Division of Decimals
Decimals are used to represent the values between two whole numbers. These values between whole numbers can be represented in the fractional form and in the decimal form. To understand the decimal form, take a look at the pattern of division provided here.
689 ÷ 10 = 68.9
689 ÷ 100 = 6.89
689 ÷ 1000 = 0.689
689 ÷ 10000 = 0.0689
As you can see, when the number is multiplied by \(\frac{1}{10}\) or divided by 10 with each increasing zero, the decimal place in the quotient also moves one place to the left. The number of decimal places is equal to the number of zeros.
Now, what happens when a number is divided by a number with a decimal number?
689 ÷ 0.1 = 6890
689 ÷ 0.01 = 68900
689 ÷ 0.001 = 689000
689 ÷ 0.0001 = 6890000
689 ÷ 0.00001 = 68900000
When multiplied by a power of 10 or divided by a power of 0.1, the decimal in the answer jumps to the right, depending on the number of decimal places the divisor covers.
Estimation of Decimal Quotients
This is the process of selecting numbers closest to the dividend and the divisor, such that they are divisible and we arrive at a quotient that is an approximate value of the required quotient. This will help in making the calculation much easier. Take a look at the example given below.
31.25 ÷ 5.9
Let’s first consider the divisor. The value is 5.9. We can round it off to 6. We can also approximate 31.25 that is the dividend to 30. We chose 30 because it is easily divisible by 6.
30 ÷ 6 = 5
So, the estimate of 31.5 ÷ 5.9 is about 5. Now, let’s examine how to divide two numbers when the dividend is smaller in value than the divisor.
4.3 ÷ 8.7
8.7 can be approximate as 9.0 or 90 tenths. In this case, we will consider 4.3 to be 43 tenths. Since 45 is divisible by 9, let’s approximate 43 tenths to be 45 tenths.
45 ÷ 90 = 0.5
So, 4.3 divided by 8.7 is about 0.5.
Solved Examples
Example 1:
Estimate the quotient: 3.9 ÷ 8.3
Solution:
Since the dividend is less than the divisor,
3.9 is 39 tenths which can be rounded as 40 tenths
8.3 is 83 tenths which can be rounded as 80 tenths
So,40 ÷ 80 = 0.5
Hence the 3.9 ÷ 8.3 is about 0.5
Example 2:
Estimate the quotient: 18.9 ÷ 3.7
Solution:
In this case, the dividend is greater than the divisor,
18.9 can be rounded as 20
3.7 can be rounded as 4
So, 20 ÷ 4 = 5
Hence 18.9 ÷ 3.7 is about 5
Example 3:
Ray can walk 3.7 miles at a stretch. He takes a break when he covers 3.7 miles. If the distance he has to cover is 28.3 miles, provide the approximate number of breaks that he will take?
Solution:
The total distance Ray has to cover is 28.3 miles. Ray can walk 3.7 miles at a stretch. We need to find the result of the following
28.3 ÷ 3.7
We can approximate 3.7 to 4 and we will also approximate 28.3 to 28 as it is divisible by 4.
28 ÷ 4 = 7
So, Ray takes about seven breaks before he can cover the entire distance.
Precision helps us determine the exact quantity required in certain scenarios. Now, it’s not necessary that we need the precise number or result accurate to its decimal point. But, there are times when we just need a quick calculation. Hence, we use estimation from time to time.
Approximation might be counterproductive or even dangerous in areas that require extreme precision, such as in different experiments or labs or in the production of medicine, for example. Determining the precise values is very important in these situations.
Compatible numbers are numbers that are easier to work with, and which make calculations much simpler. For instance, to divide 23 by 4, a compatible number of 23 can be 20 as 20 is divisible by 4. Hence, division of 20 by 4 is 5 which is much easier than the division of 23 by 4. This is the reason why compatible numbers are used in estimations.