Numerical Relations and Reasoning

This part of Analytical Aptitude covers puzzles where the answer comes from combining several small facts about order, position or number: who sits where, who is older, how far and in which direction someone ends up, what the time or the day is, which statement about an inequality chain must hold, or which number satisfies a set of digit conditions. No single formula solves them. What solves them is a disciplined representation (a grid, a number line, a coordinate plane, an equation) and a rule for using the clues in the right order.

For each family the note gives that representation, the arithmetic facts it relies on (divisibility, HCF and LCM, clock and calendar rules), and worked examples.

Direction convention. People in a row face north unless told otherwise, so their right is east (the higher slot number). Turning "right" from north means facing east.

Divisibility rules worth using. By 3 or 9: digit sum divisible by 3 or 9. By 4: last two digits divisible by 4. By 8: last three digits divisible by 8. By 11: alternating digit sum divisible by 11. And $\gcd(a, b) \times \text{lcm}(a, b) = ab$ for positive integers $a, b$.

Ranking. If a person is $r$-th from the left and $s$-th from the right, there are $r - 1$ people to the left and $s - 1$ to the right, so the row has $(r - 1) + (s - 1) + 1 = r + s - 1$ people.

Clock angle. With the hour reduced modulo 12, $0\le h<12$, and $0\le m<60$ minutes the minute hand is at $6m$ degrees from 12 and the hour hand at $30h + 0.5m$ degrees. The angle between them is $|30h - 5.5m|$, replaced by $360$ minus that value if it exceeds $180$. The hands coincide every $\frac{720}{11} = 65\frac{5}{11}$ minutes, which is 11 times in 12 hours.

Calendars. A 365-day interval shifts the weekday by one; a 366-day interval shifts it by two. For the same date next year, check whether the actual interval contains 29 February, rather than looking only at the named year. A Gregorian year is a leap year if divisible by 4, except century years, which must be divisible by 400.

Same remainder. For a positive divisor $d$, if $a$, $b$ and $c$ leave the same remainder, then $d$ divides both differences $b-a$ and $c-b$. Provided the three integers are not all equal, the largest such divisor is $\gcd(|b-a|,|c-b|)$. If all three are equal, every positive divisor works, so there is no largest one.

Example 1 (linear arrangement). P, Q, R, S and T sit in a row facing north. R sits in the middle. T sits immediately to the right of R. Q sits at one of the ends. S does not sit at an end. P is not adjacent to S. Find the order from left to right.

Example 2 (ranking). Ravi is 7th from the left and 12th from the right in a row. The row has $7 + 12 - 1 = 18$ people.

Example 3 (inequality chain). Given $A > B$, $B \ge C$, $C = D$ and $D > E$:

Example 4 (blood relations). A is B's full sister (they have the same parents). C is B's mother. D is C's father. Then C is A's mother too (A and B are siblings), so D is A's maternal grandfather, and A is D's granddaughter.

Example 5 (directions). A walker goes 5 km north, 6 km east, 2 km south and 2 km west. Coordinates: $(0, 5)$, $(6, 5)$, $(6, 3)$, $(4, 3)$. The distance from the start is $\sqrt{4^2 + 3^2} = 5$ km, towards the north-east (both coordinates positive).

Example 6 (clock). At 3:40 the angle is $|30 \times 3 - 5.5 \times 40| = |90 - 220| = 130^\circ$.

Example 7 (calendar). In a non-leap year that begins on a Monday, 1 March falls $31 + 28 = 59$ days after 1 January. $59 = 8 \times 7 + 3$, so 1 March is 3 weekdays after Monday: a Thursday.

Example 8 (ages). The ages of a father and son are in the ratio $7 : 2$. In 10 years they will be in the ratio $9 : 4$. With present ages $7x$ and $2x$: $4(7x + 10) = 9(2x + 10)$, so $28x + 40 = 18x + 90$, $x = 5$. The ages are 35 and 10.

Example 9 (LCM). Three bells ring every 6, 8 and 12 minutes and ring together at noon. They next ring together after $\text{lcm}(6, 8, 12) = 24$ minutes.

Example 10 (same remainder). The largest number that divides 70, 100 and 130 leaving the same remainder is the HCF of the differences 30 and 30, which is 30. The common remainder is $70 - 2 \times 30 = 10$.

Example 11 (digits). A two-digit number has digit sum 9, and reversing its digits increases it by 27. With digits $a$ (tens) and $b$ (units): $a + b = 9$ and $9(b - a) = 27$, so $b - a = 3$, giving $a = 3$, $b = 6$. The number is 36.

Example 12 (divisibility). The six-digit number $52k\,746$ is divisible by 9 exactly when its digit sum $24 + k$ is a multiple of 9, so $k = 3$.

Example 13 (circular arrangement). Six friends A, B, C, D, E, F sit around a round table, evenly spaced, all facing the centre. A sits opposite D. B sits immediately to the left of A. E sits immediately to the right of D. C does not sit next to D. Who sits opposite C, and who is immediately to the right of F?

Example 14 (when the hands coincide). Between 4 and 5 o'clock, when are the hands together? The minute hand gains $5.5^\circ$ per minute on the hour hand, which starts $120^\circ$ ahead. They meet after $\frac{120}{5.5} = \frac{240}{11} = 21\frac{9}{11}$ minutes, at about 4:21:49. The tempting answer 4:20 forgets that the hour hand keeps moving.

Example 15 (same date, next year). 10 January 2025 was a Friday. A non-leap year has $365 = 52 \times 7 + 1$ days, so 10 January 2026 is one weekday later: a Saturday. Leap-day trap: from 10 January 2027 (a Sunday) to 10 January 2028 there is no 29 February in between, so the shift is 1 day (Monday); but from 10 March 2027 (a Wednesday) to 10 March 2028 the span includes 29 February 2028, so the shift is 2 days (Friday).

Example 16 (relation in a sentence). Pointing to a man in a photograph, Ravi says, "He is the son of my grandfather's only son." If the grandfather is Ravi's father's father, his only son is Ravi's father, and the son of Ravi's father is Ravi or Ravi's brother; pointing to a photograph does not exclude Ravi himself. The man is uniquely Ravi's brother under this reading only if the question additionally states that he is not Ravi. If the grandfather were the mother's father, his only son would be Ravi's maternal uncle and the man Ravi's cousin. If the options contain both brother and cousin, the question is ambiguous; otherwise only one reading matches an option. Work from the far end of the chain ("my grandfather") inwards.

Example 17 (direction after turns). A person faces north, turns $90^\circ$ to the right, then $135^\circ$ to the left. Measure headings clockwise from north: $0^\circ + 90^\circ - 135^\circ = -45^\circ$, which is $315^\circ$: north-west.