Symbolic calculations — SSC CGL Reasoning Notes

Concepts and rules

In these questions the printed symbols do not have their usual meaning. The question gives a dictionary such as '+ means ÷' or 'P means ×', and you must calculate with the meanings, not the printed signs.

Order of operations (BODMAS): brackets first; then division and multiplication, which have equal priority and are done from left to right; then addition and subtraction, also equal and done from left to right.

Because ÷ and × are equal in priority, 24 ÷ 4 × 2 = 6 × 2 = 12, not 24 ÷ 8 = 3. Similarly 10 − 4 + 3 = 9, not 3.

Sign substitution: replace every printed sign by its meaning in one pass, writing the new expression fully before calculating anything.

Letter-coded operations: letters such as P, Q, R, S stand for operations. Treat them exactly like substituted signs.

Interchange questions: 'if + and × are interchanged' means every + becomes × and every × becomes + at the same time. When numbers are interchanged too (say 3 and 5), swap every 3 and 5 as well.

Correcting an equation: when asked which two signs must be interchanged to make an equation correct, apply each option to the left side and check which one gives the right side.

Balancing: when signs are missing (shown as *), test each option's sequence of signs in order from left to right.

Formula operators: a symbol may be defined by a formula, such as a # b = 2a + 3b. Then a # b and b # a are usually different, so keep the order of the inputs.

Nested formula operators are evaluated from the innermost bracket outward, using the result of the inner operation as an input of the outer one.

Method and shortcuts

Step 1: Write the dictionary in a line, for example + → ÷, − → ×, × → +, ÷ → −.

Step 2: Rewrite the expression sign by sign under the original, so that no sign is converted twice.

Step 3: Do all ÷ and × from left to right, then all + and − from left to right.

Step 4: In option-checking questions, compute the left side for each option and stop at the first exact match; still glance at the rest if time allows, because only one option should work.

Shortcut: a division that does not come out exact (like 59.5) rules an option out at once, because SSC answers in these questions are integers.

Shortcut: estimate size first. If the right side is small, an option that turns a + into × between two large numbers is usually wrong.

Example 1: sign substitution

Question: If + means ÷, − means ×, × means + and ÷ means −, find 36 + 4 − 3 × 5 ÷ 2.

Solution: Rewrite: 36 ÷ 4 × 3 + 5 − 2. Left to right for ÷ and ×: 36 ÷ 4 = 9, 9 × 3 = 27. Then 27 + 5 − 2 = 30.

Answer: 30.

Example 2: letter-coded operations

Question: If P means ×, Q means ÷, R means + and S means −, find 18 Q 3 P 4 R 7 S 5.

Solution: 18 ÷ 3 × 4 + 7 − 5 = 6 × 4 + 7 − 5 = 24 + 7 − 5 = 26.

Answer: 26.

Example 3: nested formula operators

Question: If a # b = 2a + 3b and a @ b = a × b − b, find (4 # 2) @ 3.

Solution: Inner bracket: 4 # 2 = 2 × 4 + 3 × 2 = 8 + 6 = 14. Outer: 14 @ 3 = 14 × 3 − 3 = 42 − 3 = 39.

Answer: 39.

Example 4: which signs to interchange

Question: Which two signs should be interchanged to make the equation correct? 18 × 3 + 6 ÷ 2 − 4 = 23 (a) + and × (b) + and − (c) × and ÷ (d) − and ÷

Solution: (a) 18 + 3 × 6 ÷ 2 − 4 = 18 + 9 − 4 = 23, correct. Check the others: (b) 18 × 3 − 6 ÷ 2 + 4 = 55; (c) 18 ÷ 3 + 6 × 2 − 4 = 14; (d) 18 × 3 + 6 − 2 ÷ 4 = 59.5.

Answer: (a) + and ×.

Example 5: signs and numbers interchanged

Question: If + and × are interchanged and the numbers 3 and 5 are interchanged, which equation becomes correct? (a) 5 + 3 × 2 = 13 (b) 5 × 6 + 3 = 33 (c) 3 + 7 × 5 = 50 (d) 3 × 8 + 5 = 39

Solution: Change both signs and numbers on the left side only. (a) 3 × 5 + 2 = 17, not 13. (b) 3 + 6 × 5 = 3 + 30 = 33, correct. (c) 5 × 7 + 3 = 38, not 50. (d) 5 + 8 × 3 = 29, not 39.

Note: the right side is a result, so it is not altered even though 33 contains the digit 3; only the numbers written as 3 and 5 in the expression are swapped.

Answer: (b).

