Clocks, Calendars & Time

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

1. 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 $-270^\circ$).
2. For weekdays, count the days between the two dates exactly, find the remainder mod $7$, and step forward or backward in the right direction.
3. 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)