Clock questions use angular speed; calendar questions use cycles of weekdays. Both reward a precise reference point. A clock hand's position is measured from 12 o'clock, and a weekday is found by counting complete weeks plus the remainder of days.
1. Clock-hand angular speeds
The minute hand turns 360° in 60 minutes, or 6° per minute. The hour hand turns 360° in 12 hours, or 0.5° per minute. At H hours and M minutes after 12, the hour hand is at 30H+0.5M degrees, and the minute hand at 6M degrees. Their smaller angle is the smaller of the absolute difference and 360° minus that difference.
Worked example 1. At 3:20, the hour hand is at 3×30+20×0.5=100°. The minute hand is at 20×6=120°. Their smaller angle is 20°.
At exactly 3:00 the hands are 90° apart, but the hour hand moves after 3:00. Treating it as stationary at the 3 marker is the usual clock mistake. For an overlap, set their angles equal modulo 360. Relative angular speed is 6−0.5=5.5° per minute.
2. Meeting and opposition
Starting together at 12:00, the minute hand gains a full 360° on the hour hand in 360/5.5 = 720/11 minutes, about 65 minutes 27 seconds. It gains 180° in half that interval for the first opposition. Across a 12-hour cycle, the hands coincide 11 times when the starting and ending 12 o'clock positions are not double-counted.
When solving between named hours, keep the hour hand's initial 30H° lead in the equation. For instance, between 4 and 5 o'clock, overlap occurs when 6M=120+0.5M; therefore M=120/5.5=240/11 minutes after 4.
3. Fast and slow clocks
A clock gaining five minutes per real hour displays 65 minutes of movement for each 60 real minutes. Convert between displayed and real elapsed time with a ratio. Do not add the gain once if several hours pass. If a clock loses two minutes per real hour, it displays 58 minutes per real hour. The starting display time matters if a specific final clock reading is requested.
Worked example 2. A clock gains 3 minutes every real hour and is correct at noon. After 10 real hours it is 30 minutes fast, so when actual time is 10:00 p.m. it shows 10:30 p.m.
4. Weekday cycles
Every seven days the weekday repeats. Divide the number of elapsed days by 7 and use the remainder to shift the weekday. A common year has 365 days = 52 weeks + 1 day, so the same date in the following year usually shifts by one weekday. A leap year has 366 = 52 weeks + 2 days, but date-specific shifts depend on whether February 29 lies in the interval.
The Gregorian leap-year rule: a year divisible by 4 is a leap year, except century years must also be divisible by 400. Thus 2000 was a leap year, while 1900 was not. This rule matters when counting days across centuries.
Worked example 3. If 1 January of a non-leap year is Monday, 1 January of the next year is Tuesday because 365 mod 7=1. If the first year is leap, it is Wednesday because 366 mod 7=2.
5. Inclusive counting and date spans
“Days after” and “the nth day including today” count differently. Seven days after Monday is Monday; the seventh day including Monday is Sunday. Write a small timeline when wording could change the count by one. For dates in different months, sum the remaining days in the starting month, the complete intervening months and the days in the ending month. Apply February's correct length for that year.
6. Error checks
A smaller clock angle lies from 0° to 180°. The minute hand cannot gain a full turn on the hour hand in exactly 60 minutes because the hour hand also moves. For calendars, a shift of 0 is possible after a multiple of seven days. Always write the reference weekday before calculating the remainder.
Recall before practice
1. State the angular speed of each clock hand. 2. Find the hour hand's position at 2:30. 3. Explain the Gregorian century leap-year exception. 4. Distinguish “seven days after” from “seventh day including today”.
Chapter 15 practice — 50 questions
Choose one option for each item. Keep a separate answer list; explanations follow this set.
