Problem
Find the daily gap between a clock and true time
A clock runs fast by a fixed amount every three days and is set correct at 2 p.m. We need the time it shows exactly one day later.
Number systemRatios
Your answer
How to solve
Strategy Identify Subproblems — The gain is given over three days but asked about over one, so find the one-day amount first. It comes out as a fraction of a minute, which then has to be read back as seconds before it can be written as a clock time.
1STEP 1
Find one day's gain
The clock gains 5 1/4 minutes over 3 days at a steady rate, so divide by 3 to get a single day's gain.
21/4 ÷ 3 = 7/4
It gains 7/4 minutes a day.
6.NS.A.1Identify SubproblemsThe clock gains minutes in one day.
Why?
The gain grows by the same amount every day, so three days' gain is three equal daily gains, and one day's is that total shared into three.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Turn the fraction of a minute into seconds
7/4 minutes is 1 whole minute and 3/4 of another. Three quarters of 60 seconds is 45 seconds.
3/4 × 60 = 45
One day's gain is 1 minute 45 seconds.
6.RP.A.3Analyze The Units3STEP 3
Add the gain to the true time
A clock that runs fast shows a time later than the true one, so add the gain to 2 p.m.
2:00:00 + 0:01:45 = 2:01:45
The clock reads 2:01:45 p.m.
6.RP.A.3Work BackwardsAnswer
2:01:45 p.m.
Check by tripling: three days of 1 minute 45 seconds is 5 minutes 15 seconds, and 5 1/4 minutes is exactly 5 minutes 15 seconds.
Takeaway
When a rate is given over several days but asked about one, find the one-day amount first.
- Find one day's gain
- Turn the fraction of a minute into seconds
- Add the gain to the true time