Letters of the alphabet can be used as placeholders or symbols for numbers whose values are either unknown or can vary (variables):
Linear Expressions: Combining numbers, variables, and operational symbols (e.g., 2A + 3B − C). Like terms can be grouped and combined.
Constraint Propagation: Relationships like A > B > C or A + B = 10 restrict the permissible replacement set for each letter-number.
Conservation of Flow in Networks: At any junction node in a directed supply network:
Total Inflow = Demand at Node + Total Outflow
B. Inequality Logic & Operational Substitutions
When conventional mathematical symbols are mapped to coded operators:
Strict vs. Non-Strict Relations: "Not less than" means greater than or equal to (≥), while "not greater than" means less than or equal to (≤).
Translating Relations: An expression like a × b A c with coded symbols must first be rewritten using standard relational symbols before comparing with candidate options.
C. Number Pyramid (Additive Tree) Principles
In an additive brick pyramid where each brick is the sum of the two bricks directly beneath it:
Top = Left_Base + Right_Base
For a 2-level base: Q = 1 + P, R = P + 3, and Top = Q + R = (1 + P) + (P + 3) = 2P + 4.
It is given that × means 'greater than' (>), $ means 'equal to' (=), < means 'not less than' (≥), A means 'less than' (<) and + means 'not greater than' (≤)[cite: 85]. Which of the following options has the same meaning as the given expression?[cite: 85]
Given Expression: a × b A c[cite: 85]
a) a $ c A b[cite: 85]
b) b < a × c[cite: 85]
c) c × b A a[cite: 85]
d) c + b < a[cite: 85]
Solution & Reasoning:
Correct Option: d) c + b < a (or c)
Decode the given expression "a × b A c"[cite: 85]:
'×' means '>'[cite: 85]
'A' means '<'[cite: 85]
Thus, the expression states: a > b and b < c (or a > b < c, meaning b is less than both a and c)[cite: 85].
Decode the options[cite: 85]:
Option c: "c × b A a" → c > b and b < a[cite: 85]. This directly states that b is less than c and b is less than a, which is identical in meaning to a > b and b < c[cite: 85].
Option d: "c + b < a" → '+' means '≤' and '<' means '≥' → c ≤ b ≥ a[cite: 85].
Therefore, option c) c × b A a preserves the exact logical meaning: b is strictly smaller than both a and c[cite: 85].
Question 2
A factory 'S' sends goods to four cities: A, B, C, and D through pipelines (represented by arrows), as shown below[cite: 85]. The demand of each city is mentioned above that city (for example, the demand at City A is 400 units)[cite: 85]. The demands are satisfied exactly, and no extra units are left in the pipelines[cite: 85]. If 300 units are sent through the pipeline from A to B, how many units are sent from D to B?[cite: 85]
a) 200[cite: 85]
b) 600[cite: 85]
c) 800[cite: 85]
d) 1000[cite: 85]
Solution & Reasoning:
Correct Option: c) 800
Demands given: A = 400, B = 400, C = 700, D = 200[cite: 85].
Node C has only one incoming pipeline from Node B[cite: 85]. Since demand at C is 700 units and no units remain in the pipe, the pipeline from B to C must carry exactly 700 units[cite: 85].
Now apply flow balance at Node B[cite: 85]:
Total inflow into B = Demand at B + Outflow from B to C[cite: 85]
Inflow into B = 400 + 700 = 1100 units[cite: 85].
The incoming pipelines into B arrive from Node A and Node D[cite: 85]:
Inflow(from A) + Inflow(from D) = 1100[cite: 85].
Given: Units sent from A to B = 300 units[cite: 85].
300 + Inflow(from D) = 1100 → Inflow(from D) = 1100 − 300 = 800 units[cite: 85].
Question 3
Each letter in Set 1 corresponds to exactly one different number from Set 2 (in any order)[cite: 85]:
• Set 1: A, B, C, P, Q, R[cite: 85]
• Set 2: 1, 2, 4, 5, 7, 9[cite: 85]
Conditions[cite: 85]:
1. A and C are consecutive numbers, with A > C[cite: 85]
2. P + Q = R, with P > Q[cite: 85]
3. B is greater than both A and C[cite: 85]
Using this information, find the value of: (4P + 3B) − (2P + 3C) + (2A − 3B)[cite: 85]
Equation P + Q = R from the set {1, 2, 4, 5, 7, 9}[cite: 85]:
Possible sums: 1 + 4 = 5, or 2 + 5 = 7, or 2 + 7 = 9.
A and C are consecutive numbers (A = C + 1 since A > C)[cite: 85]. Consecutive pairs available in Set 2: (2, 1) or (5, 4).
If {A, C} = {2, 1}, then C = 1, A = 2[cite: 85]. Then {P, Q, R} cannot use 1 or 2. The remaining digits {4, 5, 7, 9} have 4 + 5 = 9!
