## What is an estimate?

Let us consider these two statements:

- Approximately 4000 people were injured and more than 6000 houses had to be evacuated because of the earthquake that took place in the main lands of India
- More than 3000 people were injured and 500 people lost their lives due to the bomb blast in Delhi

Can we confidently say that those were the exact number of people who were either injured or killed due to the accidents mentioned above?

No, we cannot. The word “approximately’’ clearly says that, the numbers mentioned above are just an estimate of the number of people and not the exact number. For example, in the first sentence, of the 4000 people that were injured, it could have been 3999 or 4001 or 4050, in the same way, in the second statement, 3000 is just an estimate or an approximation of the number of people injured, because it may have been 3045 or 3006.

These two statements are very good examples of estimation or approximation.

In simple words, estimation means approximation.

## Why and where do we estimate?

Suppose, there is going to be a big event at your workplace or your school. And you have to make an estimate the number of guests that will come to the event, because food, place etc need to be arranged for them. Is it possible to know the exact number of guests that will be coming? No that is practically impossible. Hence in such situations you need to estimate.

For example, let us consider the situation where a housewife is preparing her household budget for the month. She has to allot a certain amount of money under the head “food”. Is it possible for her to know the exact amount of money that will be required for food? No, hence she just creates a rough estimate or an approximation, to serve her purpose.

Think about the situations where we need to have the exact numbers and compare them with situations where you can do with only an approximately estimated number.

## Estimating

#### To nearest tens by rounding off:

Step 1: See the digit in the units place of the given number

Step 2: If the units’ digit is less than 5, then replace the unit digit by 0, and keep the other digits just the same

Step 3: If the units’ digit is 5 or more than 5, then increase digit in the ten’s place by 1 and replace digit in the unit’s place by 0.

So now, lest consider the number 24, it lies between 20 and 30. However 24 is closer to 20 than it is to 30. Why? Because, as mentioned, notice the digit in the units place, which is 4, it is less than 5, hence the number is to be rounded off to 20.

**Examples: Round off to the nearest ten.**

**Round off 43, 66, 89, and 65.**

**Solution:**

**To solve these sums, we can go through the simple, three step process:**

For 43, the digit in the unit’s place 3, which is less than 5. So 43 rounded off to the nearest ten is 40

For 66, the digit in the unit’s place is 6,which is more than 5. So 66 rounded off to the nearest ten is 70

For 89, the digit in the unit’s place is 9, which is more than 5. So 89 rounded off to the nearest ten is 90

For 65, the digit in the unit’s place 5, so 65 rounded off to the nearest ten is 70

#### To the nearest hundreds by rounding off:

Step 1: Notice the digit in the ten’s place of the given number

Step 2: If the digit in the ten’s place is less than 5, replace each of the digits of the units and tens place by 0, and keep the other digits just the same.

Step 3: If the digit is 5 or more, increase the digit in the hundreds place by 1 and replace each digit on its right by 0.

Is 640 nearer to 600 or 700?

This can be easily solved by the three steps:

640 is nearer to 600, why? Notice the digit in the tens place, it is 4 , which is less than 5. Hence the number will be rounded off to 600.

In the same way,

579 lies between 500 and 600. It is nearer to 600 because the digit in the ten’s place is 7, which Is more than 5, so it is rounded off as 600 correct to nearest hundred.

Numbers 1 to 49 are closer to 0 than to 100 and so are rounded off to 0

**Examples: Rounding off to the nearest hundreds.**

**Round off 782, 241, 655, and 987.**

**Solution:**

For 786, the last two digits are 86 the digit in the tens place is 8, so as mentioned above in the three steps, this number will be rounded off to 800, correct to the nearest hundred.

In the same way,

For 241, the digit in the ten’s place is 4 which is less than 5. So 241 rounded off to the nearest hundred is 200

For 655, the digit in the ten’s place is 5. So 655 rounded off to the nearest hundred is 700

For 987, the digit in the ten’s place is 8 which is more than 5. So 987 rounded off to the nearest hundred is 1000

#### To the nearest thousands by rounding off:

Step 1: In the given number, notice the digit in the hundreds place.

Step 2: If the digit in the hundreds place is less than 5, replace each one of the digits in the hundreds, tens and units place by 0 and keep the other digits as they are.

