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# Estimate|Definition & Meaning

## Definition

The term “estimate” can either mean a **value** that is very **close** to the correct answer (also called an **approximation**) or the process of finding such a value. We do it all the time! However, it does require some **information** about the subject to be a decent guess. For example, if a person barely fits through a **6.5** ft high doorway, they are around **~6.5** feet tall, which is rather rare.

**Figure 1** shows some small **triangles** in a box and the **estimated** result of the number of triangles in the box without counting.

## Estimation in Real-life Calculations

Estimation is performed so often in real life as it helps to form an **idea** about a certain thing in a very few seconds without doing **calculations**. For example, we can **estimate** the total number of **books** present on the shelf in a library.

The **estimated** result, if the person is good at estimation, is often **close** or near to the actual result. The difference is that the estimated result is calculated when there is a time **constraint**, and the time to calculate the actual result is insufficient.

For** example**, if you want to buy **6** books from the bookshop, that costs **$2.95** each. The seller says the total price of **6** books equals **$20**. Now, you can check the total price by quickly **estimating** the result in your mind as:

2.95 dollars is almost equal to 3 dollars

So, when **$3** is multiplied by **6** books, the total price comes out to be** $18**. The seller was telling you **$20**, which is obviously more than the actual price, as the estimated **$18** price comes from approximately increasing the price, i.e., to **3** dollars instead of **$2.95**. Hence, you tell the seller to recheck the total selling price.

Here, the power of **estimation** saves you with the help of **approximating** some numbers in the calculations, i.e., in the above case, the price of **one** book.

## Approximation

An approximation is defined as something that is **similar** but not exactly equal to the **actual** value, description, quantity, image, etc. It is abbreviated as “**approx.**” and usually applies to numbers. For example, if the time is **8:55 a.m.**, it can be approximated to **9:00** **a.m.** as it is close to the actual time.

The **approximation** is similar to estimation, but it is used in **scientific** terminology, whereas estimation can be used in any **real-life** problem.

The approximation is used if the required data is not **completely** available, the problem’s **sensitivity** to this data, the time constraint, and how much **accuracy** is required is the solution.

Approximations are also made in **scientific** experiments. This is because some **factors** from the real situation are left out to simplify the calculations. For example, the **frictional** force is not included when calculating the **force** applied on an object which makes the result an approximation.

### Different Symbols for Approximation

It is denoted by “,**≈**” which is read as “almost equal to.” This symbol is usually used to indicate **approximation** between numbers such as **e ≈ 2.718**. The symbol “**≌**” means “approximately equal to” and approximates the **congruence** in figures, e.g., triangle LMN ≌ triangle ABC. The symbol “**≉**” denotes “**not** almost equal to,” e.g., 3 ≉ 5 but 3 ≈ 3.2.

## Equality

Equality is defined as the **relationship** between two or more quantities or mathematical expressions, saying that the two **expressions** or objects represent the **same** value. The symbol of equality is “**=**,” and C = D is read as “C is **equal** to D.”

In estimation, equality has an important role as the **estimated** value should be **close** to the actual value. If the actual value is not known, an estimation of the value can be found.

## Exact and Estimated Value

The **exact** and estimated value comes from the concept of **equality** and approximation. The **estimated** value approximates the exact value, i.e., it is close to the **actual** value. For example, the estimated value of the length of a line, equal to **12.7 cm**(exact value), is **13 cm**.

## Advantages of Estimation

The following are some **advantages** of estimation.

### Time-Saving

Estimated results are always calculated when the **time** required for the answer is very **short**. It saves a lot of time while also checking the **accuracy** of results to a point. For example, roses that are **7 cm** apart need to be planted in a **68.6 cm** long **row**. Here the calculation does not need to be exact, so **estimation** can save time as:

68.6 cm is almost 70 cm

70 cm divided by 7 is 10

So, **10** roses **7cm** apart should be enough to plant in a **68.6 cm** long row.

### Detection of Mistakes in Calculations

Estimation can **detect** mistakes in calculations as human **error** is possible. It comes in handy when the results need to be **cross-checked**. For example, to calculate **105** times **65**, the calculator shows **975**. To cross-check, approximating 105 to 100 and 65 to 60, the **estimated** answer is:

100 ✕ 60 = 6000

But why the **calculator** shows 975 as the product of 105 and 65? It is not even **close** to the estimated value, i.e., **6000**. It means some mistake occurred while typing the numbers in the calculator. The person typed **15** instead of **105**, whose product is 975, which led to an **error**. Hence, it is really important to cross-check your results by using **estimation**.

Also, estimation is performed as a calculator is not always **available** to do difficult calculations.

### Improves Mental Math

Estimation helps to improve the **skill** of doing math with your **mind** and not to use any calculator. The more **practice** of estimation is performed in real life, the more the person becomes an expert in it.

## Different Strategies of Estimation

An estimated result can be calculated using **different** strategies.

For example, to add **398** and **650**, one way is to approximate the number 398 to **400** and then add it to 650, i.e.,

400 + 650 = 1050.

The other **way** is to add up to the **tens** place of both the number, i.e., **98** and **50** of the numbers 398 and 650, respectively, as:

98 + 50 = 148

The number **148** is nearly equal to **150**, hence adding the hundreds of both the numbers, i.e., **300** and **600** with 150, gives the estimated result as:

300 + 600 + 150 = 1050

A person develops his/her own strategies to find the **estimate** of a problem, as different problems require different **strategies**.

## Visual Estimation

Visual estimation is the process of estimating how **big** or small, how **long** or wide, and how **many** things are present by using the **visual** sense. This includes the **estimation** of counts and lengths, as discussed below.

### Estimation of Counts

The estimation of **counts** requires finding the **approximate** number of the objects present without **counting** them. For example, many diamonds are shown in **figure 2**.

A person can estimate the **total** number of diamonds just by **looking** at them without counting them. The **estimated** value will be close to **18**(exact); e.g., **15** is an estimated value.

### Estimation of Lengths

The **estimation** of lengths involves estimating the **measurements** of lengths that are close to the exact measurement. For example, two squares are shown in **figure 3**. The length of the small square is 1 m. What is the estimated length of the big square?

The **estimated** length of the square can be **4 m**, but the exact length is **4.5 m**.

## An Example Involving Estimation

Two rectangles, **A** and **B**, are shown in **figure 4**.

If the **length** of the small rectangle is **2 cm** and its width is **1 cm**, estimate the length and width of the big **rectangle** and find its **estimated** area.

### Solution

The **estimated** length of the big rectangle is** 6 cm** as the **length** of the big rectangle seems **three** times the length of the small rectangle, i.e., **2** cm as:

Estimated Length of Rectangle B = 2 ✕ 3 = 6 cm

Its estimated width is** 2 cm** as the big rectangle’s width seems **two** times the width of the small rectangle, i.e., **1** cm.

Estimated Width of Rectangle B = 1 ✕ 2 = 2 cm

The **area** of a **rectangle** is given by the formula:

Area = Length ✕ Width

Placing the values of estimated **length** and **width** to get the **estimated** area as:

Estimated Area of Rectangle B = 6 ✕ 2 = 12 cm$^2$

*All the images are created using GeoGebra.*