Example 6: balancing with missing signs

Question: Which sequence of signs, placed in order, makes 8 * 4 * 2 * 6 * 3 = 24 correct? (a) × − + ÷ (b) + × − ÷ (c) − ÷ + × (d) ÷ − + ×

Solution: (a) 8 × 4 − 2 + 6 ÷ 3 = 32 − 2 + 2 = 32. (b) 8 + 4 × 2 − 6 ÷ 3 = 8 + 8 − 2 = 14. (c) 8 − 4 ÷ 2 + 6 × 3 = 8 − 2 + 18 = 24. (d) 8 ÷ 4 − 2 + 6 × 3 = 2 − 2 + 18 = 18.

Answer: (c) − ÷ + ×.

Common mistakes

Calculating while converting. Convert the whole expression first; otherwise you may convert a sign that you already converted.

Doing × before ÷ always. They are equal; go left to right: 36 ÷ 4 × 3 = 27, not 3.

Interchanging only one way. 'Interchange + and ×' changes both: every + to × and every × to +.

Changing digits inside the answer when numbers are interchanged. Swap only the numbers in the expression being tested.

Reversing inputs of a formula operator. 4 # 2 = 14 but 2 # 4 = 16 when a # b = 2a + 3b.

Practice questions

Q1. If A means ÷, B means +, C means − and D means ×, find 20 A 4 D 6 C 9 B 3. (a) 21 (b) 24 (c) 27 (d) 18

Q2. Which two signs should be interchanged to make the equation correct? 24 − 6 ÷ 3 + 2 × 5 = 45 (a) + and − (b) − and ÷ (c) + and × (d) − and ×

Q3. If a ★ b = a² + b² − ab, find (3 ★ 2) ★ 1. (a) 43 (b) 41 (c) 45 (d) 37

Answers

Q1. (b) 20 ÷ 4 × 6 − 9 + 3 = 5 × 6 − 9 + 3 = 30 − 9 + 3 = 24.

Q2. (d) 24 × 6 ÷ 3 + 2 − 5 = 48 + 2 − 5 = 45. The other swaps give 16, 11 and 25.

Q3. (a) 3 ★ 2 = 9 + 4 − 6 = 7; 7 ★ 1 = 49 + 1 − 7 = 43.

Quick revision

Decode everything first, then calculate.

÷ and × are equal (left to right); + and − are equal (left to right).

Interchange means both ways at once; for numbers, swap only inside the expression.

Formula operators: inner bracket first, keep input order.

Concepts and rules

This unit covers 3 kinds of reasoning with numbers: hidden-rule equations (if 3 # 4 = 25 …), short arithmetic puzzles (heads and legs, working backwards, trees on a road), and inequality chains (A > B ≥ C).

Hidden-rule equations: each line uses the same rule on its 2 numbers. Common rules are a² + b², a² − b², a × b + a + b, (a + b)², and writing two results side by side, such as (a − b) followed by (a + b).

A rule is confirmed only when it fits every given line. If the output has more digits than expected, suspect two results written side by side.

Heads and legs: if all animals had 2 legs, the legs would be 2 × heads. Each 4-legged animal adds 2 extra legs, so 4-legged animals = (legs − 2 × heads) ÷ 2.

Working backwards: when a final value is given after a chain of steps, undo the steps from the last to the first: undo 'add 10' by subtracting 10, undo 'half get off' by doubling.

Fence-post rule: on an open line of length L with a post every d metres including both ends, posts = L ÷ d + 1. On a closed loop (a circular track or the boundary of a field) posts = L ÷ d.

Inequality chains: A > B and B > C give A > C. A ≥ B and B ≥ C give A ≥ C. If any link in a same-direction chain is strict (> or <), the combined relation is strict.

If the signs between two letters point in opposite directions (A > B < C or A ≥ B ≤ C), no definite relation between A and C follows.

Either-or: when the chain gives A ≥ C, neither 'A > C' nor 'A = C' follows alone, but together they cover every case, so the answer is 'either I or II follows'.

Coded inequalities: words like 'not smaller than' mean ≥, 'not greater than' mean ≤, 'neither greater than nor equal to' mean <, 'neither smaller than nor equal to' mean >, and 'neither greater nor smaller' means =.

Method and shortcuts

Step 1 for hidden rules: test the simplest rules on the first line (sum, product, squares), then confirm the survivor on the remaining lines.

Step 2 for puzzles: write each condition as an equation or a backward step; do not try to guess the answer from the options first.

Step 3 for inequalities: decode symbols, combine all statements into one chain, then trace the path between the two letters in each conclusion.

Shortcut: a conclusion follows only if every sign along the path points the same way; one opposite sign kills it.

Shortcut: in working-backwards questions, check your start value forwards once; it takes seconds and catches reversed steps.

Example 1: a squares rule

Question: If 3 # 4 = 25 and 5 # 12 = 169, find 8 # 6.

