Decimal numbers are those numbers that contain a whole number part and a fractional part separated by a decimal point. The number of digits after the decimal point in a decimal number is the number of decimal places. Approximation of Decimals is a mathematical technique for measuring or locating estimated values and limiting the number of decimal places. 

Approximation or rounding decimals are used in various physical applications, including calculating product costs, distances between lines, object lengths, and weights. We round off decimals many times in our everyday lives. Let us understand the approximation of decimals in detail. Continue reading to know more.

Decimals

The numbers expressed in decimal forms are known as decimals. We can define decimal numbers as a number with a decimal point followed by digits showing the fractional part. The number of digits in the decimal part, i.e., after the decimal point, determines the number of decimal places.
For example, \(4.1, 2.453, 11.12,\) etc., are the decimal numbers.

The decimals can also be considered fractions when the denominators are \(10, 100, 1000,\) etc. 
A decimal number and the place values of it is shown below.

The place value of decimal numbers before the decimal point is the same as any whole number. But, for the digits after the decimal point, the place value is considered tenths, hundredths, thousandths, etc.

Methods for the Approximation of the Decimals

Approximation or the rounding off is an arithmetic method for locating an estimate of a specific number. Instead of delivering a long series of decimal places, decimal numbers are rounded off to a designated decimal place to make them easier to grasp and handle because decimal numbers with fewer decimal places are easy to process. 

Following are the rules for approximating the decimals:

Approximation of the Decimal Number to a Whole Number

When we approximate a decimal to a whole number, the tenth digit is checked to see greater than or less than \(5.\) Approximation of a decimal number where the tenth digit is greater than or equal to \(5,\) we add one to the one’s place digit or the first digit to the left of the decimal point. And if the tenth digit is less than \(5,\) we keep the one’s place unchanged and remove all the digits right to the one’s place.For example, \(24.77\) can be written as \(25\) (approximately), as \(25\) is the nearest whole number of \(24.77.\)

Learn in detail about Decimal Numbers here

For example, \(35.24\) can be written as \(35\) (approximately), as \(35\) is the nearest whole number of \(35.24.\)

Approximation of a Decimal Number to Tenths

Approximating or rounding off a number to the nearest tenths is equivalent to rounding it to one decimal place. In this case, the digit in the \({100^{{\rm{th}}}}\) position is examined. When the digit in the hundredth place is greater than or equal to \(5,\) the digit in the tenths place is increased by one unit. After getting the new tenth digit, the rest of the digits to the right will be removed. When the digit in the hundredth place is less than \(5,\) the digit in the tenth place remains unchanged, and we will remove the digits to the right of the tenth digit.
For example, in \(4.569, 5\) is the tenth place digit of it. The hundredths position is \(6,\) greater than \(5,\) so the tenth place digit increases by \(1\), and it becomes \(6.\)
Hence, the approximation of \(4.569\) to the closest tenths is \(4.6.\)

For example, in \(6.248, 2\) is the tenth place digit of it. The hundredths position is \(4,\) which is less than \(5,\) so the tenth place digit remains the same, and we will remove the rest of the digits right to the tenth place. Hence, the approximation of \(6.248\) to the tenths is \(6.2.\)

Approximation of a Decimal Number to the Hundredths

Approximation or rounding off to the closest or nearest hundredths is equivalent to two decimal places. Let us look at the thousandths place digit to round off a number to two decimal places. If the thousandths digit is greater than or equal to \(5,\) the hundredths digit is increased by one unit. Furthermore, if the digit in the thousandths place is less than \(5,\) the digit in the hundredths place remains unchanged, and we will remove the rest of the digits right after the hundredths place. 

