Estimation

If you ask "how many people live in your city?" almost no one knows the exact number. But everyone can give a sensible guess: "about a million", "around three lakh", "roughly forty thousand". That sensible guess is called an estimate, and it is one of the most useful skills in everyday mathematics.

Concept

An estimate is a quick, roughly-right answer. We use estimates when:

• An exact number is not needed (e.g. "is this packet about $200$ g or $2$ kg?").
• The exact number cannot be measured easily (e.g. how many leaves on a tree).
• We want to check whether a careful calculation looks reasonable.

The basic trick of estimation is rounding. You replace each number with a nearby round number , usually the nearest ten, hundred, or thousand , and then compute mentally.

For example, to estimate $487 + 312 + 198$:

• Round to nearest hundred: $487 \to 500$, $312 \to 300$, $198 \to 200$.
• Mental sum: $500 + 300 + 200 = 1000$.
• The true answer is $997$ , our estimate is wonderfully close.

Estimation is also helpful for multiplication. To estimate $48 \times 23$:

• $48 \approx 50$, $23 \approx 20$.
• $50 \times 20 = 1000$.
• True answer: $1104$. Estimate is in the right ballpark.

There are different levels of estimation. A rough estimate is "in the right order of magnitude" , for example, "the population is about a lakh" (between, say, $50{,}000$ and $200{,}000$). A closer estimate rounds to nearer values and gets you within $5\%$ or so.

Estimation also includes estimating quantities you cannot directly count. How many grains of rice in a kilogram bag? How many breaths you take in a day? You break the problem into easier pieces:

• Count grains in one small spoon (say $200$).
• A bag has roughly $80$ spoons (rough guess).
• So grains $\approx 200 \times 80 = 16{,}000$ per kg.

The exact number may be different, but the order of magnitude is right.

A famous kind of estimate is the Fermi problem, named after a physicist who loved them. "How many barbers in Mumbai?" "How many tabla players in India?" By breaking the question into smaller steps with rough numbers, you can usually reach an answer that is within a factor of $10$ of the truth , pretty good for a question you cannot look up.

The lesson: a good estimate is often better than a wrong exact calculation. Always estimate before you compute, so you know what answer to expect.

Worked examples

Example 1. Estimate the sum $568 + 712 + 295$.

• Round to nearest $100$: $600 + 700 + 300 = 1600$.
• Actual: $1575$. Very close.

Example 2. Estimate $39 \times 102$.

• Round: $40 \times 100 = 4000$.
• Actual: $3978$. Good estimate.

Example 3. A teacher buys $48$ notebooks at $\rupee 47$ each. Roughly how much does she pay?

• Estimate: $50 \times 50 = 2500$.
• Actual: $48 \times 47 = 2256$.
• Estimate is within a few hundred of the right answer , useful for budgeting.

Example 4. A school has $32$ classrooms, each with about $35$ students. Estimate the total number of students.

• $30 \times 35 = 1050$, or $32 \times 35 \approx 32 \times 35 = 1120$.
• A round-number estimate: $\approx 1100$.

Example 5. Estimate how many heartbeats you would have in a single day.

• Heart rate $\approx 75$ beats per minute.
• Minutes in a day $\approx 60 \times 24 = 1440$.
• Beats per day $\approx 75 \times 1440 \approx 1{,}08{,}000$.
• So about a lakh per day.

Try it yourself

1. Estimate $498 + 712$.
2. Estimate $59 \times 41$.
3. Estimate the cost of $19$ kg of sugar at $\rupee 41$ per kg.
4. A truck carries about $2{,}500$ kg of vegetables. Roughly how many such truckloads fill a $1$-lakh-kg warehouse?
5. Estimate how many seconds you have been alive (just an order of magnitude).
6. Estimate the number of windows in your school building.
7. Estimate the number of breaths in a day for a person breathing $\sim 16$ times per minute.
8. Sanity check: a student calculates $24 \times 36$ and gets $86$. Estimate to show this must be wrong.

Activity

Fermi day. Pick a "how many" question that no one can quickly look up: how many cricket balls fit in your classroom, how many footsteps you take in a day, or how many trees are on your street. Break the problem into smaller pieces and write down a number for each piece. Multiply. Compare your estimate with a friend's and see how close you are.