1. Through what angle does the minute hand of a clock turn in 46 minutes?
A. 276° B. 230° C. 23° D. 46°
2. If 25 August 1998 was a Tuesday, what was the day of the week on 11 October 1998?
A. Friday B. Sunday C. Saturday D. Monday
3. A clock gains 12 minutes in 24 hours. It was set right at 8 a.m. What time will it show at 8 p.m. on the same day?
A. 8:00 p.m. B. 8:12 p.m. C. 7:54 p.m. D. 8:06 p.m.
4. Today is Saturday. What day of the week was it 150 days ago?
A. Wednesday B. Tuesday C. Saturday D. Thursday
5. What was the day of the week on 22 May 2010?
A. Saturday B. Sunday C. Monday D. Friday
6. How many times in a day (24 hours) are the hands of a clock in a straight line but opposite in direction?
A. 22 B. 44 C. 24 D. 12
7. If 29 August 2015 was a Saturday, what was the day of the week on 15 September 2015?
A. Monday B. Tuesday C. Thursday D. Wednesday
8. The image of a clock in a mirror shows the time as 3:21. What is the actual time?
A. 8:21 B. 8:39 C. 9:21 D. 3:39
9. Today is Friday. What day of the week will it be after 60 days?
A. Monday B. Tuesday C. Wednesday D. Saturday
10. Today is Wednesday. What day of the week was it 60 days ago?
A. Sunday B. Saturday C. Tuesday D. Friday
11. How many times do the two hands of a clock coincide in a day (24 hours)?
A. 12 B. 24 C. 11 D. 22
12. What was the day of the week on 18 May 2002?
A. Saturday B. Friday C. Tuesday D. Sunday
13. What is the angle between the hour hand and the minute hand of a clock at 4:40?
A. 100° B. 80° C. 120° D. 260°
14. At what time between 5 and 6 o'clock will the two hands of a clock coincide?
A. 21 9/11 minutes past 5 B. 27 3/11 minutes past 5 C. 25 minutes past 5 D. 32 8/11 minutes past 5
15. If 17 September 1964 was a Thursday, what was the day of the week on 16 March 1964?
A. Monday B. Tuesday C. Sunday D. Thursday
16. At what time between 6 and 7 o'clock will the minute hand be 4 minute-spaces ahead of the hour hand?
A. 28 4/11 minutes past 6 B. 37 1/11 minutes past 6 C. 34 minutes past 6 D. 32 8/11 minutes past 6
17. If 10 April 1995 was a Monday, what was the day of the week on 1 August 1995?
A. Wednesday B. Tuesday C. Monday D. Saturday
18. What was the day of the week on 1 September 1992?
A. Monday B. Tuesday C. Friday D. Wednesday
19. If 22 December 2018 was a Saturday, what was the day of the week on 11 October 2018?
A. Thursday B. Friday C. Monday D. Wednesday
20. What was the day of the week on 22 December 1903?
A. Thursday B. Tuesday C. Monday D. Wednesday
21. A clock seen in a mirror shows 9:40. What will the clock actually show 35 minutes from now?
A. 1:45 B. 2:20 C. 10:15 D. 2:55
22. How many odd days are there from 1 July 1949 to 31 October 1949?
A. 3 B. 4 C. 5 D. 1
23. The calendar of the year 1984 can be used again for which year?
A. 2012 B. 1990 C. 1995 D. 2001
24. The 14th of March 2002 was a Thursday. What day of the week was the 23rd of the same month?
A. Saturday B. Sunday C. Thursday D. Friday
25. What is the angle between the hour hand and the minute hand of a clock at 10:22?
A. 190° B. 181° C. 168° D. 179°
26. If 4 June 1958 was a Wednesday, what was the day of the week on 13 March 1958?
A. Thursday B. Friday C. Wednesday D. Saturday
27. What is the angle between the hour hand and the minute hand of a clock at 11:56?
A. 22° B. 50° C. 6° D. 338°
28. At what time between 10 and 11 o'clock will the hands of a clock be at right angles for the first time?
A. 5 5/11 minutes past 10 B. 38 2/11 minutes past 10 C. 54 6/11 minutes past 10 D. 5 minutes past 10
29. A clock seen in a mirror shows 8:09. What will the clock actually show 20 minutes from now?
A. 4:11 B. 3:51 C. 3:31 D. 8:29
30. The 24th of December 2027 was a Friday. What day of the week was the 1st of the same month?
A. Thursday B. Friday C. Wednesday D. Tuesday
31. What was the day of the week on 27 May 1973?
A. Sunday B. Saturday C. Monday D. Friday
32. Which year after 1937 will have the same calendar as 1937?
