Chapter 1: Knowing Our Numbers – Estimation (Class 6 Maths)
Table of Contents:
- An Overview: Why Do We Need Estimation?
- Definition and Understanding: What Is Estimation
- Method of Estimation to the Nearest 10: Detailed Explanation
- Understanding the Method of Estimation to the Nearest 100:
- Case 1 (Equal to or Greater than 5)
- Case 2 (Less than 5)
- Method for Estimation to the Nearest 1000: Detailed Explanation
- Case 1 (Equal to or Greater than 5)
- Case 2 (Less than 5)
- Numerical Problems Based on Estimation Method
- Word Problems Based on Estimations of Numbers:
- School Supplies (Pencil) Estimate
- Water Tank Reading (Volume Estimate)
- Distance to Grandma’s House—Distance Estimate
An Overview: Why Do We Need Estimation?
It so happens, in our lives, we may have to make quick, smart choices. This can be in regard to making a rough estimate of numbers. This is where the mathematical method of estimation will be a needed skill. Also, calculating exact answers can be tough sometimes and may consume a lot of time.
However, with the use of the concept of estimation, it may become easier to figure out a rough answer quickly.
Example: Suppose certain lab equipment is being ordered by a school. The teacher arrives at an estimated cost (the total cost). The total cost estimated is a close guesswork made by the teacher. Let us consider that the lab equipment cost is 10,001, the rough estimate can be 10,000.
This kind of budget-planning-clever-tactic helps to come at the cost and aids in understanding any kind of overspending that may happen on unnecessary items.
Similarly, an engineer may be asked to do a project, and he, too, may have to estimate the project timeline and expenditure. Suppose the engineer knows the cost is 20,00030.
Importantly, he has set a client-pleasing delivery date so that the project is not delayed. Thus, the utilization of estimation concepts may serve as a time-saving measure and clever tricks. Using the concept of estimation, he can narrow down the cost to 20,00000.
It can also be seen as an effort-reducing method. Moreover, the use of estimation helps to reach a good-enough, fast answer.
Definition and Understanding: What Is Estimation
The concept of Estimation can be used to find a near-enough value of the original number. Hence, it is a guesswork closer to the actual value.
The process of doing estimation for a given number is simple, and it is called rounding off.
In other words, all learners have to do is round off a given number to a nearby place value--nearest tens, hundreds, or thousands.
This process of rounding off makes calculations easy to carry out if many numbers are involved in the calculation.
Imagine that planning is going for a school's annual day; in this case, you estimate attendees. By doing so and finding out the number of people attending the event, it may help to book the right-sized event hall.
Let us assume that tentative attendees are 1989, which can be rounded off to 2000.
Consider another case: suppose a software developer works on a coding program; he needs to estimate the code size. By doing so, he can get the needed storage space for the program.
This type of estimation, importantly, plays a major role in preventing future problems--a system-designing approach. In all these cases, we can't mention that we have an accurate answer, but a closer estimate.
Method of Estimation to the Nearest 10: Detailed Explanation
In order to round a given number to the nearest 10, it is necessary to check out the unit’s digit (that is: the last digit).
Suppose the digit in the units place is 5 or more, then go on to round the digit in the tens place up by 1, and at the same time, change the units digit to 0.
Now, if the number is such that the digit in the units place is between 0 and 4, then let the tens' digit remain the same, but change the digits in the units' place to 0.
One of the best examples can be rounding the number 47 to the nearest 10's place.
That is: Since the units digit is 7 (which is ≥5), we can raise the number 4 (in the tens' place) to 5 and then write 0 as the units digit.
Thus, the answer becomes 50; so, 47 is approximated to 50.
Understanding the Method of Estimation to the Nearest 100:
Case 1 (Equal to or Greater than 5)
In order to round off a given number to the nearest 100, we have to see what's at the tens digit (second last digit).
Suppose that the tens digit contains 5 or exceeds it. Then, go on to round the digit at the hundreds' place---done by increasing by 1.
Then, the number has to be changed, and it happens in the tens' as well as units' digits, turning the digits to 00.
Example 1: Suppose the number is 177, then the estimate will be 200.
Example 2: Try to approximate 456 to the nearest 100's place.
It can be seen that the tens' digit is 5 (a number greater than 5). This means change the number 4 at the hundreds digit to 5. Then, once this is done, write 00 for the last two digits to get the answer as 500.
