What is Meant by Estimating a Number?
Estimating numbers helps us calculate them mentally while performing operations like addition, subtraction, multiplication, and division. Estimation can be done in various ways, like estimating to the nearest tens, hundreds, thousands, and so on.
Why Do You Need to Estimate a Number?
Estimating a number makes it easier to check the accuracy of the result obtained by actual calculation. It also helps in developing skills in mental calculation. By learning the estimation of numbers, you can learn to estimate their sum, difference, product, and quotient. If the estimated answer is close to the actual answer, we say that the answer is reasonable.
How Can You Estimate the Product of Two Numbers?
Estimation of the product of two numbers can be done by the following methods.
- Rounding off the product
- Using compatible numbers
Rounding off Numbers
A number is rounded off to the nearest tens, hundreds, thousands, and so on to ease out calculations. When we round off a number to the nearest tens, we make sure that the ones digit is 0, so that the calculations can be performed mentally. Similarly, when we round off a number to the nearest hundreds, we make sure that the tens digits and the ones digits are 0, and so on. Let us see some examples to understand this further.
For example:
Round off 123 to the nearest tens and the nearest hundreds.
Solution:
123 rounded off to the nearest tens is 120
123 rounded off to the nearest hundreds is 100.
To understand this further, let us follow some steps:
If we want to round off 2356 to the nearest thousands, then:
Step 1: Find the place value up to which we need to round off the number. Circle that digit.
Step 2: Underline or highlight the digit that is on the right of the encircled digit.
Step 3: If the underlined digit is from 0 to 4, the digit stays the same. If the underlined digit is between 5 – 9, we need to add 1 to it.
Step 4: Now change all the digits to the right of the underlined digit to 0. All the digits on the left of the underlined digit remain the same.
Rounding off Products
In this method, both the numbers are rounded off to the nearest tens, hundreds, thousands, or ten thousands to make the calculation of the product of two numbers easier. In the next step, the reasonability of the answer is checked by comparing the actual answer with the estimated answer.
For example:
Estimate the product 11 × 27 by rounding off to the nearest tens.
Solution:
Actual product:
11 × 27 = 297
Estimation by rounding off:
11 → 10 (11 rounded off to the nearest ten is 10)
27 → 30 (27 rounded off to the nearest ten is 30)
Now, the estimated product is:
10 × 30 = 300
It is close to the actual product, hence we can say that the answer is reasonable.
Compatible Numbers
Compatible numbers are numbers that are easy to manipulate mentally.
These are the numbers on which basic operations such as addition, subtraction, multiplication, or division can be performed easily.
Compatible numbers are close to the actual number, hence making the estimation of the result easier.
Usage of Compatible Numbers to Estimate the Product
In this method, both the numbers are replaced with compatible numbers, which eases the process of mental calculation.
For example:
Use compatible numbers to estimate the product 22 × 49.
Solution:
Step 1: Choose compatible numbers.
22 is close to 20, and 49 is close to 50.
22 × 49 → 20 × 50
Step 2: Multiply.
22 × 50
= 22 × 5 tens
= 110 tens
= 1100
Hence, the estimated product of 22 × 49 is 1100 .
Solved Examples
Example 1: Estimate the product:
31 × 63
Solution:
Round off the numbers to the nearest tens:
31 → 30
63 → 60
30 × 60
= 1800
Using compatible numbers
31 is very close to 30
31 → 30
63 is close to 65
63 → 65
Now,
30 × 65
= 65 × 3 tens
= 195 tens
= 1950
Example 2: Elijah uses rounding to estimate 54 × 61. He gets a product of 3000. Is his estimate correct?
Solution:
54 × 61
54 → 50 (54 rounded to the nearest ten)
61 → 60 (61 rounded to the nearest ten)
50 × 60
= 50 × 6 tens
= 300 tens
= 3000
Hence, Elijah’s estimate is correct.
Example 3: Logan uses rounding and Emma uses compatible number to estimate the product 32 and 67. Which of the estimates is close to the actual product?
Solution:
Logan(Rounding off)
32 × 67
32 → 30 (32 rounded to the nearest tens)
67 → 70 (67 rounded to the nearest tens)
30 × 70
= 30 × 7 tens
= 210 tens
= 2100
Emma (Compatible numbers)
32 × 67
32 → 30 (32 is close to 30)
67 → 65 (67 is close to 65)
30 × 65
= 65 × 3 tens
= 195 tens
= 1950
Actual product,
32 × 67 = 2144
For the above product, Logan’s method is more accurate.
The following are some of the methods used to estimate the product of two numbers:
- Rounding off a product
- Using compatible numbers
Since 38 < 40 as well as 19 < 20, the estimated product will be less than the actual product.
Look at the digit on the right of the digit you want to estimate. If you see a digit greater than 5, round it up, and if it is less than 5, round it down.