The estimation of Large Numbers is an important part of mathematics that we use daily. Sometimes, we do not use exact numbers; instead, we use numbers that are easy to work with and close enough to the exact numbers. For example, a rough or approximate expenditure estimate is required to organize an event in the school, such as an annual function or sports day.

Thus, estimation means to make a rough idea about quantities, to judge approximate size, cost population, numbers, etc. This article will discuss the basic concepts of estimation and then learn to perform addition, subtraction, division, and multiplication and then estimate the numbers.

What is Estimation? 

Let us go through the real-life examples first.

1. \(47\) people attended my birthday party, but I may say, “There were about \(50\) people at the party”.
2. There are \(2010\) students in my school, but I may say, “ There are approximately \(2000\) students in my school”.
3. I paid \(₹202\) for a novel, but I  may say, “I paid around \(₹200\) for the novel”.

In the above examples, \(50\) is the multiple of \(10\) closest to \(48, 2000\) is the multiple of \(1000\) closest to \(2010,\) and \(200\) is the multiple of \(100\) closest to \(202.\) Such numbers that are easy to remember and close enough to the exact numbers are called estimated numbers. The method of finding estimated numbers is called estimation. Finding out such multiples is called rounding off. Thus, rounding off helps in estimation.

When a number is rounded to a lower number, we say it is rounded down, and when a number is rounded to a higher number, we say it is rounded up.

General Method for Estimation

Carefully examine the digit to be rounded off. Let us call it rounding digit and the digit immediately next to it on the right-hand side; let us call it the RHS digit. You will find out that if the RHS digit is less than \(5,\) the rounding digit remains unchanged, whereas if the RHS digit is greater than \(5\) or equal to \(5,\) the rounding digit increases by \(1\) and in both cases, each digit to the right of the rounding digit is replaced by zero. 

Let us summarise the points for the method to estimate the numbers.

1. Identify the place to be rounded and underline the rounding digit.
2. Identify the RHS digit and compare it with \(5\) (i.e., the mid-way mark)
a. If the RHS digit is less than \(5,\) the rounding digit remains unchanged.
b. the RHS digit is greater than or equal to \(5,\) the rounding digit increases by \(1.\)
3. Replace each digit to the right of the rounding digit with \(0.\)

This rounding-off method can be used for any number, however large it may be.

When we are rounding off to tens, we will check whether the unit’s place digit is less than or more than \(5.\) Similarly when we are rounding off to hundreds, we will check whether the tens place digit is less than, equal to or more than \(5\) and so on.

Let us see some examples to understand better.

More About Estimation

There are many situations where we have to estimate the sum or difference or product or quotient of numbers. There are no rigid rules for these operations (sum, difference, product and quotient). However, the procedure of estimation depends upon the following:

1. Degree of accuracy required.
2. The simplicity of the computation.
3. How quickly is the estimation completed?
4. How quickly would the guessed answer be obtained?

To Estimate the Sum

We have already learned the general method of rounding off the numbers to any place. For example, \(6496\) rounded off to a hundred will be equal to \(6500\) and \(6496\) rounded off to thousand will be equal to \(6000.\)

Let us look out at some of the examples.

Example 1: Estimate the sum of \(3563\) and \(8469\) to the nearest thousand.

Solution: \(3563\) to the nearest thousand\(=4000\)
\(6496\) to the nearest thousand\(=6000\)

Therefore, the required sum\(=6000+4000=10000.\)

To Estimate the Difference

Using the rules to round off a number lets us estimate the number while subtracting the numbers.

Example1: Estimate the difference of \(56735-2394\)

1. To the nearest hundred
2. To the nearest thousand

Answer: 1. \(2394\) the nearest hundred\(=2400\)
\(56735\) to the nearest hundred\(=56700\)

Therefore, the required difference is \(56700-2400=54300\)

2. \(56735\) to the nearest thousand\(=57000\)
\(2394\) to the nearest thousand\(=2000\)

Therefore, the required difference is \(57000-2000=55000\)

To Estimate the Product

Let us understand this through an example.

Example 1: Estimate the product of \(5835\) and \(427\) by rounding of \(5385\) correct to the nearest thousand and \(427\) correct to the nearest hundred.

Answer: \(5835\) to the nearest thousand\(=6000,\) and \(427\) to the nearest hundred\(=400\)

Therefore, the required product\(=6000×400=2400000.\)

To Estimate the Quotient

Let us understand this through an example.

Example 1: Find the estimated quotient for,  \(\frac{{971}}{{461}},\) taking each number correct to the nearest hundred.

Answer:  \(\frac{{971}}{{461}},\) is approximately the nearest hundred is equal to\( = \frac{{1000}}{{500}} = 2\)

How to Estimate the Square Root of a Number?

Consider a real-life situation.

Varun has a square piece of plywood that has an area of \(20000\;{\rm{c}}{{\rm{m}}^2}.\) He wants to use the wood as a backing for the square mirror. Mirror frames come in whole numbers unit lengths. What is the largest length of frame that Varun can buy to make the most use of his piece of plywood?

Since we know that the length required is a whole number, and the square of the length should come close to \(20000\;{\rm{c}}{{\rm{m}}^2},\) we can estimate the square root of \(20000\;{\rm{c}}{{\rm{m}}^2}\) without calculating the actual square root.

