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CogAT (Grades 3+) • Quantitative Battery

Identify the relationship in a number pair

Learn to spot the hidden rule connecting two numbers so you can crack any number-pair puzzle.

Explore the lesson 10 min read
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Why Number Relationships Matter

People have been looking for patterns in numbers for thousands of years. Ancient mathematicians noticed that certain numbers were always connected by a rule — like doubling or adding a fixed amount. Finding these hidden rules is one of the oldest skills in math.

On the CogAT Quantitative Battery, you will see pairs of numbers. Your job is to figure out how the first number turns into the second number. This section shows how that idea grew over time.

~300 BC • Euclid's Ratios

The Greek mathematician Euclid studied how two quantities relate. He wrote down rules for ratios, like "3 is to 6 as 5 is to 10."

~800 AD • Al-Khwarizmi & Algebra

The Persian scholar Al-Khwarizmi developed algebra. He showed how to describe number relationships using symbols and equations.

1600s • Functions Are Born

Mathematicians like Leibniz introduced the idea of a function — a machine that takes an input number and produces an output number using a rule.

1968 • CogAT Is Created

The Cognitive Abilities Test was first published. Its Quantitative Battery tests how well students can spot number-pair relationships without being told the rule.

The big question these thinkers were all asking is the same one you will answer on the CogAT: What rule connects one number to another?

SECTION 2

Core Principles & Definitions

A number pair is simply two numbers written together, like (3, 9). The first number is the input and the second is the output. A relationship (or rule) is the operation you do to the input to get the output.

1

Addition / Subtraction Rule

A fixed number is added to or subtracted from the input. Example: (4, 11) → the rule is add 7.
2

Multiplication / Division Rule

The input is multiplied or divided by a fixed number. Example: (5, 20) → the rule is multiply by 4.
3

Power / Square Rule

The input is raised to a power (squared, cubed, etc.). Example: (3, 9) → the rule is square it (3² = 9).
4

Two-Step Rule

Two operations work together. Example: (2, 7) → the rule is multiply by 3, then add 1 (2 × 3 + 1 = 7).
✦

KEY TAKEAWAY

Think of a number pair like a vending machine. You put coins in (the input), the machine does something inside (the rule), and a snack comes out (the output). Your job on the CogAT is to figure out what the machine does on the inside — even though you can't see it!

SECTION 3

Seeing the Rule in Action

The diagram below shows how an input number goes through a rule to become an output number. Study the arrows and labels to see how each pair is connected.

Number-Pair Rule Machine4InputRULE× 3 + 214Output7RULE× 3 + 22310RULE× 3 + 232
Each input (purple border) enters the rule box ("× 3 + 2") and comes out as the output (green border). The same rule works for every pair: 4 → 14, 7 → 23, and 10 → 32.

Notice that the same rule works for every pair. That is the key idea: the rule never changes between pairs in the same problem. Once you find it for one pair, it should work for all the others.

SECTION 4

The Math Behind Number-Pair Rules

You can write any number-pair relationship as a simple formula. The input is often called a and the output is called b. Below are the most common types of rules you will see.

ADDITION RULE

b=a+kb = a + k

Here, k is the constant you add. Example: if k = 5, then (3, 8) follows this rule because 3 + 5 = 8.

MULTIPLICATION RULE

b=a×kb = a × k

Here, k is the number you multiply by. Example: if k = 4, then (6, 24) works because 6 × 4 = 24.

SQUARING RULE

b=a2b = a²

The output equals the input times itself. Example: (5, 25) works because 5 × 5 = 25.

TWO-STEP RULE

b=a×m+kb = a × m + k

First multiply by m, then add k. Example: if m = 2 and k = 3, then (4, 11) works because 4 × 2 + 3 = 11.

💡

Quick Tip

Always try the simplest rule first. Check addition or subtraction, then multiplication or division, then squaring, and finally two-step rules. This saves you time!

SECTION 5

A Step-by-Step Strategy

When you face a number-pair problem, follow this checklist. The diagram below maps out the decision process so you never get stuck.