A. 1949 B. 1944 C. 1943 D. 1948
33. What is the angle between the hour hand and the minute hand of a clock at 4:25?
A. 342.5° B. 17.5° C. 5° D. 30°
34. At what time between 7 and 8 o'clock will the hands of a clock be at right angles for the first time?
A. 21 9/11 minutes past 7 B. 20 minutes past 7 C. 38 2/11 minutes past 7 D. 54 6/11 minutes past 7
35. Through what angle does the hour hand of a clock turn from 7:10 to 11:50?
A. 120° B. 1680° C. 140° D. 280°
36. What is the angle between the hour hand and the minute hand of a clock at 9:25?
A. 132.5° B. 227.5° C. 145° D. 120°
37. At what time between 7 and 8 o'clock will the hands of a clock be in a straight line but opposite in direction?
A. 38 2/11 minutes past 7 B. 5 minutes past 7 C. 5 5/11 minutes past 7 D. 10 10/11 minutes past 7
38. Through what angle does the hour hand of a clock turn from 4:00 to 5:20?
A. 80° B. 480° C. 40° D. 30°
39. How many leap years are there from 1801 to 1920 (both inclusive)?
A. 31 B. 28 C. 30 D. 29
40. Through what angle does the hour hand of a clock turn from 2:40 to 5:10?
A. 150° B. 75° C. 90° D. 900°
41. What was the day of the week on 7 June 1700?
A. Sunday B. Wednesday C. Monday D. Tuesday
42. The 1st of August 1952 was a Friday. How many Tuesdays were there in August 1952?
A. 3 B. 5 C. 6 D. 4
43. The 1st of February 2029 was a Thursday. How many Tuesdays were there in February 2029?
A. 4 B. 3 C. 6 D. 5
44. What is the reflex angle between the hour hand and the minute hand of a clock at 9:50?
A. 355° B. 340° C. 330° D. 5°
45. What was the day of the week on 6 August 1800?
A. Wednesday B. Thursday C. Tuesday D. Monday
46. At what time between 5 and 6 o'clock will the two hands of a clock be 4 minute-spaces apart for the first time?
A. 21 minutes past 5 B. 31 7/11 minutes past 5 C. 22 10/11 minutes past 5 D. 27 3/11 minutes past 5
47. In how many minutes does the minute hand of a clock gain 20 minute-spaces over the hour hand?
A. 22 10/11 minutes B. 21 9/11 minutes C. 20 8/11 minutes D. 20 minutes
48. What is the reflex angle between the hour hand and the minute hand of a clock at 9:32?
A. 266° B. 94° C. 282° D. 250°
49. At what time between 2 and 3 o'clock will the angle between the hands of a clock be 30° for the first time?
A. 5 minutes past 2 B. 10 10/11 minutes past 2 C. 5 5/11 minutes past 2 D. 16 4/11 minutes past 2
50. Which year before 1979 will have the same calendar as 1979?
A. 1974 B. 1973 C. 1962 D. 1972
Chapter 15 — Answers and explanations
After checking the key, re-solve any miss without looking at the formula.
1. A. The minute hand turns 360° in 60 minutes, i.e. 6° per minute. In 46 minutes it turns 46 × 6 = 276° (0.5° per minute is the hour hand's rate).
APAR26-15-02 | Angle traced by a hand | Easy
2. B. Days from 25 August 1998 to 11 October 1998: August 6 + September 30 + October 11 = 47 days. 47 = 6 weeks + 5 odd days. Tuesday + 5 = Sunday.
APAR26-15-27 | Day of another date from a known day | Easy
3. D. Error is proportional to elapsed time: in 12 hours the clock gains 12 × 12/24 = 6 minutes. So at 8 p.m. it shows 8 p.m. + 6 min = 8:06 p.m. (a fast clock shows a later time, a slow clock an earlier one).
APAR26-15-03 | Time shown by a fast/slow clock | Easy
4. A. Every 7 days the weekday repeats, so only the remainder matters: 150 = 21 × 7 + 3. 3 days before Saturday is Wednesday.
APAR26-15-32 | Day after / before n days | Easy
5. A. Count odd days up to 22 May 2010. 2000 years (a multiple of 400) → 0 odd days; 9 years (2 leap + 7 ordinary) → 4 odd days; months January–April of 2010 = 120 days → 1 odd days; date 22 → 1 odd days. Total = (0 + 4 + 1 + 22) mod 7 = 6, and 0 = Sunday, 1 = Monday, …, so the day was Saturday.
APAR26-15-26 | Day of the week for a date | Easy
6. A. The hands are opposite (180° apart) once an hour, but the 5–6 and 6–7 instants are the same (6:00), so 11 times in 12 hours and 22 times in a day, not 24.
