Estimation and rounding

Sometimes you do not need the exact answer , you just need to know whether something is roughly $1$ lakh or $1$ crore. That is the power of estimation.

Idea

Rounding means replacing a number by a nearby "neat" number , one that ends in zeros. We pick the place to round to (tens, hundreds, thousands, lakhs, crores) and follow this rule:

• Look at the digit just to the right of the rounding place.
• If it is $5$ or more, round up (increase the rounding digit by $1$).
• If it is $4$ or less, round down (keep the rounding digit the same).
• All digits to the right become zeros.

So $4{,}763$ rounded to the nearest hundred: the tens digit is $6 \geq 5$, so we round up to $4{,}800$. Rounded to the nearest thousand: the hundreds digit is $7 \geq 5$, so we round up to $5{,}000$.

Estimation is the everyday use of rounding. Before computing, round each number to a nearby neat value, do the easy arithmetic in your head, and you have an answer that is close enough. For example, $4863 + 2148 \approx 5000 + 2000 = 7000$ , and indeed the exact answer is $7011$.

Estimation is most useful for checking a long calculation. If you compute $58{,}347 \times 21 = 1{,}22{,}5287$ and your quick estimate says $60{,}000 \times 20 = 12{,}00{,}000$, you know your answer should be around $12$ lakh , not $1.2$ lakh, not $1.2$ crore. So $12{,}25{,}287$ looks reasonable.

Worked examples

Example 1. Round $7{,}43{,}289$ to the nearest thousand.

The hundreds digit is $2$ (less than $5$). So we round down: the thousands stay as $43$. Result: $7{,}43{,}000$.

Example 2. Round $2{,}99{,}850$ to the nearest ten thousand.

The thousands digit is $9$ (more than $5$). Round up: ten-thousands digit goes from $9$ to $10$, which carries. So $2{,}99{,}850 \to 3{,}00{,}000$.

Example 3. Estimate $39{,}847 \times 198$.

Round: $39{,}847 \approx 40{,}000$ and $198 \approx 200$. So $39{,}847 \times 198 \approx 40{,}000 \times 200 = 80{,}00{,}000.$ The exact value is $7{,}889{,}706 \approx 79$ lakh , close to our estimate.

Example 4. A school has $1{,}235$ students. Each will get a notebook costing $\rupee 48$. Estimate the total cost.

$1{,}235 \approx 1{,}200$ and $\rupee 48 \approx \rupee 50$. So total $\approx 1200 \times 50 = \rupee 60{,}000$. (Exact: $\rupee 59{,}280$.)

Try it yourself

1. Round $86{,}794$ to the nearest (a) ten, (b) hundred, (c) thousand.
2. Round $4{,}57{,}823$ to the nearest lakh.
3. Estimate $497 + 1{,}956 + 21$.
4. Estimate $608 \times 312$ to the nearest lakh.
5. The price of a phone is $\rupee 21{,}780$. About how many can a school buy with a budget of $\rupee 5{,}00{,}000$?
6. Round $9{,}99{,}900$ to the nearest lakh. (Watch the carry!)
7. A factory makes $2840$ pens a day. Estimate how many it makes in a (non-leap) year.
8. Is $4{,}99{,}500$ closer to $4{,}99{,}000$ or to $5{,}00{,}000$?

Activity

Pick five products in a grocery shop. Round each price to the nearest $\rupee 10$ and add them mentally to estimate the bill. Then compare with the actual total at the counter , how close were you?

In a notebook, list five distances (city to city, planet to planet, etc.) found online. Round each to a "feel-able" power of $10$.