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Input-Output
Rules

Learn to find the secret rule that turns each input number into its output number.

ISEE Quantitative Reasoning

The Number Machine

Imagine a machine. You put a number in, the machine follows a secret rule, and a new number comes out.

RULE add 3 INPUT 5 OUTPUT 8

Three Important Words

Input
The number you start with, before the rule is used.
Rule
The math step (like \( +3 \) or \( \times 2 \)) the machine does.
Output
The number that comes out after the rule is used.

Reading a Table

A table shows several inputs and their outputs. Look across each row to see one pair.

InputOutput
26
37
59

The Big Idea

The same rule must turn each input into its matching output. If a rule works for one row but fails on another, it is the wrong rule.

Two Common Rule Types

Adding / Subtracting
The output is always a fixed amount more (or less) than the input, like \( +4 \) or \( -2 \).
Multiplying / Dividing
The output is the input times (or divided by) a number, like \( \times 3 \) or \( \div 2 \).

Add vs. Multiply

How can you tell which kind of rule a table uses? Look at the gap between input and output as numbers grow.

Adding RuleMultiplying Rule
Example\( +5 \)\( \times 3 \)
Input 2becomes 7becomes 6
Input 10becomes 15becomes 30
The gapstays the samegrows bigger

How to Find the Rule

1
Pick one row. Look at the first input and its output.
2
Ask what changed. Did the number get bigger or smaller? By how much?
3
Guess a rule such as \( +3 \) or \( \times 2 \).
4
Test every row. The rule must work for all of them.

Worked Example 1

InputOutput
15
48
711

Worked Example 2

InputOutput
26
39
515

Worked Example 3

InputOutput
13
25
49

Common Mistakes

✕ Only checking one row
A rule that fits row 1 may fail on row 2. Always test every row.
✕ Mixing up the direction
Read input on the left going to output on the right, not backwards.
✕ Forgetting to multiply
If adding fails, try multiplying before giving up on the table.

Quick Summary

StepWhat to do
1. CompareLook at input and output in one row
2. GuessTry \( +n \) first, then \( \times n \)
3. TestCheck the rule on every row
4. UseApply the rule to find a missing number

Question 1 of 5

A table shows: \( 3\to 8 \), \( 5\to 10 \), \( 9\to 14 \). What is the rule?

Question 2 of 5

A table shows: \( 2\to 8 \), \( 3\to 12 \), \( 5\to 20 \). What is the rule?

Question 3 of 5

The rule is add 7. If the input is \( 6 \), what is the output?

Question 4 of 5

A table shows: \( 1\to 4 \), \( 2\to 7 \), \( 3\to 10 \). What is the rule?

Question 5 of 5

The rule is multiply by 2. The table has \( 3\to 6 \), \( 4\to 8 \), \( 7\to\;? \). What is the missing output?

You've Got It!

1
Input goes in, output comes out following one secret rule.
2
Compare a row to guess \( +n \) or \( \times n \).
3
Always test every row before trusting your rule.

Keep Practicing!

Every table hides a rule. Compare, guess, and test until you find it. You are now a rule detective!