Number-Pair Strategy FlowchartSTART: Look at the pairStep 1: Find the difference(output − input)Step 2: Check — does input + difference = output?YES → Rule is "add (difference)"NO ↓Step 3: Check — does input × something = output?YES → Rule is "multiply by (that number)"NO ↓Step 4: Check — does input × input = output?YES → Rule is "square it"NO ↓Step 5: Try a two-step rule(multiply then add/subtract)
Follow the flowchart from top to bottom. At each step, test one type of rule. If it works, you're done. If not, move to the next step.
  1. Step 1: Subtract the input from the output. This gives you the difference.
  2. Step 2: Check if that difference works as an addition rule for all pairs.
  3. Step 3: Divide the output by the input. If you get a whole number, try a multiplication rule.
  4. Step 4: Check if the input squared equals the output.
  5. Step 5: If nothing else fits, combine two operations (like multiply then add).

SECTION 6

Worked Example

Let's walk through a full problem together. Suppose you see these number pairs on a test: (2, 9), (5, 21), and (3, 13). You need to find the rule that turns each input into its output.

Finding a Two-Step Rule

  1. 1

    Step 1 — Find the Difference

    For the pair (2, 9), subtract: 9 − 2 = 7. For (5, 21), subtract: 21 − 5 = 16. The differences are not the same (7 ≠ 16), so it is not a simple addition rule.
    Addition rule? ✗ No — differences are unequal.
  2. 2

    Step 2 — Try Multiplication

    For (2, 9), divide: 9 ÷ 2 = 4.5. That is not a whole number. For (5, 21), divide: 21 ÷ 5 = 4.2. These quotients are not the same whole number, so it is not a simple multiplication rule.
    Multiplication rule? ✗ No — quotients are not equal whole numbers.
  3. 3

    Step 3 — Try Squaring

    For (2, 9), check: 2² = 4, but the output is 9. So the squaring rule does not fit.
    Squaring rule? ✗ No.
  4. 4

    Step 4 — Try a Two-Step Rule (Multiply Then Add)

    Notice that 9 ÷ 2 ≈ 4.5 and 21 ÷ 5 ≈ 4.2. These are close to 4. Let's guess "multiply by 4, then add something." For (2, 9): 2 × 4 = 8, and 9 − 8 = 1. For (5, 21): 5 × 4 = 20, and 21 − 20 = 1. The leftover is 1 both times!
    Possible rule: multiply by 4, then add 1.
  5. 5

    Step 5 — Verify With the Third Pair

    For (3, 13): 3 × 4 = 12, then 12 + 1 = 13. ✓ It works! The rule is confirmed: b = a × 4 + 1.
    Rule: b = a × 4 + 1 ✓

SECTION 7

Common Mistakes & How to Avoid Them

Even strong math students can slip up on number-pair problems. The table below shows the most common errors and how to fix them.

MistakeWhy It HappensHow to Fix It
Only checking one pairYou find a rule that works for one pair and assume it works for all of them.Always test your rule on every pair before choosing your answer.
Confusing × with +For small numbers, adding and multiplying can give similar results (e.g., 2 + 3 = 5 and 2 × 2 + 1 = 5).Check the rule on a larger pair where addition and multiplication give very different outputs.
Ignoring the orderYou try output → input instead of input → output.Remember: the rule always goes from the first number to the second number in the pair.
Skipping two-step rulesYou give up when simple rules don't work.Use the flowchart strategy. If one-step rules fail, try multiply-then-add or multiply-then-subtract.

Common mistakes on number-pair CogAT questions

✦

KEY TAKEAWAY

Think of verifying your rule like checking your work on a recipe. If you only taste-test one cookie, you might miss that the whole batch is too salty. Always test your rule on every pair to make sure it works everywhere.

SECTION 8

Connection to Number Series & Functions

Number-pair rules are actually the beginning of a bigger idea in math called functions. In algebra, a function like f(x) = 2x + 3 does the exact same thing: it takes an input (x), applies a rule (multiply by 2, add 3), and gives an output.

