Numerical and Logical Reasoning

A four-step plan for any reasoning question

Whenever a question looks busy or wordy, run it through these four steps. They keep you from guessing and stop you from holding everything in your head at once.

• Read and list what you know. Pull every given fact out of the words and write it down as a short list. Getting it onto paper is half the battle.
• Find what is being asked. Underline the actual question. Are you finding a missing number, a total, the next term, or the one option that fits?
• Work step by step. Solve one piece at a time. For patterns, find the rule. For clues, test each option and cross out what cannot work.
• Check it against the clues. Put your answer back into the original facts. If it satisfies all of them, you are done.

Worked example #1: continue the number pattern

What number comes next? 3, 7, 11, 15, ___. To extend a pattern, the first job is always to find the rule, what happens to get from one term to the next.

Look at the gaps between the terms. Subtract each term from the one after it:

• Find the rule: 7 − 3 = 4, 11 − 7 = 4, 15 − 11 = 4. The gap is always 4, so the rule is "add 4."
• Apply it: 15 + 4 = 19.

  3,  7,  11,  15,  ?
    +4  +4   +4   +4
  next term = 15 + 4 = 19

Worked example #2: deduce the number from clues

Some questions hand you several clues and ask you to narrow a list of choices down to the answer. The trick is to use each clue to cross out what cannot work, and to keep going until just one choice is left.

• Clue 1 (even): cross out 51, because it is odd. Left with 42, 24, 60.
• Clue 2 (greater than 40): cross out 24, because it is below 40. Left with 42 and 60.
• Clue 3 (digits add to 6): for 42, 4 + 2 = 6. For 60, 6 + 0 = 6. Both pass this clue too.
• Decide: check both survivors against all three clues. 42 is even, greater than 40, and its digits add to 6. 60 is also even, greater than 40, and its digits add to 6. Both 42 and 60 satisfy every clue, so these three clues are not enough to pick a single answer, the puzzle would need one more distinguishing clue to separate them.

  Choices:  42   51   24   60
  even?     ✓    ✗    ✓    ✓
  > 40?     ✓    ✓    ✗    ✓
  digits=6? ✓    ✗    ✓    ✓
  ─────────────────────────────
  passes all three:  42  and  60

Worked example #3: a multi-step word problem

These are the questions that feel longest, so the list-it-out step matters most. Read slowly and capture every number with its meaning.

• List what you know: 3 notebooks at $4.50, 2 pens at $1.25, paid with $20.
• What is asked: the change, so we need total cost first, then subtract from $20.
• Step by step: notebooks cost 3 × $4.50 = $13.50. Pens cost 2 × $1.25 = $2.50. Total = $13.50 + $2.50 = $16.00.
• Finish and check: change = $20.00 − $16.00 = $4.00. Does it make sense? She spent $16 of her $20, so $4 back is reasonable.

  notebooks:  3 × $4.50 = $13.50
  pens:       2 × $1.25 =  $2.50
              ─────────────────
  total cost           = $16.00
  change: $20.00 − $16.00 = $4.00

Worked example #4: a pattern that grows

Not every pattern adds the same amount. What comes next in 2, 6, 12, 20, ___? The gaps are not equal, so look at how the gaps themselves change.

• Find the gaps: 6 − 2 = 4, 12 − 6 = 6, 20 − 12 = 8.
• Look at the gaps: the gaps are 4, 6, 8, they go up by 2 each time. So the next gap is 10.
• Apply it: 20 + 10 = 30.

  2,   6,   12,   20,   ?
    +4   +6    +8   +10
  gaps grow by 2 → next gap = 10
  next term = 20 + 10 = 30

Tips that make reasoning questions easier

• Write the clues as a list. The moment the words feel overwhelming, turn them into short bullet facts. A tidy list turns a scary paragraph into a simple to-do.
• For patterns, check the gaps first. Constant gap means add (or subtract) the same number. If the gaps change, look at how they change.
• Use process of elimination. On a multiple-choice clue puzzle, test each option against each clue and cross out anything that fails. Aim for exactly one survivor.
• Do one step at a time. Multi-step problems are just several small problems in a row. Solve, write the result, then move to the next step.
• Sanity-check the answer. Ask "does this number make sense?" A change of $400 from a $20 bill should immediately feel wrong, that instinct catches careless slips.

Your turn: practice problems

Use the four-step plan: list what you know, find what is asked, work step by step, then check. Calculator allowed. No peeking until you have tried.

1. What number comes next? 5, 10, 20, 40, ___
2. I am thinking of a number under 30. It is odd, it is a multiple of 5, and it is greater than 10. What is it?
3. A bus has 18 seats. 14 are taken, then 6 people get on and 3 get off. How many seats are now empty?
4. Find the missing number: 1, 4, 9, 16, ___