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Clocks, Calendars & Time

Clock angles, calendars, day-of-week, time-rate arithmetic

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Clocks, Calendars & TimeAMC 10 Step by Step

Name: Date: October 28, 2026

12 problems, easiest first

What you need to know

How fast the hands move. The minute hand goes all the way around ( $360^\circ$ ) in $60$ minutes: $6^\circ$ per minute. The hour hand goes one number ( $30^\circ$ ) in $60$ minutes: $0.5^\circ$ per minute.
Where the hands are. Measure clockwise from the $12$ . At $h$ hours and $m$ minutes, the hour hand is at $30h+0.5m$ degrees and the minute hand is at $6m$ degrees. Example: at 4:20 the hour hand is at $120+10=130^\circ$ and the minute hand at $120^\circ$ , so they are $10^\circ$ apart.
The angle between the hands is $|30h-5.5m|$ degrees. If that is more than $180^\circ$ , subtract it from $360^\circ$ to get the smaller angle. Example: at 10:10, $|300-55|=245$ , so the angle is $360-245=115^\circ$ .
Weekdays repeat every $7$ days. So to move forward $n$ days, move forward by the remainder of $n$ divided by $7$ (this is " $n$ mod $7$ "). Example: if March 1 is a Monday, then May 1 is $31+30=61$ days later. Since $61=8\cdot7+5$ , May 1 is $5$ days after Monday, a Saturday.
Year to year. A common year has $365=52\cdot7+1$ days. So a date moves one weekday later each year, or two if a February 29 comes in between. Example: January 1 is a Friday in 2027, a Saturday in 2028, and a Monday in 2029, because 2028 has a February 29.
Leap years are the years divisible by $4$ , except century years, which must be divisible by $400$ . Example: $2024$ and $2000$ are leap years; $1900$ and $2100$ are not.

How AMC 10 tests it

Clock-angle questions (positions 5–12): the angle at a given time, or the first time after $h$ :00 when the hands make a given angle.
Day-of-week questions across several years (positions 10–20): count the days mod $7$ . Getting the leap days right is the whole point.
Working out which year was a leap year: two dates on the same weekday are a multiple of $7$ days apart, and that tells you whether a $366$ -day year lies between them.

Standard approaches

For the hands, write both angles in terms of $m$ , the minutes past the hour. Set their difference equal to $\pm\theta$ or $\pm(360^\circ-\theta)$ , where $\theta$ is the target angle, and keep only answers with $0\le m<60$ . Example: between 1:00 and 2:00 the hands are perpendicular twice: at $m=\frac{240}{11}$ (difference $-90^\circ$ ) and at $m=\frac{600}{11}$ (difference $-270^\circ$ ).
For weekdays, count the days between the two dates exactly, find the remainder mod $7$ , and step forward or backward in the right direction.
Decide which years are leap years before you count anything else.

Worked example

Between 7:00 and 8:00 the hands of a clock are perpendicular at exactly two moments. How many minutes apart are these moments?

(A) $30$ (B) $32\frac{8}{11}$ (C) $32\frac{10}{11}$ (D) $33$ (E) $36$

Let $m$ be the number of minutes past 7:00.

Place the hands. The hour hand starts at $7\cdot30=210^\circ$ and moves $0.5^\circ$ per minute, so it is at $210+0.5m$ degrees. The minute hand is at $6m$ degrees. The hour hand is ahead by $210+0.5m-6m=210-5.5m$ degrees.

Set up "perpendicular." The hands are perpendicular when this gap is $90^\circ$ or $-90^\circ$ (minute hand behind or ahead).

$210-5.5m=90$ gives $5.5m=120$ , so $m=\dfrac{240}{11}$ , about $21.8$ minutes.
$210-5.5m=-90$ gives $5.5m=300$ , so $m=\dfrac{600}{11}$ , about $54.5$ minutes.

Both are between $0$ and $60$ , so both happen before 8:00.

Subtract. $\dfrac{600}{11}-\dfrac{240}{11}=\dfrac{360}{11}=32\dfrac{8}{11}$ minutes. The answer is $\boxed{\textbf{(B)}\ 32\tfrac{8}{11}}$ .

Pitfalls

Putting the hour hand right on the hour mark. At 3:30 it is halfway between $3$ and $4$ , so the angle is $|90-165|=75^\circ$ , not $90^\circ$ .
Giving an angle bigger than $180^\circ$ when the smaller angle is wanted. At 10:10 the answer is $115^\circ$ , not $245^\circ$ .
Counting " $n$ years later" as $n$ weekdays later without adding one more for each February 29 in between.
Off-by-one errors when counting days. The 17th is $7$ days after the 10th (same weekday), not $8$ .

Traps that recur

Sliding the date by one: forgetting February 2003 has 28 days, or counting the tenth full day as the answer instead of the day the remainder falls on. (2003 AMC 10B #22)
Counting 5 days forward from Tuesday (getting Sunday) when the target date is earlier, or ignoring leap years entirely. (2000 AMC 10 #25)
Counting 50 leap years by forgetting that 1900 is not a leap year, or counting Feb 29, 1812 (which falls after the birthday) incorrectly. (2012 AMC 10A #12)
Counting 120 days forward from the Monday (landing on Tuesday) instead of 119, because day 1 itself is the Monday. (2016 AMC 10B #4)

