Depth · one concept from first principles · step 2 of 4
Write the daily gain or loss as a proportion
A clock gains a steady 6 minutes a day. It is set exactly right at 8 a.m. today. We need the time it shows at 12 noon the next day.
Ratios
Your answer
How to solve
Strategy Convert to Algebra — The gain and the time passed grow together at a fixed rate, so one proportion links them. The only care needed is counting the hours across midnight -- and remembering the clock shows the real time plus the gain.
1STEP 1
Count the hours that pass
From 8 a.m. today to 8 a.m. tomorrow is a full day, and then four more hours to noon.
24 + 4 = 28
28 hours pass in all.
7.RP.A.3Analyze The Units2STEP 2
Write the proportion
Six minutes are gained in every 24 hours, so the gain over 28 hours is in the same ratio.
24 : 6 = 28 : ■
One unknown, one equation.
7.RP.A.3Convert To AlgebraThe gain and the time that has passed are in the same ratio.
Why?
The clock gains at a steady rate, so each hour adds the same amount of gain and the total gain is a count of equal hourly gains.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Both the time and the gain are counted in those same hourly units, so doubling one doubles the other and the two stay in step.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Solve it
In a proportion the outer two multiply to the same as the inner two.
6 × 28 = 168, 168 ÷ 24 = 7
The clock gains 7 minutes.
7.RP.A.2Identify Subproblems4STEP 4
Add the gain to the real time
A clock running fast shows a time later than the real one.
It shows 12:07 p.m.
7.RP.A.3Identify SubproblemsAnswer
12:07 p.m.
Scale it down: 28 hours is seven sixths of a day, and seven sixths of 6 minutes is 7 minutes.
Takeaway
A steady gain is proportional to the time that has passed -- count that time carefully first.
- Count the hours that pass
- Write the proportion
- Solve it
- Add the gain to the real time
Where next?
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