Step 3: If the digit in the hundred’s place is 5 or more, increase the digit in the thousands place by 1 and replace the other entire digit in the hundreds, tens and units place by 0. We know that numbers 1 to 499 are nearer to 0 than 1000, so these numbers are rounded off as 0.

The numbers 501 to 999 are nearer to 1000 than 0 so they are rounded off as 1000.

Number 500 is also rounded off as 1000.

**Examples: Rounding off to the nearest thousands.**

**Round off: 1648, 1121, 3950, and 9351.**

**Solution:**

1560, the the number in the hundreds place is 5, hence according to the three steps, the number will be rounded off to 2000.

For 2020, digit in the hundred’s place is 0, which is less than 5. So 2020 rounded off to the nearest thousand is 2000.

For 4951, the digit in the hundred’s place is 9 which is more than 5. So 4951 rounded off to the nearest thousand is 5000.

For 9556, the digit in the hundred’s place is 5. So 9556 rounded off to the nearest thousand is 10,000.

## Estimating outcomes of number situations

Generally when we add numbers, the placement of the numbers plays a very important part. So we follow the basic algorithmic pattern. We place the numbers in such a way that the digits in the units place are in the same column, the digits in the tens place are in the same column and so on.

For example if we are adding 9876+1022+2045

Place all the digits in their respective columns and add. We carry forward 1 (in this case) to the next column.

But implementing this method is not always convenient. Sometimes we need answers faster, for example when we are in the grocers. Then we make an estimation of all the numbers and then add, which makes it easier. We then get an estimated sum.

There are no specific rules or regulations when it comes to estimation of sums. It depends on the situation, the accuracy required and the amount of time.

## To Estimate a Sum or a Difference

When we are estimating a certain sum or difference we need to know why we are estimating, so that we can estimate accordingly and accurately. Estimating has no fixed rules and regulations and hence varies in situations. We have to keep in mind, that we have to estimate accurately or we don’t get the desired result and it does not serve our purpose.

For numbers with two digits, there can only be one estimate because you can only round to the tens place. For numbers with three digits, you can get two estimates, and so forth.

**For example: **

**Example: Estimate 5225+16985**

You can clearly see that 16985 is greater than 5225. Hence we have to round off the numbers to the nearest thousands;

Rounding off to thousands,

16985 is rounded off to 17,000 and 5225 is rounded off to 5000.

Now, we add both these estimated numbers and get the result which is 22,000

Therefore, estimated sum=22,000

**Example: Estimate 8976-987**

From the numbers above, we can clearly see that 8976 is greater than 987.

We estimate or round off the numbers to the nearest thousands;

8976 when rounded off to the nearest hundred is 9000 and 987 when rounded off is 1000.

Therefore, the estimated difference is 9000-1000, which is equal to 8000.

**To Estimate Product**

How do we usually estimate products? Why do we estimate products? We do so because it helps us calculate faster.

For example, let us consider

If we try and multiply these numbers, it will take some time. That’s why it is easier if we round off the numbers

We round off 32 and 87 to the nearest tens.

32 rounded off to the nearest ten is 30 and 87 rounded off to the nearest ten is 90.

the estimated product is which is 270.

Let us consider

The usual approach would be to round off the numbers to the nearest hundred and then multiply, but if we do that, then the estimated product is much larger than the original product. Hence we have to round off the numbers to the nearest ten and then multiply. If we estimate 193 to 190 and 74 to its nearest ten, which is 70 and then multiply, we get, 13300 which is an accurate estimate and we also get the answer faster.

Now let us estimate

Using the same rule;

We round off 896 to 900 (rounding off to nearest hundred) and 74 to 70 (rounding off to nearest ten.

So, now the estimated product is

Estimation helps us in various situations. It helps us to check our answers without a calculator. For example we multiply and we are not very sure about our answer, we can just round off 31 to 30 and 64 to 60. We then get the estimated product, which is 1,800. If our answer is close to 1800, we are right, otherwise not; and we can check again. (We can say all this, assuming that the rounding off, of the numbers is correct)

Same general rule may be followed by addition and subtraction of two or more numbers.

