What is meant by Estimation?
Estimation (or estimating) is the process of arriving at an estimate or approximation. The estimate is close to an exact value.
For example, the estimation of 52 is 50. As we can observe, 50 is close to 52.
Estimation can be used to calculate the value that is nearest to the exact value of the result of a multiplication equation. This can be done using rounding.
Example: Each jar has 94 chocolates. Estimate the total number of chocolates.
Here we can estimate 94 in two ways:
1) Estimate in tens: as we know, 94 is nearest to 90, as it is only 4 more than 90.
2) Estimate in hundreds: 94 is nearest to 100, as it is only 6 less than 100.
So, 94 can be estimated as either 90 or 100.
What do you mean by rounding of a number?
A number can be rounded to its nearest tens, hundreds, thousands, and so on. For this, we simply need to replace the number with its nearest multiple of ten, hundred or thousand, as the case may be.
For example, 24 can be rounded to 20, 29 can be rounded to 30.
Let’s take another example,
14554 rounded off to nearest tens, hundredths and thousands,
1. If 14554 is rounded to nearest tenths, the number will be 14550.
2. If 14554 is rounded to nearest hundredths, the number will be 14600.
3. If 14554 is rounded to nearest thousandths, the number will be 14000.
4. Also, if 14554 is rounded to nearest ten-thousandths, the number will be 10000.
How to estimate a product using rounding?
The following steps can be applied to estimate the product using rounding:
1. Firstly, round off the number to the nearest thousands, hundreds or tens.
2. Multiply the rounded number with the other factor to get the result.
3. Use symbol to show that it is an estimate of the actual multiplication result.
Example: Estimate 192 × 8.
Answer:
Rounding 192 to the nearest tens will be 190.
So the equation becomes,
190 × 8 = 1520 [Multiply]
So, 192 × 8 \(\approx\) 1520 or 192 × 8 , is about 1520.
How to estimate two numbers between which the estimated product lies
The following steps can be followed to estimate the two numbers between which the estimated product of a multiplication equation lies:
1. Firstly, round off one of the factors to the nearest thousands or hundreds or tens. Take two rounded values of the factor, one which is less than the factor and second which is greater than the factor.
For example: 13 can be round off to nearest tens as 10 and 20. 10 less than 13 and 20 greater than 13.
2. Multiply the rounded number values with the other factor to get the result. Here we will get two multiplication results.
3. The exact answer or the product will always lie between these two estimated results.
4. This method can be used to find whether the estimated product of a multiplication equation is reasonable or not.
Example: 192 × 8, estimate the two numbers the given product is between.
Answer:
We know that 192 can be rounded off to nearest tens as 190 and nearest hundreds as 200. So, we have two estimations.
Case 1: When we choose 190,
190 × 8 = 1520 [Multiply]
Case 2: When we choose 200
200 × 8 = 1600 [Multiply]
So the two estimates are 1520 and 1600.
Hence, the product is between 1520 and 1600.
Solved Examples on Estimating the Product
Example 1: A student finds a product as given. Find whether the given answer is reasonable or not. Estimate the product.
263 × 3 = 869
Answer:
We know that 263 can be rounded off to nearest tens as 260 and 270. Here we have two estimations.
Case 1: When we choose 260
260 × 3 = 780 [Multiply]
Case 2: When we choose 270
270 × 3 = 810 [Multiply]
So, The product 263 × 3 will lie between 780 and 810.
It is given that 263 × 3 is equal to 869 which does not lie between 780 and 810
Hence, the given answer is not reasonable.
Example 2:
Thomas earns $ 4,554 each month. We estimate that he will earn $ 23,000 in 5 months. Determine whether the estimation is greater or less than the actual earning?
Answer:
Thomas’ earning in a month = $ 4,554
Thomas’ earning in 5 months = $ 4,554 × 5 = $ 22,770 [Multiply]
$ 23,000 is greater than $ 22,770.
So, the estimation of 5 months earnings is greater than the actual amount of earnings.
Example 3: Find the two estimates that the product 849 × 4 is between.
Answer:
We know that 849 can be rounded off to nearest tens as 840 and 850. Here we have two estimations.
Case 1: When we choose 840
840 × 4 = 3360 [multiply]
Case 2: When we choose 850
850 × 4 = 3400 [multiply]
Hence, the product of 849 × 4 will lie between 3360 and 3400.
Example 4: Estimate the product 62331 × 3.
Answer:
Round off 62331 nearest to the thousands. So 62331 is estimated as 62000.
62000 × 3 = 186000
So, 62331 × 3 \(\approx\) 186000 or 62331 × 3 , is about 186000.
Example 5: Which Automobile has a selling price twice as that of Bike’s.
Answer:
From the given graph we can see that,
Selling price of car = $25000
Selling price of Bus = $20000
Selling price of truck = $30000
Selling price of bike = $15000
Twice of bike’s price = $15000 × 2 = $30000 [Multiply]
We can see that a truck’s price is $30000 which is twice of $15000. Hence, the correct answer is truck.
A number can be rounded off to a value which is close to the exact number. Rounding off a number is used to make complex mathematical calculations simple.
For example 21 can be rounded to nearest tens as 20. Similarly, a number can be rounded off to nearest hundreds, thousands and so on.
By determining the two estimates between which the result of a multiplication equation lies we can tell whether a product is reasonable or not. If the product is in between the two estimates it is reasonable, otherwise not.