Reason about a fast or slow clock

3.MD.A.1 · take · grade 3

Archetype: Elapsed Time and Base-Sixty Regrouping · step in a 10-type progression

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $2{:}00$ and then looked at it $3$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $2{:}00$ , and the clock after the arrow labeled " $3$ hours later" reads $5{:}09$ .

Show solution

Understand

Hana set her clock to exactly 2:00. After 3 real hours it read 5:09 instead of 5:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 2:00.
After 3 actual hours the clock displays 5:09.
The figure shows a digital display 02:00 with an arrow labeled '3 hours later' pointing to a display reading 05:09.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

Plan

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole 3 hours (subproblem 1), then share that extra time equally across the 3 hours to get the gain per hour (subproblem 2).

Execute

#7 Identify Subproblems 3.MD.A.1

Three hours after 2:00, a correct clock would read 5:00.

2{:}00 + 3 \text{ hours} = 5{:}00

Adding 3 hours to the hour shows where a good clock should be.

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 5:09 but should show 5:00, so it has gained 9 minutes over the 3 hours.

5{:}09 - 5{:}00 = 9 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

#8 Analyze the Units 3.MD.A.1

The 9 extra minutes were gained evenly across 3 hours, so divide 9 by 3.

9 \div 3 = 3

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 3 minutes per hour

Review

Check by building up: 3 minutes fast each hour for 3 hours is 3 + 3 + 3 = 9 minutes, so the clock shows 5:00 + 9 min = 5:09, matching the figure.

Look for a pattern (tool 5): after 1 hour it would read 3 minutes fast (3:03), after 2 hours 6 minutes fast (4:06), after 3 hours 9 minutes fast (5:09) — confirming 3 minutes per hour.

Standards · min grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡 Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!