Estimation in Maths

Have you ever observed the headlines of any news channels closely?

Let's look at a couple of examples. 

• The US drew with Canada in a football match watched by 51,000 spectators in the stadium and 40 million TV viewers worldwide.
• Approximately, 2000 people were killed and more than 50,000 injured in a cyclonic storm in the coastal areas.
• Over 13 million passengers are carried over the 63,000-kilometer route of railways every day.

Can we firmly say that the numbers quoted in the news are the exact match to the number of people in these scenarios? 

No! We know that these are not the exact numbers.

The word "approximately" indicates that the number (of people) was close to these given numbers.

Clearly, 51,000 could be 50,800 or 51,300, but definitely not 70,000

In the same way, 13 million passengers could indicate more than 12 million but less than 14 million people, but definitely not 20 million.

The quantities given in the examples above are not exact counts but are estimated to give an idea of the quantity.

This indicates that we round numbers to make them easier to work with or to express a number to an acceptable level of accuracy. 

This natural and quick process of assessing data is called estimation.

Let's learn how to estimate and solve problems involving estimation.

Table of Contents 

Estimation of a number is a reasonable guess of the actual value to make calculations easier and realistic.

Estimation means approximating a quantity to the required accuracy.

This is obtained by rounding off the numbers involved in the calculation and getting a quick and rough answer. 

We subconsciously involve estimation in all our day-to-day activities.

We typically estimate numbers to a certain level of accuracy.

Numbers are rounded to a particular place value to make them as close to a whole number as possible.

In all cases of rounding, we need to do the following:

1. Find the digit in the place we want to round to.
2. Observe the digit to its right to decide how to round:
   • If the digit to the right is 0-4 i.e., 0, 1, 2, 3, 4: we leave the digit alone (round down).
   • If the digit to the right is 5-9 i.e., 5, 6, 7, 8, 9: we increase the digit by 1 (round up).
3. Finally, we remove any extra digits that have not been rounded off. 

Let's assume that you want to estimate the total cost of several different items in your grocery list or that there are a few gift items you want to buy.

How would you roughly calculate the amount of money you'd need?

You would round the prices to the nearest tens place because it involves money and it becomes easier to calculate when you round off in this way.

If you are working with extremely large numbers, you might want to round the number to the nearest one million or one billion.

Did we use the same rounding off techniques in all these situations?

There are no hard and fast rules when you want to estimate the outcomes of numbers.

The procedure varies with the degree of accuracy required as well as how quickly we need the estimate.

Example:  We need to estimate \(854917.632\)

(a) to the nearest whole number.

(b) to the nearest thousand.

(c) to the nearest tenth.

(a) To estimate \(854917.632\) to the nearest whole number, we need to check the number in the tenth place. 

The digit in the tenths place is 6, so we need to round up.

Hence, we increase the digit in the units place by 1

Then we get the number 854917 + 1 = 854918 

\(\therefore\) The estimation of \(854917.632\) is \(854918\)

(b) To estimate \(854917.632\) to the nearest thousand, we need to check the number in the hundreds place.

4 is in the thousands place. The digit to its right is 9

So, we round up 4 i.e., increase the digit in the thousands place by 1

Then we get the number 854000 + 1000 = 855000 

(c) \(854917.632\) to the nearest tenth: we need to check the number in the hundredths place.

The digit in the hundredths place is 3

So, we round down 6 i.e., we leave the digit alone.

Let us see some real-life scenarios involving estimation, such as:

• counting numbers
• measuring the length, breadth, or other dimensions of a place or figure
• totaling the amount in a bill etc.

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Example 1

 

 

Tia and John collect marbles.

Tia has collected 335 and John has collected 442 marbles.

Choose a better estimate to find the number of marbles they own together. 

Solution

Here, a better estimate can be obtained by rounding off the digits to the nearest tens.

To get an estimate of the sum, round off all the numbers to the same place value, then add them.

