 2 then add 1).

6 ÷ 2 = 3 and then 3 + 1 = 4.

4 is the output for 6.

Input
Output
20
11
6
4
10
-
18
-

For the next input (10), 10 ÷ 2 = 5 and then 5 + 1 = 6.

6 is the next output.

10 ÷ 2 = 5 and then 5 + 1 = 6.

6 is the next output.

Input
Output
20
11
6
4
10
6
18
-

For the last input (18), 18 ÷ 2 = 9 and then 9 + 1 = 10. 10 is the last output.

18 ÷ 2 = 9 and then 9 + 1 = 10.

10 is the last output.

Input
Output
20
11
6
4
10
6
18
10
 

Example 4

 

Which rule was made to create this table? y = 3x + 4, y = 6x + 2, y = 2x + 5

Input (x)
Output (y)
1
7
2
9
3
11
4
13

Notice that the input is x and the output is y. To see which rule was used to create the table, we must test the pairs of input/outputs (x/y's) in the different equations.

Let's test the first pair: x = 1 and y = 7. Substitute this into the first equation

y = 3x + 4

7 = 3(1) + 4

7 = 3 + 4

7 = 7

This works, so this could be the right equation. But we're not sure yet.

Let's try the second pair in this equation: x = 2 and y = 9

y = 3x + 4

9 = 3(2) + 4

9 = 6 + 4

9 = 10

Wait, this isn't right.

So, this was not the rule used to create the table.

Let's go back and try the first pair in the second equation:

y = 6x + 2

7 = 6(1) + 2

7 = 6 + 2

7 = 8

This doesn't work, so this cannot be the right rule.

This cannot be the right rule.

It seems as if the table must made from the last rule, but we want to make sure. So let's try the first pair (x = 1 and y = 7) in the last equation: y = 2x + 5

y = 2x + 5

y = 2x + 5

y = 2x + 5

y = 2x + 5

7 = 2(1) + 5

7 = 2 + 5

7 = 7

This works. Let's try another pair.

Let's try x = 2 and y = 9

y = 2x + 5

9 = 2(2) + 5

9 = 4 + 5

9 = 9

This works again, so it appears this is the rule.

To be positive, let's check the last two pairs.

Test x = 3 and y = 11

y = 2x + 5:

11 = 2(3) + 5

11 = 6 + 5

11 = 11 True!

Test x = 4 and y = 13

13 = 2(4) + 5

13 = 8 + 5

13 = 13

True!

The rule used to create the table was y = 2x + 5

Summary
  • An input/output table gives pairs of numbers that follow a certain pattern or rule.
  • To determine the rule for a table, simply look at one pair of input/output numbers to figure out the operation involved.
  • Once you know the rule for a table, you can finish filling in the rest of it by merely looking at the input and apply the rule.

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