The Estimation of Quotients
This page emphasizes the concept of the estimation of quotients. Two methods of estimation — using division facts and using compatible numbers — are explained in detail. Each method is supported by solved examples.
What is a Quotient?
A quotient is a number determined by dividing a number by another number.
For example, if we divide 10 slices of pizza among 5 children, the result will be 2 slices of pizza, which means each child will get 2 slices of pizza. In this example, 2 is called the quotient. This is mathematically represented as
10 ÷ 5 = 2
Where 10 is the dividend, 5 is the divisor, 2 is the quotient, and ÷ is the symbol for division.
What is Meant by the Estimation of Quotients?
Estimation refers to obtaining a result that is close to the correct or actual result. It involves drawing some conclusions or rounding a number to the nearest decimal place or to the nearest ones, tens, hundreds, and so on.
For various reasons, estimation may be important. It is especially useful when there is not enough information to calculate an exact value. Accountants use estimates when they want to calculate transaction amounts.
- To estimate the quotient, we round the dividend to the nearest tens, hundreds, or thousands.
- There are two methods for estimating the quotient:
- Division facts.
- Compatible numbers.
How do We Estimate Quotients Using Division Facts?
The division facts of various numbers can be used to estimate quotients in a division problem.
To calculate the quotient, we must first look at the first two or more digits of the dividend, depending on the divisor, and then use the basic division facts.
For example, let us estimate \(820\div~9\) using division facts.
Here the dividend is 820. The first two digits of this number form the number 82. 82 is close to the multiple of 9, that is,
\(9\times~9~=~81\) or \(81\div~9~=~9\) (division fact of number 9).
So, \(9\times~90~=~810\) or \(810\div~9~=~90\).
Therefore, after applying the division fact of number 9, we can say that the quotient of \(810\div~9\) will be close to 90. Hence, the estimated quotient is 90.
This is how we can estimate the quotient using basic division facts of various numbers.
How do We Estimate Quotients Using Compatible Numbers?
Compatible numbers are those numbers that are simple to divide. Such numbers are close to the exact value of the actual numbers, making it easier to estimate the answer and solve problems. To make the numbers compatible, we can round them to the nearest tens, hundreds, thousands, ten thousands, and so on.
We can check whether an answer is reasonable when solving division problems by estimating two numbers between which the quotient lies.
For example, let us estimate \(164\div~4\) by using compatible numbers.
We have to think of the numbers that are close to 164 and are easily divisible by 4.
Use 160 and 200 because they are close to 164 and also easily divisible by 4.
For 160, \(16\div~4~=~4\), so \(160\div~4~=~40\).
For 200, \(20\div~4~=~5\), so \(200\div~4~=~50\).
Choose 160 because 164 is closer to 160.
So, the estimate of \(164\div~4\) is about 40.
Here, the numbers 160 and 200 are the compatible numbers which are divisible by 4.
How can We Use Estimation to Compare?
Estimation can be used to compare the estimated quotient with the exact value of the quotient. We can determine if an answer to a division problem is reasonable or not. In other words, we can estimate the two numbers between which the exact quotient is.
For this, we need to determine two rounded values of the dividend, which are easily divisible. These values can be the dividend rounded to the nearest 10 or 100 or 1000 and so on, depending on the value of the dividend.
For example, let us find 2 numbers that the quotient \(264\div~2\) is between.
We have to think of the numbers that are close to 264 and easily divided by 2.
These numbers are 260 and 300 (here 260 is 264 rounded to the nearest tens and 300 is 264 rounded to the nearest hundreds).
Trying 260 , \(26\div~2~=~13\), So that \(260\div~2~=~130\)
Trying 300 , \(30\div~2~=~15\), So that \(300\div~2~=~150\)
264 is between 260 and 300.
So, the quotient \(264\div~2\) is between 130 and 150.
Solved Estimation Quotient Examples
1. Estimate the quotient: \(463\div~5\).
Solution:
Look at the first two digits of the dividend and use basic division facts. We have to think of the numbers that are close to 463 and easily divided by 9.
Trying 450 , \(45\div~5~=~9\), So that \(450\div~5~=~90\)
Trying 500 , \(50\div~5~=~10\), So that \(500\div~5~=~100\)
Choose 450 because 463 is closer to 450.
So, \(463\div~5\) is about 90.
2. For \(5184\div~6\), find the two numbers that the quotient is in between.
Solution:
We have to think of the numbers that are close to 5184 and easily divided by 6.
