About right is enough
A rough estimate can tell us whether an answer is reasonable.
Counting your breaths
Nobody counts every breath for a whole day. Yet we can still say roughly how many litres of air a person breathes in a day.
How could we begin?
What this lesson covers
The idea
Understand the situation, identify the quantities that matter, then make a rough estimate; an approximate value is often enough to tell whether a result is reasonable or impossible.
Counting your breaths
Nobody counts every breath for a whole day. Yet we can still say roughly how many litres of air a person breathes in a day.
How could we begin?
- Give up, because we cannot count them all
- Break it into small parts we can estimate
- Pick any big number
Build an estimate
Choose sensible numbers for the three parts. Then cross-check by a second route.
Reason first, calculate roughly
First understand the situation, then identify the quantities that matter, then make a rough estimate. An approximate value is often enough to tell whether a result is reasonable or impossible.
A second route (balloons) gave 8.6 thousand litres. Two different routes landing close together makes us trust the estimate. Careful reasoning matters even more than exact calculation.
- Step | Breathing example
- Understand the situation | We breathe air in and out all day
- Identify the quantities | breaths a minute, litres a breath, minutes a day
- Rough estimate | 14 × 0.5 × 1440 is about 10 thousand litres
Notes
To estimate: understand the situation, identify the quantities that matter, then make a rough estimate. An approximate value often tells us whether a result is reasonable or impossible.
Check yourself
At rest you take 12 breaths a minute. A day has 1440 minutes. How many breaths do you take in a day?
Answer: 17280
12 × 1440 = 17,280 breaths. That is close to the “about 20 thousand” in our estimate.
You take about 20 thousand breaths a day, and one breath is about 0.5 litre. About how many litres of air is that a day?
Answer: 10000 litres
20,000 × 0.5 = 10,000 litres a day.
Someone claims a person breathes in 1,000,000 litres of air a day. Our estimate was about 10,000 litres. Is the claim reasonable?
Cross-check: blow up 3 balloons a minute, each holding 2 litres, for 1440 minutes. How many litres is that?
Answer: 8640 litres
6 × 1440 = 8,640 litres, close to our 10,000 litre estimate.
- No. It is 100 times bigger: each of 20,000 breaths would have to be about 50 litres — correct. Yes. 1,000,000 ÷ 20,000 = 50 litres a breath, far more than a balloon holds.
- Yes, because estimates are never exact. Estimates are rough, but not 100 times off.
- We cannot say without counting every breath. A quick estimate is enough to see this claim is impossible.