Number analogies

Learning outcome

Find the single arithmetic rule that turns the first number into the second, then apply that same rule to the third number.

Concepts and reasons

A number analogy has the form p : q :: r : ? The rule must explain q from p and produce exactly one result from r. Check the rule on the given pair before testing options.

The rules used most often in this module are:

• Multiply or divide by a fixed number. 4 : 12 :: 6 : 18 uses ×3.
• Add or subtract a fixed number. 9 : 14 :: 6 : 11 uses +5.
• Square, or square and then add or subtract 1. 5 : 25 uses 5 squared. 5 : 24 uses 5 squared minus 1.
• Reverse the digits. 18 : 81 reverses 18. This is not the same as adding the difference between the two numbers.

If two different rules both fit the first pair, use a rule that still fits and that the options can distinguish. A rule is not established by the answer you hoped for. It is established by the given pair.

Worked examples

Example 1. 7 : 49 :: 9 : ? 49 is 7 squared, so the rule is "square the number". Then 9 squared is 81. The product 7 × 9 = 63 uses both given numbers, so it is not a rule from the first pair alone.

Example 2. 5 : 24 :: 8 : ? 5 squared is 25, and 25 − 1 = 24. The same rule gives 8 squared minus 1 = 63. The square 64 is the trap for stopping one step early.

Example 3. 18 : 81 :: 27 : ? Reversing the digits of 18 gives 81. Reversing 27 gives 72. Adding 63, the gap from 18 to 81, would give 90, which is a different rule and is not the digit reversal.

Common traps

Do not mix the third number into the rule. The rule is fully visible in the first pair. Do not stop at the square when the given second number is one less or one more than the square. Digit reversal is a different operation from adding a fixed gap.

Practice questions

1. 6 : 36 :: 11 : ?
2. 4 : 15 :: 7 : ? Use square minus 1.
3. 12 : 21 :: 47 : ? Why is 56 the wrong answer?

Answers and explanations

1. 36 is 6 squared, so 11 squared is 121.
2. 4 squared minus 1 is 15. Then 7 squared minus 1 is 48.
3. 21 is 12 with the digits reversed, so 47 reverses to 74. The number 56 is 47 + 9, and 9 happens to be 21 − 12, but that addition rule is not digit reversal.

Analogy

5 : 24 :: 8 : 63.

5 squared is 25, and 25 minus 1 is 24. The same rule gives 64 minus 1 = 63. Stopping at 64 misses the last step.

Quick reference

• The rule must be visible in the first pair alone.
• Check square, square plus or minus 1, a fixed sum, and digit reversal.
• Do not fold the third number into the rule.
• If two rules fit, use the one the options can separate.