**Example 1: Estimate the product of 42 and 58**

**Solution:**

42 estimated to the nearest ten, is equal to 40

58 estimated to the nearest ten is equal to 60

the required estimation is

**Example 2: Estimate the product of 367 and 231 by rounding off each number to the nearest hundred**

**Solution: **

367 estimated to the nearest hundred which is equal to 400

231 estimated to the nearest hundred which is equal to 2000

the required estimated product is equal to

**To Estimate Quotient**

When we are given a certain sum, where we have to divide, just like in estimation of products, we round off the divisor and dividend and then find out the quotient. This serves the same purpose as estimation of products. Just as we find out the estimation of products to verify our answers, in the same way, we find out the estimation of quotients. We round off the divisor and dividend to their nearest hundreds or tens or thousands and then find out the estimated quotient.

**Example: Find the estimated quotient of **

**Solution: **

627 when rounded off to its nearest thousand is equal to 600

23 when rounded to its nearest ten is 20

the required estimated quotient is is approximately equal to

**Example: Find the estimated quotient of **

**Solution: **

985 rounded off to its nearest thousand is 1000

48 rounded off to its nearest ten is 50

the required estimated quotient is is approximately equal to

**Example: Find the estimated quotient of **

**Solution: **

74 rounded off to its nearest ten is 70

34 rounded off to its nearest ten is 30

the required estimated quotient is and is approximately equal to which is approximately equal to 2.

**Example: Find the estimated quotient of **

Solution: When 694 is rounded off to its nearest hundred is 700

58 rounded off to its nearest ten is 60

the required estimated quotient is is approximately equal to (approximately)

## Cluster Estimation

For estimating sums and products another method of estimation may be used known as Cluster Estimation. Cluster estimation is used when the numbers involved, more or less, are all near about a common number, i.e., the numbers involved tend to cluster around a particular value. Two things need to be kept in mind when using this method of estimation:

1. This method can only be used to estimate sums and products.

2. This method will help you only if the numbers cluster around a particular number, i.e., the usability of this method depends solely on the type of sum or product you are dealing with and the numbers associated in the sum or product.

**Example: Estimate the sum 578+598+625+609+637**

**Solution:**

578 rounded off to its nearest hundred is 600.

598 rounded off to its nearest hundred is 600.

625 rounded off to its nearest hundred is 600.

609 rounded off to its nearest hundred is 600.

637 rounded off to its nearest hundred is 600.

So we can clearly see that the numbers associated in the given sum all cluster around the value 600.

So, we can replace all the given numbers in the sum by 600.

we have 578+598+625+609+637=600+600+600+600+600=3000.

So, the estimated sum is 3000.

**Example: Estimate the product $latex 38 times 42 times 36 times 43 $**

**Solution:**

38 rounded off to its nearest ten is 40.

42 rounded off to its nearest ten is 40.

36 rounded off to its nearest ten is 40.

43 rounded off to its nearest ten is 40.

So we can clearly see that the numbers associated in the given product all cluster around the value 40.

So, we can replace all the given numbers in the product by 40.

we have

We multiply all the 4’s to get

Now, we place four zeros(one for each 40) next to 256 on the right.

the estimated product =2560000

**Exercise**

- Round off each of the following numbers to the nearest ten:
- 53
- 287
- 364
- 2045

- Round off each of the following numbers to the nearest hundred:
- 648
- 2356
- 13768
- 1249

- Round of each of the following numbers to the nearest thousands:
- 5486
- 6823
- 14380
- 23569

- There are 54 balls in box A and 79 balls in box B. Estimate the total number of balls in both the boxes taken together. Estimate to the nearest thousand.
- Estimate the difference to the nearest ten: (53-18)
- Estimate each difference to the nearest thousand:
- (35863-27677)
- (47005-39488)

- Estimate the sum to the nearest hundred: (458+324)
- Estimate the following products by rounding off the first number upwards and the second number downwards:
- Find the estimated quotient for each of the following:
- Estimate the sum to the nearest thousand: (2343+1211+3218)
- In a Bucket A there are 104 blue balls, in Bucket B there are 232 red balls. If I take all the balls from Bucket A and Bucket B, approximately how many balls will I have? (Estimate to the nearest 100)
- Two green kites, seven yellow kites, eleven blue kites are flying in the sky. Two kites fall to the ground, estimate the number of kites in the sky to the nearest ten.
- Estimate using the method of Cluster Estimation:
- (13+15+8+9)
- (48+54+45+55)

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