Estimation

🏆Practice 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. 

For example:

It can be estimated that half of 1603 is approximately 800.

B - 50% of 1603  Estimation

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Test yourself on estimation!

einstein

Approximately what is \( \frac{6}{25} \) as a percentage?

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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=96024\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


Do you think you will be able to solve it?

Examples with solutions for Estimation

Exercise #1

Approximately what is 625 \frac{6}{25} as a percentage?

Video Solution

Step-by-Step Solution

We look for the closest fraction to be able to divide the numerator by the denominator:

525 \frac{5}{25}

We break down the denominator into a multiplication exercise:

55×5 \frac{5}{5\times5}

We simplify:

15 \frac{1}{5}

We convert the fraction into a percentage

15×100=1005= \frac{1}{5}\times100=\frac{100}{5}=

20% 20\%

Answer

20%

Exercise #2

Approximately what is 3470 \frac{34}{70} written as a percentage?

Video Solution

Step-by-Step Solution

To solve this problem, we'll convert the fraction 3470 \frac{34}{70} into a percentage by following these steps:

  • Step 1: Calculate the decimal equivalent of 3470 \frac{34}{70} .

  • Step 2: Multiply the decimal by 100 to convert it into a percentage.

Now, let's work through each step:
Step 1: Perform the division 3470 \frac{34}{70} . The result of dividing 34 by 70 is approximately 0.4857.
Step 2: Convert this decimal into a percentage by multiplying by 100:
0.4857×100%=48.57% 0.4857 \times 100\% = 48.57\% .

Since we want an approximate percentage from the given choices, we round 48.57% 48.57\% to the nearest whole number, which is 50% 50\% .

Therefore, the solution to the problem is 50% 50\% .

Answer

50% 50\%

Exercise #3

Approximately what is 8011 \frac{80}{11} written as a percentage?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Convert the fraction 8011 \frac{80}{11} into a decimal number.
  • Step 2: Multiply the resulting decimal by 100 to find the percentage.
  • Step 3: Identify the correct choice from the provided options.

Now, let's work through each step:
Step 1: Divide 80 80 by 11 11 . The result is approximately 7.27 7.27 , as 11 11 goes into 80 80 7 times with a remainder.
Step 2: Multiply 7.27 7.27 by 100 100 to convert it to a percentage: 7.27×100=727% 7.27 \times 100 = 727\% . However, since the question asks for an approximation, use rounding to the nearest hundred percent: approximately 800% 800\%.
Step 3: Among the choices provided, the closest approximate percentage is 800%, corresponding to choice 4.

Therefore, the solution to the problem is 800%.

Answer

800%

Exercise #4

Approximately what is 145 \frac{14}{5} as a percentage?

Video Solution

Step-by-Step Solution

We are looking for the closest fraction to be able to divide the numerator by the denominator:

155=3 \frac{15}{5}=3

Convert the fraction into percentage:

3×100=300 3\times100=300

300% 300\%

Answer

300%

Exercise #5

Approximately what is 19 \frac{1}{9} as a percentage?

Video Solution

Step-by-Step Solution

To find the approximate percentage of 19 \frac{1}{9} , we follow these steps:

  • Step 1: Use the formula to convert the fraction to a percentage. This involves multiplying the fraction by 100:
Percentage=19×100 \text{Percentage} = \frac{1}{9} \times 100

Calculating the above expression gives us:

Percentage11.11% \text{Percentage} \approx 11.11\%

This result is approximately equal to 10% when rounding to the nearest whole number, which corresponds to choice 1 in the list of possible answers.

Therefore, the approximate percentage of 19 \frac{1}{9} is 10%.

Answer

10%

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