Use 5100 and 5220 because they are close to 5184 and are easily divided by 6.
\(510\div~6~=~85\), so \(5100\div~6~=~850\).
\(522\div~6~=~87\), so \(5220\div~6~=~870\).
5184 is in between 5100 and 5220.
So the quotient, \(5184\div~6\) is between 850 and 870.
(Here, 5100 and 5220 are the compatible numbers which are easy to divide.)
3. A gallon contains 3,795 milliliters approximately. A gallon contains 5 times the number of milliliters as a quart. How many milliliters does a quart contain?
Solution:
We have to estimate \(3795~\text{milliliters}\div~5\) to know how many milliliters are in 1 quart.
Look at the first three digits of the dividend and use basic division facts. The first three digits of the dividend form the number 379. So we have to think of a number that is close to 379 and easily divided by 5.
Trying 3750 , \(375\div~5~=~75\), So that \(3750\div~5~=~750\)
Trying 3800 , \(380\div~5~=~76\), So that \(3800\div~5~=~760\)
Choose 3800 because 3795 is closer to 3800.
So, \(3795\div~5\) is about 760.
Therefore, there are about 760 milliliters in 1 quart.
4. For four months, a teenager works as a waiter in a restaurant and earns $3178. Every month, he earns the same amount. How much money does he make on a monthly basis?
Solution:
We have to estimate $\(3178\div~4\) to know the monthly income.
We have to think of the numbers that are close to 3178 and easily divided by 4.
Use 3080 and 3120 because they are close to 3178 and easily divided by 4.
\(308\div~4~=~77\), so \(3080\div~4~=~770\).
\(312\div~4~=~78\), so \(3120\div~4~=~780\).
3178 is in between 3080 and 3120. So the quotient, \(3178\div~4\) is between 770 and 780.
Therefore, the teenager earns between $770 to $780 per month.
5. Describe how can we come up with a better estimate for \(4250\div~5\) than the one provided. (Round numbers 4520 to 5000. Estimating \(5000\div~5\).
\(5000\div~5~=1000\), so \(4250\div~5\) is approximately 1000).
Solution:
We have to think of the numbers that are close to 4520 and easily divided by 5.
Use 4500 and 5000 because they are close to 4520 and easily divided by 5.
\(450\div~5~=~9\), so \(4500\div~5~=~900\).
\(500\div~5~=~10\), so \(5000\div~5~=~1000\).
4520 falls between 4500 and 5000. So the quotient of \(4250\div~5\)is between 900 and 1000.
(Here, 4500 and 5000 are the compatible numbers that are easy to divide.)
Choose 4500 because 4500 is closer to 4520 as compared to 5000.
So, the quotient \(4250\div~5\) is about 900.
6. Estimate to compare: \(25\div~9 \) _ 2.
Solution: Look at the two digits of the dividend and use basic division facts. We have to think of a number that is close to 25 and easily divided by 9.
Try 18 , \(18\div~9~=~2 \)
Try 27 , \(27\div~9~=~3 \)
Choose 27 because 27 is closer to 25.
So, \(25\div~9~=~3 \) is about 3.
Since \(27~> ~25 \)
Therefore, \(25\div~9> 2 \).
7. Estimate to compare: \(143\div~3 \) _ 50.
Solution:
Look at the first two digits of the dividend and use basic division facts. We have to think of a number that is close to 14 and easily divided by 3.
Trying 120 , \(12\div~3~=~4\) so \(120\div~3~=~40\)
Trying 150 , \(15\div~3~=~5\) so \(150\div~3~=~50\)
Choose 150 because 150 is closer to 143.
So, \(143\div~3\) is about 50.
Since \(143< ~50\).
Therefore, \(143\div~3~< ~50\).
Compatible numbers are easy to add, subtract, multiply, or divide. If a division problem has a complex or large value of dividend that is not easily divisible, we can replace it with a compatible number to make the division operation easy. This compatible number will be close to the actual number and is easy to divide. However, the result of this division will always be an approximation or an estimation of the exact result.
For example, in \(450\div~11\), 450 is not divisible by 11. In this case, we can replace it with a compatible number, that is 440.
\(440\div~11~=4\).
Hence, \(450\div~11\) is about 4.
Estimating quotients can assist you in double-checking your work and solving more difficult division problems. Before evaluating an exact value, or if an exact value is not required, an estimation is frequently used. An estimate is a close approximation of the true answer.