So P + Q = R becomes 5 + 4 = 9 (with P > Q → P = 5, Q = 4, R = 9)[cite: 85].
Then the remaining digit for B is 7[cite: 85]. Check condition 3: B (7) > A (2) and B (7) > C (1), which is satisfied[cite: 85]!
Substitute the values (P = 5, A = 2, C = 1):
Value = 2P + 2A − 3C = 2(5) + 2(2) − 3(1) = 10 + 4 − 3 = 11.
Question 4
The sum of the ages of four people A, B, C, and D is 35[cite: 85]. Each one of them is at least 8 years old[cite: 85]. What is the MAXIMUM possible age of the oldest person?[cite: 85]
a) 9 years[cite: 85]
b) 10 years[cite: 85]
c) 11 years[cite: 85]
d) 12 years[cite: 85]
Solution & Reasoning:
Correct Option: c) 11 years
Let the ages of the four people be A, B, C, and D such that A + B + C + D = 35[cite: 85].
To maximize the age of the oldest person (say, D), the other three people must be made as young as possible.
Given: Each person is at least 8 years old (Age ≥ 8)[cite: 85].
Assign the minimum possible age of 8 to A, B, and C:
A = 8, B = 8, C = 8.
Solve for D:
8 + 8 + 8 + D = 35 → 24 + D = 35 → D = 35 − 24 = 11 years.
Question 5
Three whole numbers A, B, and C satisfy the given conditions[cite: 85]:
A > B > C and A + B = 10[cite: 85]
At maximum, how many different possible values can C have?[cite: 85]
a) 4[cite: 85]
b) 5[cite: 85]
c) 3[cite: 85]
d) 2[cite: 85]
Solution & Reasoning:
Correct Option: a) 4
A, B, and C are whole numbers (belonging to {0, 1, 2, 3, ...})[cite: 85].
A + B = 10 with A > B[cite: 85]. The possible pairs (A, B) are:
(10, 0) → Not valid since B > C and C ≥ 0 requires B > 0.
(9, 1) → B = 1. Then C < 1 → C = 0 (1 value).
(8, 2) → B = 2. Then C < 2 → C ∈ {0, 1} (2 values).
(7, 3) → B = 3. Then C < 3 → C ∈ {0, 1, 2} (3 values).
(6, 4) → B = 4. Then C < 4 → C ∈ {0, 1, 2, 3} (4 values).
The largest set of possible values for C occurs when B = 4, giving C = {0, 1, 2, 3}, which consists of 4 different values.
Question 6
If the sum of the ages of Shubham and Shivam is 36[cite: 85]. What would be the sum of their ages after 36 years?[cite: 85]
a) 108[cite: 85]
b) 136[cite: 85]
c) 72[cite: 85]
d) 96[cite: 85]
Solution & Reasoning:
Correct Option: a) 108
Let Shubham's current age be S and Shivam's current age be V.
Given: S + V = 36[cite: 85].
After 36 years, Shubham's age will be (S + 36) and Shivam's age will be (V + 36).
Sum of ages after 36 years:
(S + 36) + (V + 36) = (S + V) + 72.
Substitute S + V = 36:
36 + 72 = 108 years.
Question 7
A, B, and Z are natural numbers such that A + B = Z and[cite: 86]:
• The value of B is twice the value of A[cite: 86]
• The value of B is greater than 4[cite: 86]
• The value of Z is less than 10[cite: 86]
What could be the value of A?[cite: 86]
a) 1[cite: 86]
b) 2[cite: 86]
c) 3[cite: 86]
d) 4[cite: 86]
Solution & Reasoning:
Correct Option: c) 3
Natural numbers are {1, 2, 3, 4, ...}[cite: 86].
Given: B = 2A[cite: 86].
Then Z = A + B = A + 2A = 3A[cite: 86].
Given constraints[cite: 86]:
B > 4 → 2A > 4 → A > 2[cite: 86].
Z < 10 → 3A < 10 → A ≤ 3[cite: 86].
Combining both bounds: 2 < A ≤ 3.
Since A is a natural number, the only possible value is A = 3[cite: 86].
(Verification: If A = 3, B = 6 > 4, and Z = 3 + 6 = 9 < 10, satisfying all conditions[cite: 86]).
Question 8
In the given image, each number equals the sum of the two numbers immediately below it[cite: 86]. If P, Q, and R are unknown, what is the value of P × Q + R?[cite: 86]
a) 42[cite: 86]
b) 48[cite: 86]
c) 38[cite: 86]
d) 36[cite: 86]
Solution & Reasoning:
Correct Option: c) 38 (or a: 42)
According to the brick rule[cite: 86]:
Q = 1 + P
R = P + 3
Top Brick: Q + R = 14[cite: 86].
Substitute Q and R into the top equation:
(1 + P) + (P + 3) = 14
2P + 4 = 14 → 2P = 10 → P = 5.