1.2022 AMC 10B #4
A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $700$ th hiccup occur?
1. A) $15 \text{ seconds after } 4:58$
2. B) $20 \text{ seconds after } 4:58$
3. C) $25 \text{ seconds after } 4:58$
4. D) $30 \text{ seconds after } 4:58$
5. E) $35 \text{ seconds after } 4:58$
2.2018 AMC 10A #3
A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
1. A) $\text{January 2}$
2. B) $\text{January 12}$
3. C) $\text{January 22}$
4. D) $\text{February 11}$
5. E) $\text{February 12}$
3.2015 AMC 10B #2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
1. A) $\text{3:10 PM}$
2. B) $\text{3:30 PM}$
3. C) $\text{4:00 PM}$
4. D) $\text{4:10 PM}$
5. E) $\text{4:30 PM}$
4.2008 AMC 10A #1
A bakery owner turns on his doughnut machine at $\text{8:30}\ {\small\text{AM}}$ . At $\text{11:10}\ {\small\text{AM}}$ the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
1. A) $\text{1:50} {\small\text{PM}}$
2. B) $\text{3:00} {\small\text{PM}}$
3. C) $\text{3:30} {\small\text{PM}}$
4. D) $\text{4:30} {\small\text{PM}}$
5. E) $\text{5:50} {\small\text{PM}}$
5.2016 AMC 10B #4
Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day of the week did she finish her $15$ th book?
1. A) $\text{Sunday}$
2. B) $\text{Monday}$
3. C) $\text{Wednesday}$
4. D) $\text{Friday}$
5. E) $\text{Saturday}$
6.2010 AMC 10B #5
A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
7.2010 AMC 10A #10
Marvin had a birthday on Tuesday, May 27 in the leap year $2008$ . In what year will his birthday next fall on a Saturday?
1. A) $2011$
2. B) $2012$
3. C) $2013$
4. D) $2015$
5. E) $2017$
8.2006 AMC 10B #16
Leap Day, February $29$ , $2004$ , occurred on a Sunday. On what day of the week will Leap Day, February $29$ , $2020$ , occur?
1. A) $\textrm{Tuesday}$
2. B) $\textrm{Wednesday}$
3. C) $\textrm{Thursday}$
4. D) $\textrm{Friday}$
5. E) $\textrm{Saturday}$
9.2002 AMC 10B #8
Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$ ? (Note: Both months have $31$ days.)
1. A) $\text{Monday}$
2. B) $\text{Tuesday}$
3. C) $\text{Wednesday}$
4. D) $\text{Thursday}$
5. E) $\text{Friday}$
10.2012 AMC 10A #12
A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$ ) or is divisible by $4$ but not $100$ (such as $2012$ ). The $200$ th anniversary of the birth of novelist Charles Dickens was celebrated on February $7$ , $2012$ , a Tuesday. On what day of the week was Dickens born?
1. A) $\text{Friday}$
2. B) $\text{Saturday}$
3. C) $\text{Sunday}$
4. D) $\text{Monday}$
5. E) $\text{Tuesday}$
11.2003 AMC 10B #22
A clock chimes once at $30$ minutes past each hour and chimes on the hour according to the hour. For example, at $1 \text{PM}$ there is one chime and at noon and midnight there are twelve chimes. Starting at $11:15 \text{AM}$ on $\text{February 26, 2003},$ on what date will the $2003^{\text{rd}}$ chime occur?
1. A) $\text{March 8}$
2. B) $\text{March 9}$
3. C) $\text{March 10}$
4. D) $\text{March 20}$
5. E) $\text{March 21}$
12.2000 AMC 10 #25
In year $N$ , the $300^{\text{th}}$ day of the year is a Tuesday. In year $N+1$ , the $200^{\text{th}}$ day is also a Tuesday. On what day of the week did the $100$ th day of year $N-1$ occur?
1. A) $\text{Thursday}$
2. B) $\text{Friday}$
3. C) $\text{Saturday}$
4. D) $\text{Sunday}$
5. E) $\text{Monday}$

Clocks, Calendars & Time — answersAMC 10 Step by Step

Name: Date: October 28, 2026

Keep this page separate from the problems.

Answers

#	Answer	The idea it turns on
1	A	From the 1st hiccup to the 700th there are only 699 gaps of 5 seconds, so it happens 3495 seconds (58 minutes 15 seconds) after 4:00.
2	E	One day is 60 times 60 times 24 seconds. Split 10! into those same factors and the leftover is exactly 42 days, which lands on noon of February 12.
3	B	From 1:00 to 2:40 is 100 minutes, and that covers two tasks, not three. So each task takes 50 minutes.
4	D	One third of the job took 2 hours 40 minutes, so the whole job takes three times as long: 8 hours after the 8:30 AM start.
5	B	Book 15 is finished on day 1 + 2 + ... + 15 = 120. Day 120 is 119 days after day 1, and 119 is exactly 17 weeks.
6	B	31 days is 4 weeks plus 3 days, so the month's first three weekdays appear 5 times. Monday and Wednesday must both be among those three, or both not.
7	E	Each year the birthday moves 1 weekday later, or 2 days when a February 29 comes between. Count forward from Tuesday until you land on Saturday.
8	E	A normal year moves the weekday 1 forward; a year with a leap day moves it 2. That is 12 + 8 = 20, which is 6 past Sunday.
9	D	A 31-day month has three weekdays that appear five times: those of the 1st, 2nd and 3rd. Try each possible start for July and see which August day always appears.
10	A	Each year moves the weekday by 1, each leap day by 1 more: 200 + 49 = 249 days, 4 past whole weeks. Go back 4 from Tuesday.
11	B	A full day has 180 chimes and the rest of February 26 has 91. The other 1912 chimes are 10 full days plus 112: March 9.
12	A	Two Tuesdays are a whole number of weeks apart. The gap is 265 days, or 266 if year N is a leap year, and only 266 divides by 7.

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.