Case 2 (Less than 5)
If the tens' digit is between 0 and 4, then keep the hundreds digit the same and change the last two digits to 00.
Example: Try to approximate 416 to the nearest 100's place.
It can be seen that the tens' digit is a number less than 5. This means retain the number 4 at the hundreds digit. Then, once this is done, write 00 for the last two digits to get the answer as 400.
Method for Estimation to the Nearest 1000: Detailed Explanation
For rounding a given number to the nearest 1000, have a close look at the digit in the hundreds place (that is: the third last digit).
Case 1 (Equal to or Greater than 5)
Suppose in the hundreds place, the digit is equivalent to the number 5 or exceeds 5, then rounding the digit at the thousands place (by increasing it) by 1 makes sense.
Example: Round 7,632 to the nearest 1000.
Hundreds digit = 6 (which is greater than 5), so we increase 7 by 1. The number in the thousandths place becomes 8, and the remaining digits turn to 000.
Answer: 8,000.
Case 2 (Less than 5)
Consider a case wherein the hundreds digit lies between 0 and 4. In such situations, refrain from changing the thousands' digit. Instead, changes can be brought in the last three digits by making them 000.
Example: Round 7,432 to the nearest 1000.
Hundreds digit = 4 (which is ≤5), so we keep 7 as the thousands digit and write 000.
Answer: 7,000.
Numerical Problems Based on Estimation Method
- Round the number 68 to the nearest 10th place.
Answer: 70
Explanation: Since the units digit is 8 (that is greater than 5), so all we have to do is add 1 to 6 (the tens’ digit). Thus, 6 will become 7, and the last digit turns into 0, giving 70.
- What can be the estimate (rounding off) of the number 123, considering the nearest 100th place?
Answer: 100
Explanation: As it can be seen that the tens’ digit is 2 (less than 4). So, the hundred’s digit does not change and remains 1. But the last two digits change to 00, making the answer 100.
- Approximate the number 289 to the nearest 10.
Answer: 290
Explanation: It can be noticed that the unit’s digit is 9 (greater than 5). Hence, the tens’ digit 8 will have to be changed to 9. At the same time, the units’ digit will become 0, making the answer 290. - Round off the number 6,547 to the nearest 1000.
Answer: 7,000
Explanation: In the problem, the digit at the hundreds’ place is 5 (≥5). Hence, it becomes necessary to add 1 to the thousands’ digit 6. This will change 6 (1000th digit) to 7. Also, the last three digits will change to 000, and hence, we get the answer as 7,000. - Round the number 3,412 to the nearest 100.
Answer: 3,400
Explanation: In this case, we notice that the digit at the tens’ place is 1 (≤4). Hence, the digit at the hundreds' place remains as 4 (same number). However, the last two digits will become 00, and the answer will be 3,400.
Word Problems Based on Estimations of Numbers:
- School Supplies (Pencil) Estimate
- Meena needs to take about 47 pencils for her class for students. But the problem lies in carrying exactly 47 items, as she isn't able to decide whether she should carry 40 or 50. Estimate to the nearest 10, how many pencils she should pack.
Answer: 50 pencils.
Explanation: We look at 47’s unit's digit (7) and notice that it’s ≥5. Hence, we can go on to round up the 4 tens to 5. This will change 47 pencils to 50 pencils. So, Meena has to pack exactly 50 pencils to be safe and ensure that all students get a pencil.
Water Tank Reading (Volume Estimate)- A water tank shows a reading of about 1,238 liters on its gauge. In order to quickly understand whether it is almost full or not, round the given number to the nearest 100.
Answer: 1,200 liters.
Explanation: It can be noticed that the tens' digit is 3 (which is less than 5). Hence, the hundreds digit does not change and remains 2. However, the last two digits will go to 00, giving the answer as 1,200 liters.
Distance to Grandma’s House—Distance Estimate
The map indicates that it is about 8,765 meters to reach Grandma’s house. To have a rough estimate of the distance (how far it is), approximate the number to the nearest 1000 meters.
Answer: 9,000 meters.
Explanation: When we look at the digit at the hundreds' place, which is 7 (as the 765 hundreds digit is 7), it’s ≥5. So, we can round the number to 8 thousand up to 9 and write the remaining digits as 000, making the answer 9,000 meters.