APAR26-15-04 | Clock facts | Easy
7. B. Days from 29 August 2015 to 15 September 2015: August 2 + September 15 = 17 days. 17 = 2 weeks + 3 odd days. Saturday + 3 = Tuesday.
APAR26-15-29 | Day of another date from a known day | Easy
8. B. Mirror image and real time add up to 12:00, so real time = 11:60 − 3:21 = 8:39 (subtracting from 12:00 with the hour off by one, e.g. 9:21, is the usual slip). Rule: subtract from 11:60.
APAR26-15-01 | Mirror image of a clock | Easy
9. B. Every 7 days the weekday repeats, so only the remainder matters: 60 = 8 × 7 + 4. 4 days after Friday is Tuesday.
APAR26-15-31 | Day after / before n days | Easy
10. B. Every 7 days the weekday repeats, so only the remainder matters: 60 = 8 × 7 + 4. 4 days before Wednesday is Saturday.
APAR26-15-30 | Day after / before n days | Easy
11. D. The hands coincide 11 times in 12 hours (only once between 11 and 1, at 12 o'clock), so in 24 hours they coincide 2 × 11 = 22 times, not 24.
APAR26-15-06 | Clock facts | Easy
12. A. Count odd days up to 18 May 2002. 2000 years (a multiple of 400) → 0 odd days; 1 years (0 leap + 1 ordinary) → 1 odd days; months January–April of 2002 = 120 days → 1 odd days; date 18 → 4 odd days. Total = (0 + 1 + 1 + 18) mod 7 = 6, and 0 = Sunday, 1 = Monday, …, so the day was Saturday.
APAR26-15-28 | Day of the week for a date | Easy
13. A. Minute hand from 12 = 6° × 40 = 240°. Hour hand from 12 = 30° × 4 + 40/2 = 140° (the hour hand moves 0.5° per minute; forgetting this gives 120°). Difference = |140 − 240| = 100°. The smaller angle between the hands is 100°.
APAR26-15-05 | Angle between hands | Easy
14. B. At 5 o'clock the minute hand is 5 × 5 = 25 minute-spaces behind the hour hand; to coincide it must gain 25 spaces. The minute hand gains 55 spaces in 60 minutes, so 25 spaces take 25 × 60/55 = 25 × 12/11 = 27 3/11 minutes. Required time: 27 3/11 minutes past 5 (not 25 minutes — the 12/11 factor must be applied).
APAR26-15-07 | Hands coincide | Easy
15. A. Days from 16 March 1964 to 17 September 1964: March 15 + April 30 + May 31 + June 30 + July 31 + August 31 + September 17 = 185 days. 185 = 26 weeks + 3 odd days. Thursday − 3 = Monday.
APAR26-15-39 | Day of another date from a known day | Medium
16. B. At 6 o'clock the minute hand is 5 × 6 = 30 minute-spaces behind the hour hand; to be 4 spaces ahead it must gain 30 + 4 = 34 spaces. The minute hand gains 55 spaces in 60 minutes, so 34 spaces take 34 × 60/55 = 34 × 12/11 = 37 1/11 minutes. Required time: 37 1/11 minutes past 6 (not 34 minutes — the 12/11 factor must be applied).
APAR26-15-16 | Hands a given distance apart | Medium
17. B. Days from 10 April 1995 to 1 August 1995: April 20 + May 31 + June 30 + July 31 + August 1 = 113 days. 113 = 16 weeks + 1 odd day. Monday + 1 = Tuesday.
APAR26-15-43 | Day of another date from a known day | Medium
18. B. Count odd days up to 1 September 1992. 1900 years → 1 odd days (100 yrs = 5, 200 = 3, 300 = 1, 400 = 0); 91 years (22 leap + 69 ordinary) → 1 odd days; months January–August of 1992 = 244 days → 6 odd days; date 1 → 1 odd days. Total = (1 + 1 + 6 + 1) mod 7 = 2, and 0 = Sunday, 1 = Monday, …, so the day was Tuesday.
APAR26-15-38 | Day of the week for a date | Medium
19. A. Days from 11 October 2018 to 22 December 2018: October 20 + November 30 + December 22 = 72 days. 72 = 10 weeks + 2 odd days. Saturday − 2 = Thursday.
