Apply the same relationship to complete a number analogy - CogAT Grades 3 Plus

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Where Do Number Analogies Come From?

People have compared numbers and looked for patterns for thousands of years. Ancient mathematicians noticed that numbers can be connected by rules, like doubling or adding a certain amount. Number analogies (puzzles where you find a hidden rule between numbers) grew out of this love of patterns.

In the 1900s, psychologists began using pattern puzzles to measure how well people reason with numbers. These puzzles became part of cognitive ability tests (tests that check how you think and solve problems). The CogAT is one of the most popular of these tests today.

~300 BCE • Ancient Greek Proportions

Greek mathematicians like Euclid studied how numbers relate through ratios and proportions.

1905 • First Intelligence Tests

Alfred Binet created the first modern reasoning test in France, including number pattern questions.

1954 • CogAT Is Born

Researchers at the University of Iowa developed the Cognitive Abilities Test, featuring number analogies in the Quantitative Battery.

Today • Millions of Students

The CogAT is used in schools across the country to help understand how students reason with numbers, shapes, and words.

So here is the big question: when you see a pair of numbers connected by a rule, can you figure out that rule and apply it to a new pair? That is exactly what number analogy problems ask you to do.

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Core Principles of Number Analogies

A number analogy is a puzzle that looks like this: 3 → 9 :: 5 → ?. The symbol "→" means "goes to," and "::" means "in the same way as." Your job is to find the rule that turns the first number into the second, then use that same rule on the new number.

Spot the Pair

Look at the first two numbers. They are connected by a hidden math operation, like adding, subtracting, multiplying, or dividing.

Name the Rule

Ask yourself: what do I do to the first number to get the second? Write the rule in your own words, like "multiply by 3" or "subtract 7."

Apply the Rule

Take the same rule and use it on the new starting number. The answer you get is the missing piece of the analogy.

Check Your Work

Plug your answer back in. Does the second pair follow the exact same rule as the first pair? If yes, you are done!

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KEY TAKEAWAY

Think of a number analogy like a recipe. If the recipe says "add 2 cups of sugar to every batch of cookie dough," then every batch gets 2 cups — not just the first one. The rule stays the same no matter which numbers you start with.

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Seeing the Pattern

A picture can make number analogies much easier to understand. The diagram below shows how the same rule — multiply by 3 — connects both pairs of numbers in the analogy 2 → 6 :: 4 → 12.

Both pairs use the same rule (× 3). The first pair confirms the rule. The second pair uses it to find the missing number.

Notice how Pair 1 (cyan boxes) and Pair 2 (violet boxes) both have the same arrow labeled "× 3." That is the heart of every number analogy: the operation stays the same, even though the starting numbers are different.

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The Math Behind Number Analogies

Every number analogy hides a math operation. Here are the most common ones you will see on the CogAT. Knowing these makes it much faster to spot the rule.

ADDITION RULE

a + n = b → c + n = ?

Find what number n is added to the first number to get the second. Then add the same n to the new number. Example: 5 → 12 :: 8 → 15 (n = 7)

SUBTRACTION RULE

a − n = b → c − n = ?

Find what number n is subtracted. Then subtract the same amount from the new number. Example: 20 → 14 :: 30 → 24 (n = 6)

MULTIPLICATION RULE

a × n = b → c × n = ?

Find the number that the first value is multiplied by. Then multiply the new number by the same amount. Example: 3 → 15 :: 7 → 35 (n = 5)

SQUARING RULE

a² = b → c² = ?

The second number is the first number multiplied by itself. Example: 4 → 16 :: 6 → 36 (because 4² = 16 and 6² = 36)

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Pro Tip

If the second number is bigger than the first, try addition or multiplication. If it is smaller, try subtraction or division. This simple check can save you time!

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A Closer Look at Common Rule Types

Not every number analogy uses a single, simple operation. Some use two steps, and others use less obvious patterns. The diagram below organizes the most common rule types from easiest to hardest.

Rule types range from simple one-step operations (top row) to trickier two-step combinations (bottom row). Start by checking the easy rules first.

Rule Type	Example Analogy	Hidden Operation
Add	7 → 13 :: 10 → 16	+ 6
Subtract	18 → 11 :: 25 → 18	− 7
Multiply	6 → 24 :: 9 → 36	× 4
Divide	36 → 9 :: 48 → 12	÷ 4
Square	3 → 9 :: 7 → 49	n²
Two-step	2 → 7 :: 4 → 13	× 3 + 1

Quick-reference table of the most common number analogy rules

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Worked Example: Solving a Number Analogy Step by Step

Let's walk through a complete problem together. Follow each step carefully.

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Problem

8 → 19 :: 12 → ?

Solving 8 → 19 :: 12 → ?

1
Step 1 — Look at the First Pair

The first pair is 8 and 19. Notice that 19 is bigger than 8, so the rule probably involves addition or multiplication.
2
Step 2 — Test Addition

What do you add to 8 to get 19? Calculate 19 − 8 = 11. So the rule might be "add 11."

Possible rule: + 11
3
Step 3 — Test Multiplication

Does 8 times any whole number equal 19? Try 8 × 2 = 16 (too small) and 8 × 3 = 24 (too big). So multiplication by a whole number does not work here. We will stick with addition.
4
Step 4 — Apply the Rule to the Second Pair

The rule is "add 11." Apply it to 12: 12 + 11 = 23.