We know that, \({140^2} = 19600,\) and \({150^2} = 22500.\) Since \(19600 < 20000 < 22500,\) and \(20000\) is closer to \(19600\) (or \({140^2}\)), we can estimate that \({\rm{140}}\,{\rm{cm}}\) will be the length of the largest frame that Varun can buy to make the most use of his plywood of \(20000\;{\rm{c}}{{\rm{m}}^2}\)

In mathematical terms, \(\sqrt 2 0000 \cong 140,\) read as the square root of \(20000,\) is approximately equal to \(140.\)

Solved Examples

Q.1. Estimate the sum of \(82456, 80326\) and \(5555\)
Ans: \(82456\) to the nearest thousand\(=82000\)
\(80326\) to the nearest thousand\(=80000\)
\(5555\) to the nearest thousand\(=5555\)
Therefore, the required sum\(=82000+80000+6000=168000\)

Q.2. Round off \(6532471\) to the nearest lakh.
Ans: The given number is \(6532471.\)
Here the rounding digit is \(5,\) and the RHS digit is \(3.\)
As the RHS digit \(3<5,\) so the rounding digit \(5\) remains unchanged. Now, replace each digit on the right of the rounding digit with \(0.\)
Thus, \(6532471\) rounds off to the nearest lakh\(=6500000.\)

Q.3. Vivek had \(₹68515\) with him. He purchased some home appliances for \(₹52109.\) What is the approximate amount of money left with him?
Ans: Round off each number to its greatest place value and then perform subtraction.
Amount Vivek had\(=₹68515\) the estimated price will be\(=₹70000\)
Amount spent on home appliances\(=₹52109,\) and the estimated price will be\(=50000\)
Therefore, the amount left with him\(=70000-50000=₹20000\)
Hence, the approximate amount left with Vivek is \(₹20000.\)

Q.4. \(275\) apples are kept in \(42\) boxes. Approximately how many apples are there in each box?
Ans: Round off each number to its greatest place value and then perform division.
The number of balls in \(42\) boxes (rounded off to \(40)=275\) (rounded off to \(300).\)
Therefore, the number of balls in \(1\) box\(=300/40=7\)
Hence, there will be approximately \(7\) apples in each box.

Q.5. Explain the statement.
In a football match between Germany and Spain, \(63000\) watched the match sitting in the stadium and \(53\) million television viewers worldwide.
Does the above state that exactly \(63000\) spectators were in the stadium, or did exactly \(53\) million viewers watch the match on television?
Ans: Certainly not. It states that approximately \(63000\) spectators were in the stadium, and approximately \(53\) million viewers watched the match on television. \(63000\) could be \(62680\) or \(63500\) or \(63890\) etc., but in no case it can be taken as \(70000\) or \(73000\) or \(75000,\) etc. In the same way, \(53\) million maybe \(52.7\) million or \(53.8\) million etc., and not \(57\) million or \(48\) million etc.,
Thus, the quantities \(63000\) and \(53\) million given above, are in fact, not the exact counts but estimates to give an idea of these quantities.

Summary

In this article, we learned the concept of estimation. Estimation is also known as rounding off the numbers. Rounding off can be to the nearest tens, hundreds, thousands, ten thousand, lakhs and so on. We also learned the need to estimate the numbers in real life, and then we learned the rules while estimating the numbers.
In addition to this, we solved the handful of examples to strengthen our hold on the estimation of numbers.

Learn About Uncertainty in Measurement

FAQs

Q.1. How do you estimate and round numbers?
Ans: Carefully examine the digit to be rounded off (let us call it rounding digit) and the digit immediately next to it on the right-hand side (let us call it the RHS digit). You will find out that if the RHS digit is less than \(5,\) the rounding digit remains unchanged, whereas if the RHS digit is greater than \(5\) or equal to \(5,\) the rounding digit increases by \(1\) and in both cases, each digit to the right of the rounding digit is replaced by zero.

Q.2. What is the estimation of numbers?
Ans: Such numbers that are easy to remember and close enough to the exact numbers are called estimated numbers. The method of finding estimated numbers is called estimation. Thus, estimation means to make a rough idea about quantities, to judge approximate size, cost population, numbers, etc.

Q.3. Define rounded down and rounded up in the estimation of numbers.
Ans: When a number is rounded to a lower number, we say it is rounded down, and when a number is rounded to a higher number, we say it is rounded up.

Q.4. How do you estimate large numbers?
Ans: We can make use of the following methods to estimate the numbers.
1. Identify the place to be rounded and underline the rounding digit.
2. Identify the RHS digit and compare it with \(5\) (i.e., the mid-way mark)
a. If the RHS digit is less than \(5,\) the rounding digit remains unchanged.
b. If the RHS digit is greater than or equal to \(5,\) the rounding digit increases by \(1.\)
3. Replace each digit to the right of the rounding digit with \(0.\)

Q.5. How do you solve for estimation?
Ans: To estimate the sum, difference, product and quotient of two given numbers, we round off the given numbers to their greater place values and then find the sum, difference, product and quotient.

We hope this detailed article on the estimation of large numbers helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!