Solution: 3² + 4² = 9 + 16 = 25 and 5² + 12² = 25 + 144 = 169. So a # b = a² + b², and 8 # 6 = 64 + 36 = 100.

Answer: 100.

Example 2: heads and legs

Question: A farm has hens and cows with 50 heads and 140 legs in all. How many cows are there?

Solution: If all 50 were hens, legs = 100. The extra 40 legs come from cows, each giving 2 extra, so cows = 40 ÷ 2 = 20 and hens = 30. Check: 20 × 4 + 30 × 2 = 80 + 60 = 140.

Answer: 20 cows.

Example 3: reading an inequality chain

Question: Given A > B ≥ C = D < E, which of these are definitely true: A > D, B ≥ D, A > E?

Solution: A > B ≥ C = D gives A > D (one strict link). B ≥ C = D gives B ≥ D. Between A and E the path is A > … = D < E, with signs in opposite directions, so A and E have no definite relation.

Answer: A > D and B ≥ D are true; A > E is not certain.

Example 4: two results side by side

Question: If 5 + 3 = 28, 9 + 1 = 810 and 8 + 6 = 214, then 7 + 3 = ? (a) 104 (b) 410 (c) 421 (d) 310

Solution: 5 − 3 = 2 and 5 + 3 = 8 give 28; 9 − 1 = 8 and 9 + 1 = 10 give 810; 8 − 6 = 2 and 8 + 6 = 14 give 214. So write (a − b) then (a + b). For 7 and 3: 4 and 10, giving 410. Option (a) writes them in the wrong order.

Answer: (b) 410.

Example 5: working backwards

Question: A bus leaves the depot with some passengers. At the first stop half the passengers get off and 10 get on. At the second stop half get off and 5 get on. Now there are 25 passengers. How many were there at the start? (a) 70 (b) 50 (c) 80 (d) 60

Solution: Undo the second stop: 25 − 5 = 20 stayed, which was half, so 40 were on board before it. Undo the first stop: 40 − 10 = 30 stayed, which was half, so 60 at the start.

Check forwards: 60 → 30 + 10 = 40 → 20 + 5 = 25. Correct.

Answer: (d) 60.

Example 6: coded inequalities

Question: P @ Q means P is not smaller than Q; P # Q means P is neither smaller than nor equal to Q; P & Q means P is neither greater than nor smaller than Q; P % Q means P is not greater than Q. Given M @ N, N # R, R & T, T % K, which conclusion follows? I. M # T II. K @ N (a) Only I (b) Only II (c) Both I and II (d) Neither I nor II

Solution: Decode: @ is ≥, # is >, & is =, % is ≤. Chain: M ≥ N > R = T ≤ K.

I. M # T means M > T: the path M ≥ N > R = T has one strict link, so M > T is true. II. K @ N means K ≥ N: the path N > R = T ≤ K changes direction at T, so it is not certain.

Answer: (a) Only I.

Common mistakes

Accepting a hidden rule after checking one line. a² + b² and (a + b)² − 2ab agree, but many rules fit only the first line; check all lines.

Undoing steps in the forward order. In working backwards, the last step is undone first.

Using L ÷ d for an open road. An open line with both ends planted needs L ÷ d + 1 trees.

Writing A ≥ C as A > C. A single strict sign is needed for a strict conclusion.

Marking 'either-or' for any 2 conclusions that are both doubtful. Either-or needs the same 2 letters and the 2 conclusions must together cover every case (like > and = when ≥ is known).

Practice questions

Q1. If 2 * 3 = 11, 4 * 5 = 29 and 3 * 6 = 27, then 6 * 7 = ? (a) 49 (b) 54 (c) 55 (d) 56

Q2. Trees are planted every 5 m along one side of a straight 100 m road, including one at each end. How many trees are there? (a) 20 (b) 21 (c) 19 (d) 22

Q3. Given P ≥ Q = R > S and T ≤ R, which follows? I. P > T II. P = T (a) Only I (b) Only II (c) Either I or II (d) Neither I nor II

Answers

Q1. (c) Rule a × b + a + b: 6 + 5 = 11, 20 + 9 = 29, 18 + 9 = 27. So 6 * 7 = 42 + 13 = 55.

Q2. (b) Open line with both ends: 100 ÷ 5 + 1 = 21.

Q3. (c) P ≥ Q = R ≥ T gives P ≥ T only. Neither P > T nor P = T is certain, but one of them must hold, so either I or II follows.

Quick revision

Hidden rules: try a² ± b², ab + a + b, and two results written side by side; check every line.

Heads and legs: 4-legged = (legs − 2 × heads) ÷ 2.

Open line: L ÷ d + 1 posts; closed loop: L ÷ d posts.

Inequalities: same direction along the path, and at least one strict sign for a strict result; ≥ known means either > or =.