12 + 11 = 23
5
Step 5 — Check Your Answer

Does Pair 1 follow the same rule as Pair 2? 8 + 11 = 19 ✓ and 12 + 11 = 23 ✓. Both pairs use "add 11," so the answer is confirmed.

Answer: 23

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Strategies and Common Mistakes

Even strong math students can slip up on number analogies. Knowing the most common traps helps you avoid them. The table below compares good strategies with common mistakes.

Good Strategy ✓	Common Mistake ✗
Try subtraction first to check for an addition rule.	Guessing the answer without testing the rule on the first pair.
Try division to check for a multiplication rule.	Only testing one operation and picking it without verifying.
Check for two-step rules if one-step rules don't fit.	Forgetting that rules can involve more than one step.
Always verify your answer by plugging it back in.	Stopping after finding the answer without double-checking.

Strategies vs. common mistakes

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KEY TAKEAWAY

Imagine you are a detective at a crime scene. You wouldn't arrest someone based on one clue. You would gather evidence and make sure everything fits. Do the same with number analogies: test multiple operations before you commit to an answer.

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Connecting to More Advanced Thinking

Number analogies are actually a stepping stone to bigger math ideas. When you identify a rule and apply it, you are doing something very similar to using a function in algebra. A function is just a rule that takes an input number and turns it into an output number — exactly like a number analogy!

Number Analogy Skill	Advanced Math Connection
Find the rule between two numbers	Write a function rule like f(x) = 2x + 3
Apply the same rule to a new number	Evaluate a function for a given input
Check that both pairs follow the rule	Verify solutions by substitution
Recognize two-step rules	Compose functions (do one rule, then another)

How number analogy skills connect to algebra

So every time you solve a number analogy, you are building the same thinking muscles you will use in algebra, data science, and even computer programming. These skills will stay with you for years to come!

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Practice Problems

Try these five problems on your own. They start easy and get harder. After each one, check your answer and read the explanation.

Question 1: CONCEPTUAL

In the analogy 5 → 10 :: 9 → ?, what is the hidden rule and what is the missing number?

Show answer

The rule is "multiply by 2." Check: 5 × 2 = 10. Apply the same rule: 9 × 2 = 18. The answer is 18. This is a simple one-step multiplication rule. Always start by asking, "What do I do to the first number to get the second?"

Question 2: BASIC CALCULATION

Solve: 15 → 9 :: 22 → ?

Show answer

The second number is smaller, so try subtraction. 15 − 9 = 6, so the rule is "subtract 6." Apply it: 22 − 6 = 16. The answer is 16. When the output is smaller than the input, subtraction or division is likely the rule. Here, subtraction fits perfectly because 15 − 6 = 9.

Question 3: INTERMEDIATE

Solve: 4 → 16 :: 7 → ?

Show answer

Test addition: 16 − 4 = 12, so add 12? Then 7 + 12 = 19. But also test multiplication: 4 × 4 = 16, and squaring: 4² = 16. Both × 4 and squaring give 16. Try both on 7: 7 × 4 = 28, and 7² = 49. Which rule was intended? On the CogAT, look at the answer choices to decide. If 49 is a choice, the rule is squaring. If 28 is a choice, the rule is × 4. Without choices, squaring (answer 49) is the more elegant pattern. This problem shows why testing multiple rules matters.

Question 4: APPLIED

A CogAT question shows: 3 → 10 :: 5 → 26 :: 7 → ?. What is the missing number? (Hint: the rule uses two steps.)

Show answer

Test the first pair. 3² = 9, and 9 + 1 = 10. So the rule might be "square, then add 1." Check with the second pair: 5² = 25, and 25 + 1 = 26 ✓. Now apply to 7: 7² = 49, and 49 + 1 = 50. The answer is 50. Two-step rules become easier when you have two known pairs to verify your guess. Always check the rule against every pair before applying it.

Question 5: CRITICAL THINKING

Create your own number analogy that uses a two-step rule. Write the analogy, state the rule, and explain how someone could figure it out.

Show answer

Sample analogy: 2 → 9 :: 4 → 17. The rule is "multiply by 4, then add 1" (2 × 4 + 1 = 9, 4 × 4 + 1 = 17). To figure it out, a solver would try one-step rules first. Addition gives + 7 for the first pair, but 4 + 7 = 11 ≠ 17, so addition alone fails. Then the solver might try multiplication: 2 × 4.5 = 9, but 4.5 is not a clean number. Trying × 4 gives 8, which is 1 short of 9, suggesting "× 4 + 1." Checking: 4 × 4 + 1 = 17 ✓. Creating analogies is a great way to deepen your understanding!

SUMMARY

Putting It All Together

Summary

A number analogy gives you a pair of numbers connected by a hidden rule and asks you to apply that same rule to a new number. The most common rules involve addition, subtraction, multiplication, division, and squaring. Harder problems may combine two operations into a two-step rule.

To solve any number analogy, follow four steps: spot the pair, name the rule, apply the rule, and check your work. Always test more than one operation, especially when your first guess does not fit perfectly. These skills build the same thinking you will use with functions in algebra and beyond.