Number of marbles Tia has collected = 335 rounded off to 340

Number of marbles John has collected = 442 rounded off to 440

\(\therefore\) Total number of marbles = 340 + 440 = 780

\(\therefore\) The estimated answer is 780

Example 2

 

 

A train terminal has 12 platforms.

Each platform has about 115 trains departing each day.

Find an estimate of the number of trains leaving the terminal every day.

Solution

This is an estimation problem involving multiplication.

The rule is to estimate all the numbers that we need to multiply to the nearest round number. 

The total number of trains = Number of platforms \(\times\) Number of trains leaving each platform

 \begin{eqnarray}\text{Total number of trains} &=& 12 \times 115\\ \text{Estimated value}&=&10 \times120\end{eqnarray}

\(\therefore\) Total number of trains is 1200

Example 3

 

 

In a town, there are 1,759 families.

An average of two children from each family will attend a welfare program organized at various schools.

Each school can accommodate 320 children.

Calculate the minimum number of school avenues that are needed in the region.

Solution

The total number of families = 1759 rounded off to 1800

The capacity of each school = 320 rounded off to 300

The final estimated answer =  \( 1800\div 300 = 6\)

\(\therefore\) 6 school avenues are required.

Example 4

 

 

Ron is planning a major yard renovation.

He wants to reseed his entire lawn. 

Here is an overhead view of his yard.

Help Ron find the estimate of the area of his yard.

Solution

The dimensions of the yard are 12.3 ft and 14.5 ft.

The rounded off dimensions will be 12 ft and 15 ft.

The yard is in the form of a rectangle.

\begin{eqnarray}
          \text{Area of rectangle} &=&\text {Length} \times\text {Breadth}\\ \text{Area of rectangle} &=& 12\hspace{0.2cm}\text {ft} \times 15\hspace{0.2cm}\text {ft}\\ &=&180\hspace{0.25cm} \text {sq.ft}    
\end{eqnarray}

\(\therefore\)180 sq.ft is the required estimate.

Example 5

 

 

Estimate the value of the following sum by first rounding each value to the nearest hundredth:

123.616283 + 5.123 + 6.183478

Solution

Let us first estimate each number and then we will add the numbers to get the required estimate. 

Given value	The digit in thousandth place	Round up/down	Estimated value(to nearest hundredths place)
123.616283	6	round up	123.62
5.123	3	round down	5.12
6.183478	3	round down	6.18
Estimated sum			134.92

\(\therefore\) Estimated sum = 134.92

Imagine you organize a carnival in your town.

The first thing you do is find out roughly how many guests will visit the carnival.

Can you calculate the exact number of visitors attending the carnival? It is practically impossible.

Let us consider another situation.

The finance minister of a certain country presents an annual budget.

The minister allocates a certain amount under the head "Education."

Can the amount allocated be absolutely accurate?

It can only be a reasonably good estimate of the expenditure the country needs for education during that year.

In the above-mentioned scenarios, a rough estimate will help us plan things properly.

Here we do not need exact figures.

In all these kinds of situations where the round-off value is as good as the exact value, we use estimates. 

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Why Do We Estimate in Math?

Often in math, we focus on getting an accurate answer.

But in some places, the estimation gives a quick and almost realistic answer.

Estimation saves time and we can achieve a better picture of things in a simpler way.

Hence, to get a quick and reliable answer, we estimate in math.

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How to Estimate a Number?

Here is the summary on how to estimate a number in general:

 

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Let's Summarize

This mini-lesson targeted the fascinating concept of estimation. The math journey around estimation starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

1. What is the difference between exact and estimate?

The estimate is the nearest rounded off value to the exact value.

2. Why do we estimate in math?

Estimation saves time and we can achieve a better picture of things in a simpler way.

Hence, to get a quick and reliable answer, we estimate in math.

3. What is the advantage of estimation?

Estimation helps save time and money before investing in real.

By estimating, we can benefit as we can save money and time on huge projects.

Estimation helps in doing a reality check of any project.

4. How do you estimate sums and differences?

To get an estimate of the sum, round off all the numbers to the same place value, then add them.

To estimate the difference, round off all the numbers to the same place value, then subtract them accordingly.