Number-Pair Problems (CogAT)Functions (Algebra & Beyond)
Given as pairs like (3, 11)Written as f(3) = 11
You figure out the rule by testingThe rule is given as a formula
Uses whole numbers and simple operationsCan use fractions, decimals, exponents, and more
Goal: identify the hidden ruleGoal: graph, analyze, and apply the rule

The CogAT also tests number series (also called number sequences). In those problems, you look at a list of numbers like 2, 5, 8, 11, … and find the pattern. The skill is the same: you are finding a rule. Mastering number pairs gives you a strong foundation for series problems, algebra, and beyond.

SECTION 9

Practice Problems

Try these five problems on your own. Start with the easier ones and work your way up. Use the strategy flowchart if you get stuck!

Question 1: CONCEPTUAL

The number pair is (6, 18). What is the relationship between the two numbers? Describe the rule in words.
Show answer
The rule is multiply by 3. You can check: 6 × 3 = 18. In formula form, the rule is b = a × 3. This is a simple one-step multiplication relationship.

Question 2: BASIC CALCULATION

The pairs (3, 10) and (7, 22) follow the same rule. What is the rule? Use it to find the output when the input is 5.
Show answer
Test addition: 10 − 3 = 7, but 22 − 7 = 15. The differences are not equal, so it is not simple addition. Try multiply then add: 3 × 3 = 9, and 10 − 9 = 1. Check: 7 × 3 = 21, and 22 − 21 = 1. The rule is b = a × 3 + 1. For input 5: 5 × 3 + 1 = 16. The output is 16.

Question 3: INTERMEDIATE

You are given three pairs: (2, 4), (3, 9), (5, 25). What is the rule? Is there more than one possible rule? Explain.
Show answer
Check squaring: 2² = 4 ✓, 3² = 9 ✓, 5² = 25 ✓. The rule is b = a² (square the input). You might also notice that 2 × 2 = 4 and wonder if the rule is "multiply by 2." But 3 × 2 = 6, not 9, so that does not work for all pairs. Always test your rule on every pair. The only rule that fits all three is squaring.

Question 4: APPLIED

A CogAT question shows: (4, 14), (6, 20), (10, ?). What number replaces the question mark? Show your work using the strategy flowchart.
Show answer
Step 1: Differences — 14 − 4 = 10, 20 − 6 = 14. Not equal, so no simple addition. Step 2: Division — 14 ÷ 4 = 3.5, 20 ÷ 6 ≈ 3.33. Not equal, so no simple multiplication. Step 3: Squaring — 4² = 16 ≠ 14. No. Step 4: Two-step rule — try multiply by 3: 4 × 3 = 12, 14 − 12 = 2. Try 6 × 3 = 18, 20 − 18 = 2. Both have a leftover of 2! The rule is b = a × 3 + 2. For input 10: 10 × 3 + 2 = 32. The answer is 32.

Question 5: CRITICAL THINKING

Consider the pairs (1, 2), (2, 6), (3, 12). One student says the rule is "add 1 then multiply by that number." Another student says the rule is "multiply by the next whole number." Are both students correct? Explain, and find the output for input 4.
Show answer
Let's test both ideas. Student 1: "add 1 then multiply by that sum." For (1, 2): 1 + 1 = 2, then 1 × 2 = 2 ✓. For (2, 6): 2 + 1 = 3, then 2 × 3 = 6 ✓. For (3, 12): 3 + 1 = 4, then 3 × 4 = 12 ✓. Student 2: "multiply by the next whole number," meaning a × (a + 1). For (1, 2): 1 × 2 = 2 ✓. For (2, 6): 2 × 3 = 6 ✓. For (3, 12): 3 × 4 = 12 ✓. Both descriptions produce the same formula: b = a × (a + 1). They are just two ways of saying the same thing. For input 4: 4 × (4 + 1) = 4 × 5 = 20. This problem shows that a rule can be described in different words but still be the same math.

SUMMARY

Lesson Summary

Summary

A number-pair relationship is a hidden rule that turns an input into an output. The four main rule types are addition/subtraction, multiplication/division, squaring, and two-step combinations. Use the strategy flowchart to test rules in order from simplest to most complex.

Always verify your rule on every pair, not just one. This skill is the foundation for number series problems and connects directly to the concept of functions in algebra. With practice, you will spot the pattern faster and feel more confident on the CogAT Quantitative Battery.

Varsity Tutors • CogAT (Grades 3+) • Identify the relationship in a number pair