CA Foundation · Quantitative Aptitude

Number Series, Coding-Decoding and Odd Man Out: formula sheet

Full chapter guide

Key formulas

Alphabet position: A = 1, B = 2, ..., Z = 26; Memory aid: EJOTY gives E = 5, J = 10, O = 15, T = 20, Y = 25. Count from the nearest anchor.
Opposite letter (mirror): Position of opposite letter = 27 − position of the letter; A ↔ Z, B ↔ Y, M ↔ N. The two positions always add up to 27.
Forward or backward shift: New position = old position ± n; Use + for moving toward Z and − for moving toward A.
Wrap-around rule: If the result is more than 26, subtract 26. If it is less than 1, add 26.; Example: Y (25) + 3 = 28, and 28 − 26 = 2, so the letter is B.
Gap between terms: Gap = position of next term − position of current term; Check whether the gap is constant, growing, or alternating before deciding on the rule.
Alternating series: Odd-numbered terms form one series, even-numbered terms form another; Use this when a single gap does not fit all terms.
Letter position: A = 1, B = 2, ... Z = 26; Memorise EJOTY as 5, 10, 15, 20, 25 to find any position fast.
Mirror (reverse) letter: Reverse position = 27 − position; Equivalent check: a letter and its mirror always have positions that add up to 27.
Mirror pairs: A↔Z, B↔Y, C↔X, D↔W, E↔V, F↔U, G↔T, H↔S, I↔R, J↔Q, K↔P, L↔O, M↔N; The middle of the alphabet is between M (13) and N (14).
Forward shift: New position = old position + k; If the result is above 26, subtract 26.
Backward shift: New position = old position − k; If the result is below 1, add 26.
Rule check: Difference = coded position − original position; Find it for every letter. Same difference means a uniform shift. A repeating pattern of differences means an alternating rule.
Alphabet position: A=1, B=2, C=3 ... M=13, N=14 ... Y=25, Z=26; Memorise this. Use the anchors E=5, J=10, O=15, T=20, Y=25 to find any letter fast.
Opposite (reverse) position: Position from the end = 27 − position from the start; For example, D is 4th from the start, so it is 23rd from the end (W). Use it when the code is based on reverse alphabet order.
Sum of letter values: Code = sum of the positions of all letters; For example, CAT = 3 + 1 + 20 = 24. Check whether the question uses sum, product or side-by-side digits.
Shifted position code: Code of letter = position ± k; First find k from the given example, then apply the same k to every letter. Check it on all examples.
Digit placement: Side-by-side coding: CAB → 3, 1, 2 → 312; Do not add the digits unless the question says the code is a sum.
Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47; 2 is the only even prime. 1 is neither prime nor composite.
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400; A perfect square never ends in 2, 3, 7 or 8.
Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000; Learn these ten by heart. A cube ends in the same digit as its root only for roots ending in 0, 1, 4, 5, 6, 9.
Divisibility by 3 and 9: Divisible by 3 if digit sum is divisible by 3; by 9 if digit sum is divisible by 9; Use the digit sum for both digit-sum patterns and divisibility checks.
Divisibility by 4, 8, 11: 4: last two digits divisible by 4. 8: last three digits divisible by 8. 11: (sum of digits at odd places) - (sum at even places) is 0 or a multiple of 11; Apply these to find a number that is not a multiple of the common factor.
Common number-form patterns: n² ± 1, n³ ± 1, n² + n, n(n + 1); If the numbers look 'one off' from squares or cubes, test these forms.
Alphabet position: A=1, B=2, C=3, ... M=13, N=14, ... Z=26; Memorise M = 13 and N = 14. Every other position can be reached from these two quickly.
Opposite (mirror) letter: Position of opposite letter = 27 − position of the letter; A pairs with Z, B with Y, M with N. Two letters that are mirror pairs have positions adding up to 27.
Vowels: Vowels = A, E, I, O, U; All other letters are consonants. Count vowels in each group or word when positions give no pattern.
Gap between letters: Gap = position of next letter − position of current letter; Compare the gaps within each group. A group with different gaps from the others is often the odd one.
Sum of positions: Group value = sum of positions of its letters; Use it for pairs and triples when gaps do not match. Check whether the sums are equal, all even, or all multiples of some number.

Quick revision

Test differences first, then second differences, then ratios.
Common series rules: add a constant, multiply by a constant, add squares or cubes, alternate two patterns.
Wrong term: find the rule from the terms that agree, then see which term breaks it.
Letter positions: A = 1, M = 13, N = 14, Z = 26. A and Z, B and Y and so on add up to 27.
Letter coding: check for a fixed forward or backward shift, or a reversed alphabet.
Always verify the coding rule on every given example, not just one.
Number coding: look for a letter-to-number map, or an operation on positions.
Odd man out in numbers: check prime, square, cube, odd or even and divisibility.
Odd man out in words: check meaning or category first, then letter count and letter patterns.
Use the options to test a rule quickly instead of building the full answer.
Skip if no rule appears in about a minute, because each wrong answer loses 0.25 marks.

Common mistakes

Counting from A every time and miscounting positions. Fix: Learn EJOTY (E 5, J 10, O 15, T 20, Y 25) and count from the nearest anchor.
Forgetting to wrap around after Z (or before A). Fix: If the position goes above 26, subtract 26. If it goes below 1, add 26.
Forgetting wrap-around past Z or before A Fix: If a result is above 26, subtract 26. If it is below 1, add 26. For example, A − 2 gives Y.
Confirming the rule on only the first letter Fix: Test the rule on every letter of the given pair before applying it. Alternating rules show up only from the second or third letter.
Using the wrong alphabet position, such as taking M as 12 or counting from the wrong end. Fix: Use the anchors E=5, J=10, O=15, T=20, Y=25 and count from the nearest one.
Confusing the sum of values with side-by-side digits. Fix: Check the given example. If CAB is 312, the digits are side by side. If CAT is 24, the values are added.
Calling 1 a prime number. Fix: A prime has exactly two distinct factors. 1 has only one, so it is not prime.
Forgetting that 2 is prime and even. Fix: 2 is the only even prime. In a set like 2, 3, 5, 9, the odd one is 9, not 2.
Picking an answer after finding one rule that fits only two options. Fix: Test the rule on all four options. It must fit exactly three and fail for one.
Miscounting alphabet positions, especially after M. Fix: Anchor on M = 13 and N = 14. For letters after N, count forward from N, or use 27 − position of the mirror letter.

Exam tips

Memorise the positions of A to Z, and the EJOTY anchors, before you practise. This alone saves most of your time.
Check the options early. If three options have a different first letter from the likely answer, you can often finish the question after finding one letter.
When the gaps do not fit, try splitting the series into alternate terms before trying anything complex.
Watch for wrap-around in long series. If the answer looks odd, check whether the position crossed 26.
Do not spend more than about a minute on a stubborn question. Skip it and return later, since wrong answers carry 0.25 negative marking.
Questions are short and worth the same as any other MCQ, so do them early and quickly to bank easy marks.
Always compute the second letter of the target word before choosing, because options often share the same first letter.
Watch for wrap-around cases: if the word contains A, B, Y or Z, expect the question to test it.