For example, in \(4.616, 1\) in the hundredths place digit of it. The thousandths position is \(6,\) which is greater than \(5\), so the digit of the hundredths place is increased by \(1\), and it becomes \(2.\)
Hence, the rounding off or the approximation of \(4.616\) to the closest tenths is \(4.62.\)

For example, in \(3.432, 3\) is the hundredths place digit of it. The thousandths position is \(2,\) which is less than \(5\), so the hundredths place digit remains the same, and we will remove the rest of the digits right to the hundredths place. Hence, the approximation of \(3.432\) to the hundredths place is \(3.43.\)

Solved Examples: Approximation of Decimals

Q.1. What is 129.257 rounded to the nearest hundredth?
Ans: In \(129.257, 5\) is the hundredths place digit. The thousandths position is \(7,\) which is greater than \(5,\) so the digit of the hundredths place is increased by \(1,\) and it becomes \(6.\)
Hence, the rounding off of \(129.257\) to the nearest tenths is \(129.26.\)

Q.2. What is 12.21 rounded to the nearest tenths?
Ans: in \(12.21,2\) is the tenth place digit. The hundredths position is \(1,\) which is less than \(5,\) so the tenth place digit will remain the same. Hence, the approximation of \(12.21\) to its tenth is \(12.2.\)

Q.3. What is 42.38 rounded to the nearest tenths?
Ans: in \(42.38, 3\) is the tenth place digit. The hundredths position is \(8,\) which is greater than \(5,\) so the tenth place digit is increased by \(1,\) and it becomes \(4.\)
Hence, the rounding off of \(42.38\) to the closest tenths is \(42.4.\)

Q.4. What is the nearest whole number of 81.28?
Ans: When we do rounding off or approximation of decimals to the nearest whole number, the tenth digit is checked to see if it is greater than or less than \(5.\) So, we will keep the one’s digit the same and remove the other digits right to the one’s place. Hence, the nearest whole number of \(81.28\) is \(81\) as the tenth digit is \(2,\) less than \(5.\)

Q.5. Find the nearest whole number of 69.7?
Ans: Rounding up a number where the tenth digit is greater than \(5\) equals adding one to the first digit to the left of the decimal point or one’s digit.
Hence, the nearest whole number of \(69.7\) is \(70\) as the tenth digit \(7\) is greater than \(5.\)

Summary

In this article, we have covered the approximation or rounding off of decimals and different methods of approximation of decimals such as approximation of decimal numbers to a whole number, tenths place, hundredths place, etc. In the end, we solved some examples of the approximation of decimals.

Frequently Asked Questions

Q.1. How do you find approximation in math?
Ans: Approximation or the rounding off is an arithmetic method for locating an estimate of a specific number. Instead of delivering a long series of decimal places, decimal numbers are rounded off to a designated decimal place to make them easier to grasp and handle because decimal numbers with fewer decimal places are easy to process.

Q.2. How do you calculate the approximation nearest to the tenth position of a decimal number?
Ans: To approximate a decimal number to the nearest tenths, check the digit in the hundredth position. When the hundredths place digit is greater than or equal to \(5,\) the digit in the tenths place is increased by one unit, and the rest of the digits to the right will be removed. When the digit in the hundredth place is less than \(5,\) the digit in the tenths place remains unchanged, and the digits to the right of the tenths digit are omitted.

Q.3. How do you calculate the approximation nearest to the hundredths position of a decimal number?
Ans:  Approximation to the nearest hundredths is equivalent to rounding off to two decimal places. If the thousandths digit is greater than or equal to \(5,\) the hundredths digit is increased by one unit. If the digit in the thousandths place is less than \(5,\) the digit in the hundredths place remains unchanged.

Q.4. How do you approximate to 4 decimal places?
Ans: To approximate to \(4\) decimal places, we should check whether the fifth decimal place is greater than or less than or equal to \(5.\) Look at the digit in the hundred-thousandth place. If the hundred-thousandths digit is greater than or equal to \(5,\) the ten-hundredths digit is increased by one unit. If the digit in the hundred-thousandth place is less than \(5,\) the digit in the ten-thousandth place remains unchanged.

Q.5. Is 0.5 going to be rounded up or down?
Ans: There are no digits after tenths place of \(0.5.\) So, it will be rounded down only, or we can say the tenth place remains unchanged. 

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