APAR26-15-34 | Day of another date from a known day | Medium
20. B. Count odd days up to 22 December 1903. 1900 years → 1 odd days (100 yrs = 5, 200 = 3, 300 = 1, 400 = 0); 2 years (0 leap + 2 ordinary) → 2 odd days; months January–November of 1903 = 334 days → 5 odd days; date 22 → 1 odd days. Total = (1 + 2 + 5 + 22) mod 7 = 2, and 0 = Sunday, 1 = Monday, …, so the day was Tuesday.
APAR26-15-40 | Day of the week for a date | Medium
21. D. Mirror image and real time add up to 12:00, so real time = 11:60 − 9:40 = 2:20 (subtracting from 12:00 with the hour off by one, e.g. 3:40, is the usual slip). After 35 minutes the clock shows 2:20 + 35 min = 2:55 (adding 35 minutes to the mirror reading 9:40 gives 10:15, a trap).
APAR26-15-13 | Mirror image of a clock | Medium
22. B. Days in July–October 1949: 31 + 31 + 30 + 31 = 123. 123 = 17 × 7 + 4, so the number of odd days is 4. (Taking every month as 30 days gives 1.)
APAR26-15-41 | Odd days in a range of months | Medium
23. A. Two years share a calendar when the odd days between them total a multiple of 7 and both have the same leap status (1984 is a leap year). Odd days (ordinary year 1, leap year 2): 1984: 2, 1985: 1, 1986: 1, 1987: 1, …, 2010: 1, 2011: 1; sum = 35 = 5 × 7. So 2012 (leap) has the same calendar as 1984, 28 years later. (The 6/11/28-year rules of thumb apply only in specific positions of the leap cycle.)
APAR26-15-42 | Year with the same calendar | Medium
24. A. Difference of dates = 23 − 14 = 9 days = 1 weeks + 2 days. Counting forward 2 from Thursday gives Saturday. (Counting both end dates, i.e. 10 days, is the usual off-by-one error.)
APAR26-15-36 | Another date of the same month | Medium
25. D. Minute hand from 12 = 6° × 22 = 132°. Hour hand from 12 = 30° × 10 + 22/2 = 311° (the hour hand moves 0.5° per minute; forgetting this gives 168°). Difference = |311 − 132| = 179°. The smaller angle between the hands is 179°.
APAR26-15-19 | Angle between hands | Medium
26. A. Days from 13 March 1958 to 4 June 1958: March 18 + April 30 + May 31 + June 4 = 83 days. 83 = 11 weeks + 6 odd days. Wednesday − 6 = Thursday.
APAR26-15-35 | Day of another date from a known day | Medium
27. A. Minute hand from 12 = 6° × 56 = 336°. Hour hand from 12 = 30° × 11 + 56/2 = 358° (the hour hand moves 0.5° per minute; forgetting this gives 6°). Difference = |358 − 336| = 22°. The smaller angle between the hands is 22°.
APAR26-15-09 | Angle between hands | Medium
28. A. An angle of 90° corresponds to 90/6 = 15 minute-spaces, so the hands are 90° apart when the minute hand is 15 spaces behind or 15 spaces ahead of the hour hand. At 10 o'clock the minute hand is 5 × 10 = 50 spaces behind, so the gain needed within this hour is 50 + 15 − 60 = 5 (minute hand 45 spaces behind, i.e. 15 spaces ahead round the dial) or 50 − 15 = 35 (minute hand 15 spaces behind). The first of these is a gain of 5 spaces. Gaining 55 spaces takes 60 minutes, so 5 spaces take 5 × 12/11 = 5 5/11 minutes. Answer: 5 5/11 minutes past 10. (The other time in this hour is 38 2/11 minutes past 10.)
APAR26-15-14 | Hands at right angles | Medium
29. A. Mirror image and real time add up to 12:00, so real time = 11:60 − 8:09 = 3:51 (subtracting from 12:00 with the hour off by one, e.g. 4:09, is the usual slip). After 20 minutes the clock shows 3:51 + 20 min = 4:11 (adding 20 minutes to the mirror reading 8:09 gives 8:29, a trap).
APAR26-15-17 | Mirror image of a clock | Medium
30. C. Difference of dates = 1 − 24 = -23 days = -3 weeks − 2 days. Counting back 2 from Friday gives Wednesday. (Counting both end dates, i.e. 24 days, is the usual off-by-one error.)