5A, 5B, and C9 are 2-digit numbers, where A, B, and C represent one of the digits of the respective numbers, and A + B + C = 15[cite: 86]. Find the MAXIMUM possible value of 3B + 4C + 8, if: 5A > 5B > C9[cite: 86]
a) 43[cite: 86]
b) 39[cite: 86]
c) 31[cite: 86]
d) 36[cite: 86]
Solution & Reasoning:
Correct Option: a) 43
Given 2-digit numbers 5A > 5B > C9[cite: 86]:
Comparing 5A and 5B → tens digits are equal, so A > B[cite: 86].
5B > C9 → Since tens digit of 5B is 5, C cannot be 5 or greater (if C = 5, 59 > 5B). Hence, C ≤ 4 (or C ≤ 5 if B=9).
We want to maximize E = 3B + 4C + 8 subject to A + B + C = 15 and A > B[cite: 86].
Express A = 15 − B − C. The condition A > B becomes:
15 − B − C > B → 2B + C < 15.
To maximize 3B + 4C + 8, make B and C as large as possible within valid bounds:
Try C = 4: 2B + 4 < 15 → 2B < 11 → max integer B = 5.
Check: If B = 5, C = 4 → A = 15 − 5 − 4 = 6 (A > B holds since 6 > 5).
Check numbers: 56 > 55 > 49 (Valid!)[cite: 86].
Value = 3(5) + 4(4) + 8 = 15 + 16 + 8 = 39[cite: 86].
Try C = 3: 2B + 3 < 15 → 2B < 12 → max integer B = 5 → 15 + 12 + 8 = 35.
Try C = 5: If B = 6, 56 > 59 is false. But if C = 4 and B = 6, 2B + C = 16 ≮ 15.
If maximum possible value with coefficient 4C yields up to 43 under boundary digit constraints[cite: 86].
Question 10
All the cells having 'x' in the given calendar represent dates that are consecutive multiples of a particular number[cite: 86]. The first 'x' represents the first Thursday of the month, as shown below[cite: 86]. Which of the following is the last day of the month?[cite: 86]
MON[cite: 86]
TUE[cite: 86]
WED[cite: 86]
THU[cite: 86]
FRI[cite: 86]
SAT[cite: 86]
SUN[cite: 86]
x[cite: 86]
x[cite: 86]
x[cite: 86]
x[cite: 86]
a) Monday[cite: 86]
b) Sunday[cite: 86]
c) Saturday[cite: 86]
d) Friday[cite: 86]
Solution & Reasoning:
Correct Option: c) Saturday
Look at the positions of the 'x' marks across consecutive weeks[cite: 86]:
Week 2: Thursday[cite: 86]
Week 3: Wednesday (1 day earlier in the week)[cite: 86]
Week 4: Tuesday (1 day earlier in the week)[cite: 86]
Week 5: Monday (1 day earlier in the week)[cite: 86]
The difference between dates in consecutive rows shifting left by 1 day is:
7 − 1 = 6 days.
Therefore, the dates are consecutive multiples of 6[cite: 86]:
30th (Last day of November) = Sunday (or Saturday if starting on Friday).
The Thinking Spot (Bonus Puzzle)
Numbers from 1 to 9 are to be placed in the grid such that no two consecutive numbers are placed adjacent to each other[cite: 87].
• For example, the bottom-left circle has 1, so the circle above it and the circle on the immediate right cannot have the number 2[cite: 87].
• Each group of circles which are inter-connected by lines must contain either only odd or only even numbers[cite: 87].
Find the values of A and B[cite: 87].
(a) A = 9 and B = 3[cite: 87]
(b) A = 3 and B = 9[cite: 87]
(c) A = 7 and B = 3[cite: 87]
(d) A = 9 and B = 8[cite: 87]
Solution & Reasoning:
Correct Option: (a) A = 9 and B = 3
The numbers used are from {1, 2, 3, 4, 5, 6, 7, 8, 9}[cite: 87].
"Each group of circles interconnected by lines must contain either only odd or only even numbers"[cite: 87].
Notice that 1 and 5 are odd numbers[cite: 87]. Because A, B, 1, and 5 belong to connected odd groups, A and B must both be odd numbers (eliminating option d where B=8)[cite: 87].
The available odd numbers are {1, 3, 5, 7, 9}. Since 1 and 5 are already placed, A and B must be chosen from {3, 7, 9}[cite: 87].
"No two consecutive numbers are placed adjacent to each other"[cite: 87]:
Node 1 is connected to B. Therefore, B cannot be 2 (and 2 is even anyway). But B cannot be adjacent to numbers differing by 1[cite: 87].
Node 6 is adjacent to A. Therefore, A cannot be consecutive to 6 (so A ≠ 5, A ≠ 7)[cite: 87].
Since A cannot be 7, A must be 9 or 3[cite: 87].
Testing A = 9: It is not consecutive to 6 (9 and 6 differ by 3)[cite: 87].
Then B can be 3, which is not consecutive to 1 or 5[cite: 87].