AMC 10 · 2025 · #16
Grade 10 geometry-2drate-ratioPick an answer.
The problem hands me a duration in minutes and asks for something about angles, so the whole game is a unit conversion: each hand has its own constant speed in degrees per minute, and once I write those two rates down, every position is just rate times time (Analyze the Units). That splits the work into two clean subproblems (Identify Subproblems): first the size of the angle between the hands, then the tangent of that angle. The second subproblem is where a picture earns its keep (Draw a Diagram) — the angle turns out to be exactly half of 45 degrees, so I can build it by drawing a 45-degree right triangle and bisecting it, instead of recalling a trigonometric identity.
Convert 2025 minutes to a clock reading
2025 minutes reads 9:45.
A dial has no memory of completed laps, so only the remainders after dividing by 60 and by 12 can affect where the hands point.
A dial has no memory of completed laps, so only the remainders can affect where the hands point.
▸ Why?
After a full lap everything returns to where it began, so complete laps change nothing.
▸ Why?
Any elapsed time splits into whole laps plus one remainder, and only the remainder is visible.
Place the minute hand
The minute hand sits at 270 degrees.
Once a hand's speed is written in degrees per minute, its position is nothing more than speed times time.
6.RP.A.2Analyze The UnitsPlace the hour hand
The hour hand has crept to 292.5 degrees.
The hour hand creeps the entire time, so any leftover minutes always drag it past the hour mark, never leave it sitting on one.
7.RP.A.2Analyze The UnitsSubtract to get the acute angle
The gap is the acute 22.5 degrees.
Angles measured from a common ray simply subtract, and the acute angle is whichever of the two arcs comes out under a right angle.
4.MD.C.7Identify SubproblemsBisect a 45-degree right triangle
Bisect a 45 degree right triangle.
Halving an angle you already know beats hunting for a new one, and a bisector splits the far side in a ratio you can compute.
10.G-SRT.B.5Draw A DiagramRead the tangent off the picture
Reading it off gives root two minus one.
Setting the adjacent leg to 1 makes the opposite leg equal the tangent, so a length you already computed is the answer.
10.G-SRT.C.6Draw A DiagramGive each hand its own speed in degrees per minute, subtract the two positions to get the gap, then find the tangent of that gap by bisecting a 45-degree right triangle.
- Convert 2025 minutes to a clock reading
- Place the minute hand
- Place the hour hand
- Subtract to get the acute angle
- Bisect a 45-degree right triangle
- Read the tangent off the picture