APAR26-15-44 | Another date of the same month | Medium
31. A. Count odd days up to 27 May 1973. 1900 years → 1 odd days (100 yrs = 5, 200 = 3, 300 = 1, 400 = 0); 72 years (18 leap + 54 ordinary) → 6 odd days; months January–April of 1973 = 120 days → 1 odd days; date 27 → 6 odd days. Total = (1 + 6 + 1 + 27) mod 7 = 0, and 0 = Sunday, 1 = Monday, …, so the day was Sunday.
APAR26-15-45 | Day of the week for a date | Medium
32. C. Two years share a calendar when the odd days between them total a multiple of 7 and both have the same leap status (1937 is an ordinary year). Odd days (ordinary year 1, leap year 2): 1937: 1, 1938: 1, 1939: 1, 1940: 2, 1941: 1, 1942: 1; sum = 7 = 1 × 7. So 1943 (ordinary) has the same calendar as 1937, 6 years later. (The 6/11/28-year rules of thumb apply only in specific positions of the leap cycle.)
APAR26-15-33 | Year with the same calendar | Medium
33. B. Minute hand from 12 = 6° × 25 = 150°. Hour hand from 12 = 30° × 4 + 25/2 = 132.5° (the hour hand moves 0.5° per minute; forgetting this gives 30°). Difference = |132.5 − 150| = 17.5°. The smaller angle between the hands is 17.5°.
APAR26-15-12 | Angle between hands | Medium
34. A. An angle of 90° corresponds to 90/6 = 15 minute-spaces, so the hands are 90° apart when the minute hand is 15 spaces behind or 15 spaces ahead of the hour hand. At 7 o'clock the minute hand is 5 × 7 = 35 spaces behind, so the gain needed within this hour is 35 − 15 = 20 (minute hand 15 spaces behind) or 35 + 15 = 50 (minute hand 15 spaces ahead). The first of these is a gain of 20 spaces. Gaining 55 spaces takes 60 minutes, so 20 spaces take 20 × 12/11 = 21 9/11 minutes. Answer: 21 9/11 minutes past 7. (The other time in this hour is 54 6/11 minutes past 7.)
APAR26-15-11 | Hands at right angles | Medium
35. C. Time elapsed = 11:50 − 7:10 = 280 minutes. The hour hand moves 0.5° per minute, so the angle = 280 × 0.5 = 140° (counting only whole hours gives 120°, which ignores the 40 extra minutes).
APAR26-15-08 | Angle traced by the hour hand between two times | Medium
36. A. Minute hand from 12 = 6° × 25 = 150°. Hour hand from 12 = 30° × 9 + 25/2 = 282.5° (the hour hand moves 0.5° per minute; forgetting this gives 120°). Difference = |282.5 − 150| = 132.5°. The smaller angle between the hands is 132.5°.
APAR26-15-10 | Angle between hands | Medium
37. C. An angle of 180° corresponds to 180/6 = 30 minute-spaces, so the hands are 180° apart when the minute hand is 30 spaces behind or 30 spaces ahead of the hour hand. At 7 o'clock the minute hand is 5 × 7 = 35 spaces behind, so the gain needed within this hour is 35 − 30 = 5 (minute hand 30 spaces behind) or 35 + 30 − 60 = 5 (minute hand 30 spaces behind, i.e. 30 spaces ahead round the dial). The first of these is a gain of 5 spaces. Gaining 55 spaces takes 60 minutes, so 5 spaces take 5 × 12/11 = 5 5/11 minutes. Answer: 5 5/11 minutes past 7.
APAR26-15-15 | Hands opposite | Medium
38. C. Time elapsed = 5:20 − 4:00 = 80 minutes. The hour hand moves 0.5° per minute, so the angle = 80 × 0.5 = 40° (counting only whole hours gives 30°, which ignores the 20 extra minutes).
APAR26-15-18 | Angle traced by the hour hand between two times | Medium
39. D. Multiples of 4 from 1804 to 1920: (1920 − 1804)/4 + 1 = 30. The century year 1900 is not divisible by 400, so it does not count. Leap years = 29. (Blindly taking 120/4 = 30 is the trap.)
APAR26-15-37 | Counting leap years in a range | Medium
40. B. Time elapsed = 5:10 − 2:40 = 150 minutes. The hour hand moves 0.5° per minute, so the angle = 150 × 0.5 = 75° (counting only whole hours gives 90°, which ignores the 30 extra minutes).
