Operations on Numbers

India’s population in 2018 is claimed to be around 1.35 billion. Do you think we go around counting each individual? Of course not. In fact, these are rightly known as population estimates. Hence, estimation of numbers is an essential operation. It’s time to move on from simple addition, subtraction, multiplication, and division to more complex operations on numbers.

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Operations on Numbers- Estimation

Estimating means guessing a value which is close to the exact value, can be used in its place without making the result inaccurate and is easier to calculate. The most common technique we use for estimation of numbers is rounding-off. In fact, estimation and rounding-off in maths go hand in hand. We have already discussed the importance of estimation in brief.

Let us learn about rounding-off. Precisely rounding off means making a number simpler but keeping its value close to the original value. The result is less accurate, but easier to use. We will be learning about rounding-off to nearest tens, nearest hundreds and nearest thousands.

Rounding off to Nearest Tens

To round off a number to nearest tens we look at the nearest values with a zero at the ones place. Now among these two, we choose the one which is nearest to the number e aim to round off. This chosen value is the round off of the given number.

For example, let us round off 17 correct to nearest tens. The two values with a zero at ones place nearest to 17 are 10 and 20. Obviously, 17 is nearer to 20 as compared to 10. Hence, roundoff of 17 is 20. Similarly, roundoff of 12 is 10. Note that we round off 15 to 20(although 15 is equidistant from 10 and 20, we generally round it off to 20). Note that the numbers 1, 2, 3 and 4 are nearer to 0 and 5, 6, 7, 8 and 9 are nearer to 10.

Rounding off to Nearest Hundreds

Again, when rounding off a number to nearest hundreds we choose two values, with zero at both tens and ones place, nearest to the number we aim to roundoff. Further, among these two choices, the value nearest to the number to be rounded off is the answer.

For example, let us round off 356 correct to the hundreds place. This number lies between 300 and 400(when we want to round off to hundreds). Now, 356 is nearer to 400. Hence 400 is the round off of 356. The rule of thumb is, numbers 1 to 49 are nearer to 0 than 100 and the numbers 50 to 99 are nearer to 100 than 0.

Rounding off to Nearest Thousands

If we observe the general trend, it appears that for rounding off to a place, all the places before that place must have a zero. Thus, when we want to round off a number to nearest thousands, we look at the values with zero at ones, tens and hundreds place, nearest to this number. Further, the value nearest to this number is its roundoff.

As an example let us round off 3467 correct to nearest thousands. 3467 lies between 3000 and 4000. As can be seen, 3000 is nearest to 3467. Therefore the rounded value of 3467 is 3000 correct to thousands. It is important to realize that numbers 1 to 499 are nearer to 0 than 1000 whereas numbers 500 to 999 are nearer to 1000.

Estimating Sum and Differences

In real life situations, we often need to estimate the operations on numbers in order to quickly determine a rough value. This estimation is again based on rounding off of numbers. However, we need to pay attention to the fact whether the guessed answer is sensible and very close to the exact value or not. Moreover, we need to carefully choose the place up to which we have to round off. Observe the following examples carefully.

(1) Estimate 5290+17896

Here we round off to thousands. 17896→18000 and  5290→5000. Now, 18000+5000=23000. Note that this is very close to the sum of 17896 and 5290.

(2) Estimate 5763 – 436.

Here if we round off to thousands then 5763→6000 and 436→0. The difference between 6000 and 0 will not be close to the difference of 5763 and 436. Hence, rounding off to thousands place proves to be inaccurate here.

Rather, if we round off to hundreds place then the estimate comes very close to the exact value. Therefore, 5763→5800 and 436→400. Lastly, 5800-400=5400 which is fairly close to the exact value.

Estimation of Products

Estimation of products is much simpler and ordered. Whenever we need to estimate the product of two numbers, we start off by rounding each of these numbers to their greatest places. Further, we multiply these rounded numbers. This product is our estimate.

For example, assume that you want to estimate the product of 479 and 81. The greatest place for 479 and 81 is hundreds and tens respectively. Hence, we round off 479→500 and 81→80. Now 500×80=40000. Thus 40000 is our estimate for the product of 479 and 81.

Solved Example for You

Question 1: Estimate 578×161.

Answer: Here, we round off 578→600 and 161→200. Now, the answer is 600×200=120000.

Question 2: Give an example of an estimate?

Answer: Estimate refers to finding a value that is near enough to the right answer. Furthermore, this happens usually with some calculation or thought involved. For example, Alex estimated there were 5,000 roses in the field by counting one row then undertaking multiplication of it by the total number of rows present.

Question 3: What are the various ways used for estimation?

Answer: There are three ways of estimation that are in existence. These three ways are front-end, clustering, and rounding methods.

Question 4: How come is estimation useful?

Answer: Estimation refers to the process of finding out an approximation or estimate. This value which is a value is very useful even if data is incomplete, unstable, or uncertain. The value is useful because its derivation takes place from the best information available.

Question 5: Can we say that an estimate is guess?

Answer: An estimate refers to the resulting judgment or calculation. A related term to estimate is approximation, which means close or near. Guess means to reach a conclusion without sufficient information. So, one can say that an estimate is an educated guess. Also, an educated guess is a more casual estimate.