Estimation

🏆Practice statistical estimation

Estimation

In fact, estimation allows us to guess (hence the redundancy) the supposed result, without performing the exact calculation.
That is, in certain cases, we don't need to know the solution precisely, a rough idea is sufficient to solve a particular mathematical problem.

Sometimes we are asked to compare mathematical expressions, draw deductions from one exercise to another, round numbers to simplify a calculation, and other similar tasks. 

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Test yourself on statistical estimation!

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שגיא מתאמן למשחק חשוב שקורה בעוד יומיים. בכדי להשתפר הוא מנסה לקלוע לסל כמה שיותר פעמים. באימון היום שגיא פספס 17 זריקות מתוך 89 הזריקות שזרק. מה בערך אחוז הפעמים בהם שגיא קלע?

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Let's look at some examples.

Example 1

Given the two expressions 13+6513+65 and 17+6817+68

We are asked to determine which expression is greater without solving them.
Upon observing them, we see that 1717 is larger than 1313 and also 6868 is larger than 6565.
If each term of the second expression is greater than those of the first, then the entire second expression is greater than the first.
Therefore, it is true that:

13+65<17+6813+65 < 17+68


Let's look at another example

If we know that 600×12=7200600\times 12=7200
what will be the result of the exercise 12×120012\times 1200 without calculating it?
Let's look at the data we have and compare it with the exercise we need to solve:

12×1200=12×600×2=7200×2=1440012\times 1200= 12\times 600\times 2= 7200\times 2= 14400-

In fact, what did we do? In the required exercise, we can represent the 12001200 by 600×2600\times 2. This will give us the given expression, and all we have to do is multiply the result by 22.

We can also talk about estimation when referring to percentages.
We will demonstrate this with an example.

If we need to estimate the 50%50\% of 15031503, we can say that 15031503 is quite close to 15001500, and therefore, we can say that the 50%50\% would give us approximately 750750.

That is, 50%50\% of 15037501503 ≈ 750


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Example 3

We need to fill in the greater than/less than/equal to sign between the following two expressions: 69+27 69+27 --- 66+24 66+24

Solution: 

Let's look at the two expressions and see that 6969 is larger than 6666 and also 2727 is larger than 2424.

Therefore, it is not necessary for us to calculate the exact result to determine that:

69+27>66+24 69+27 > 66+24


Example 4

We are asked to deduce from the following data 40×12=48040\times 12= 480 the result of the exercise 40×2440\times 24.

Solution: 

In this case as well, we can see that 24×40=12×40×224\times 40=12\times 40\times 2 holds true.

This means that we do not need an exact calculation, but all we need to do is to multiply the given result by 22.

Which will give us: 24×40=12×40×2=480×2=960.24\times 40=12\times 40\times 2= 480\times 2= 960.


Do you know what the answer is?

Example 5

We are asked to solve the exercise 21×4121\times 41 approximately, without a calculator and without performing an exact calculation.

Solution

In this case too, we can use estimation to make an approximate calculation.

We will round 4141 to 4040.

We will round 2121 to 2020.

Now we will multiply the estimated results and obtain the following: 40×20=80040\times 20= 800.

That is, the approximate result of the required exercise is 800800.


Example 6

If the number 400400 is increased by a number greater than 7070

What can we say about the result?

Solution: 

In this case, it's about adding or subtracting inexact numbers. 

If we add 400400 plus 7070, the result would be 470470.

Since we are adding to 400400 a number larger than 7070, the result we will obtain will be greater than 470470


Check your understanding

Review Questions

What is estimation in mathematics?

In mathematics, an estimation is an approximation of a calculation, whether it be addition, subtraction, multiplication, or any other operation; however, when estimating, it is not necessary to perform the operation as such, that is, it is not necessary to carry it out and get the exact result, but rather just to estimate an approximate result.


What is an estimation and why is it important?

As we have already mentioned, an estimation is an approximation of some result. It is very important because from a young age we are taught to perform some mental calculations of operations to reach an approximate result without the need to perform the operation as such. As this skill develops, the result will become more precise, that is, closer to the real value.


How do you estimate the result of an operation?

Let's look at some examples of how to estimate the results of some operations

  • Estimation of a sum:

Example 1

Prompt. Estimate the result of 142+256= 142+256=

Solution

Let's break down the numbers of each addend into hundreds, tens, and ones as follows:

142=100+40+2 142=100+40+2

256=200+50+6 256=200+50+6

Now let's add the hundreds, tens, and ones separately

100+200=300 100+200=300

40+50=90 40+50=90

2+6=8 2+6=8

From here we can estimate that the result will be 398 398

Result

398 398

  • Estimation of a product

Example 2

Prompt. Estimate the following operation 1923= 19\cdot23=

Solution:

To estimate the result, we will round each factor as follows

The number 19 19 will be rounded to 20 20

And the number 23 23 will be rounded to 20 20

Therefore, we perform the multiplication 2020=400 20\cdot20=400

Then an approximate result is 400 400 , it should be noted that this is not the correct result, it is just an estimation of the result.

Result

The estimation is 400 400


If you're interested in this article, you might also be interested in the following articles:

The Real Number Line

Exponents for Seventh Grade

What is a Square Root and What is it Used For?

The Multiplication Tables

On the Tutorela blog, you'll find a variety of math articles.


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