APAR26-15-20 | Angle traced by the hour hand between two times | Medium
41. C. Count odd days up to 7 June 1700. 1600 years (a multiple of 400) → 0 odd days; 99 years (24 leap + 75 ordinary) → 4 odd days; months January–May of 1700 = 151 days → 4 odd days; date 7 → 0 odd days. Total = (0 + 4 + 4 + 7) mod 7 = 1, and 0 = Sunday, 1 = Monday, …, so the day was Monday.
APAR26-15-49 | Day of the week for a date | Difficult
42. D. August 1952 has 31 days. The first Tuesday falls on the 5th, then every 7 days: 5, 12, 19, 26. That is 4 Tuesdays. (A 31-day month has 3 weekdays occurring 5 times: the first 3.)
APAR26-15-46 | Number of a given weekday in a month | Difficult
43. A. February 2029 has 28 days (ordinary year). The first Tuesday falls on the 6th, then every 7 days: 6, 13, 20, 27. That is 4 Tuesdays. (A 28-day month has 0 weekdays occurring 5 times: none.)
APAR26-15-47 | Number of a given weekday in a month | Difficult
44. A. Minute hand from 12 = 6° × 50 = 300°. Hour hand from 12 = 30° × 9 + 50/2 = 295° (the hour hand moves 0.5° per minute; forgetting this gives 30°). Difference = |295 − 300| = 5°. The reflex angle = 360° − 5° = 355°.
APAR26-15-22 | Reflex angle between hands | Difficult
45. A. Count odd days up to 6 August 1800. 1700 years → 5 odd days (100 yrs = 5, 200 = 3, 300 = 1, 400 = 0); 99 years (24 leap + 75 ordinary) → 4 odd days; months January–July of 1800 = 212 days → 2 odd days; date 6 → 6 odd days. Total = (5 + 4 + 2 + 6) mod 7 = 3, and 0 = Sunday, 1 = Monday, …, so the day was Wednesday.
APAR26-15-48 | Day of the week for a date | Difficult
46. C. At 5 o'clock the minute hand is 5 × 5 = 25 minute-spaces behind the hour hand; to be 4 spaces behind (the first time they are 4 apart) it must gain 25 − 4 = 21 spaces. The minute hand gains 55 spaces in 60 minutes, so 21 spaces take 21 × 60/55 = 21 × 12/11 = 22 10/11 minutes. Required time: 22 10/11 minutes past 5 (not 21 minutes — the 12/11 factor must be applied).
APAR26-15-25 | Hands a given distance apart | Difficult
47. B. In 60 minutes the minute hand gains 60 − 5 = 55 minute-spaces. So 20 spaces are gained in 20 × 60/55 = 20 × 12/11 = 21 9/11 minutes (not 20 minutes: the hour hand also moves).
APAR26-15-23 | Time to gain minute-spaces | Difficult
48. A. Minute hand from 12 = 6° × 32 = 192°. Hour hand from 12 = 30° × 9 + 32/2 = 286° (the hour hand moves 0.5° per minute; forgetting this gives 78°). Difference = |286 − 192| = 94°. The reflex angle = 360° − 94° = 266°.
APAR26-15-21 | Reflex angle between hands | Difficult
49. C. An angle of 30° corresponds to 30/6 = 5 minute-spaces, so the hands are 30° apart when the minute hand is 5 spaces behind or 5 spaces ahead of the hour hand. At 2 o'clock the minute hand is 5 × 2 = 10 spaces behind, so the gain needed within this hour is 10 − 5 = 5 (minute hand 5 spaces behind) or 10 + 5 = 15 (minute hand 5 spaces ahead). The first of these is a gain of 5 spaces. Gaining 55 spaces takes 60 minutes, so 5 spaces take 5 × 12/11 = 5 5/11 minutes. Answer: 5 5/11 minutes past 2. (The other time in this hour is 16 4/11 minutes past 2.)
APAR26-15-24 | Hands at a given angle | Difficult
50. B. Two years share a calendar when the odd days between them total a multiple of 7 and both have the same leap status (1979 is an ordinary year). Odd days (ordinary year 1, leap year 2): 1978: 1, 1977: 1, 1976: 2, 1975: 1, 1974: 1, 1973: 1; sum = 7 = 1 × 7. So 1973 (ordinary) has the same calendar as 1979, 6 years earlier. (The 6/11/28-year rules of thumb apply only in specific positions of the leap cycle.)
APAR26-15-50 | Year with the same calendar | Difficult