Equivalent Decimals: The word decimal comes from the Latin word Decem which means ‘Ten’. Decimal numbers are the standard form of representing non-integer numbers. We represent rational as well as irrational numbers in the form of decimals. Furthermore, a decimal number can be expressed as a fraction.

Decimal numbers are those whose whole number part and the fractional part are separated by a decimal point. Two decimals are said to be equivalent when they have the same value. The equivalent decimals are considered as, unlike fractions. Read the complete article to know more about equivalent decimals with appropriate examples.

Definition of Decimal Numbers

Decimals are the extension of the number system and can also be considered fractions. Decimals are types of numbers with a whole number and a fraction part separated by a decimal point.

For example,

\(0.1\) is a decimal number with two numbers after the decimal point.
\(2.03\) is a decimal number with two numbers after the decimal point.
\(11.123\) is a decimal number with three numbers after the decimal point.

Types of Decimal Numbers

There are different types of decimal numbers. They are:

Like Decimals

Like decimals have the same number of decimal places.

For example, \({\text{0}}{\text{.003,3}}{\text{.034,12}}{\text{.120,}}\) and \(234.104\) are like decimals as they have the same number of decimal places.

Learn All the Concepts of Decimals

Unlike Decimals

Unlike decimals have different numbers of decimal places.

For example, \({\text{1}}{\text{.045,20}}{\text{.20,0}}{\text{.1,}}\) and \({\text{0}}{\text{.1234}}\) are unlike decimals as they have different decimal places.

Place Value of Decimal Numbers

The place value system helps to denote the position of a digit in a number that helps to define its value. When we write some numbers, the position of each digit is essential.

The power of \(10\) can be determined by using the place value chart given below:

The digits present to the left of the decimal point are multiplied with the positive powers of \(10\) in increasing order from right to left. The digits present to the right of the decimal point are multiplied with the negative powers of \(10\) in increasing order from left to right.

Conversion of Decimals to Fractions

The decimal place value determines the tenths, hundredths, and thousandths.

• A tenth means one-tenth or \(\frac{1}{{10}},\) in decimal form, it is \(0.1\)
• A hundredth means \(\frac{1}{{100}},\) in decimal form, it is \(0.01\)
• A thousandth means \(\frac{1}{{1000}},\) in decimal form, it is \(0.001\)

Here, we will discuss how a decimal number can be converted into a fraction.

If we want to convert a decimal number to a fraction, we go through the following steps:

1. Remove the decimal point from the number. The number obtained will be the numerator of the fraction.
2. The denominator will be \(1,\) followed by the count of numbers after the decimal point. Ex- If there are three numbers after a decimal point, then the numerator will be \(1000.\)
3. Simplify the fraction, if possible.
   For example, \(1.3 = \frac{{13}}{{10}},25.11 = \frac{{2511}}{{100}}\)

Equivalent Decimals

Two decimals are equivalent when they have the same value (or the same amount). For example, by showing \({\text{0}}{\text{.2,0}}{\text{.20, }}\) and \({\text{0}}{\text{.200}}\) cover the same space, we can see that these two decimals have the same value. Hence, these are equivalent decimals.

If we add zeros at the end of a decimal number, the value of the decimal number will not change. Let us discuss it in detail.

Rule to Identify Equivalent Decimals

1. Checking the place value of the digits of the decimal numbers
2. Converting the decimal numbers into the fractions

Checking the Place Value of the Digits of the Decimal Numbers

We have discussed earlier that identifying the place value of the digits of any decimal number is very important. We can use this concept to identify the equivalent decimals.

Let us take some equivalent decimals examples to understand the concept better.

Case I: \(0.5,0.50\)

The decimals are said to be equivalent if their values are the same.

Both the numbers have the same digit \(\left( 0 \right)\) in the ones place. So, we will move to the right and check the digits of tenth places in both numbers. The digits in the tenth place \(\left( 5 \right)\) are also the same. If we move to the hundredth place, we observe in the first number, \(0.5,\), there are no more digits, but in \(0.50,\), there is a digit \(0.\)

Hence, \(0.5,0.50\) are equivalent fractions. In \(0.50,\) if there is any digit other than zero after the tenth place, then it can not be equivalent to \(0.5.\)

Note: If we add multiple numbers of zeros after the tenth place of \(0.5,0.50\) we can get multiple numbers of equivalent decimals such as \(0.500,0.5000,0.50000,\) and so on.

Case II: \(0.3,0.03\)

In these two decimals, we can see that the tenth place digits of both the numbers are not the same. Therefore, they are not equivalent.

Converting Decimal Numbers into Fractions

We have talked about how we can convert a decimal number into a fraction. After converting the decimal numbers, the decimal numbers are equivalent if we get the same fractions.

Case I: \(0.5,0.50\)
Converting \(0.5\) we have, \(\frac{5}{{10}}.\)
Now, converting \(0.50\) we have, \(\frac{{50}}{{100}} = \frac{5}{{10}}.\)
Therefore, \(0.5,0.50\) are the equivalent decimals representing the same fractions.

Case II: \(0.3,0.03\)
Converting \(0.3\) we have, \(\frac{3}{{10}}.\)
Now, converting \(0.03\) we have, \(\frac{3}{{100}}.\)
Therefore, \(0.3,0.03\) are not equivalent decimals as they are not representing the same fractions.

Solved Examples on Equivalent Decimals

Let us understand equivalent decimals facts using some examples.

Q.1. Write three equivalent decimal numbers of \(1.4.\)
Ans: We know that the equivalent decimals have the same value. We can form equivalent decimals of the given by adding zeros at the end of decimal numbers.
Therefore, the three equivalent decimals of \(1.4\) are \(1.40,1.400,1.4000.\)

Q.2. Check whether \(0.20\) and \(0.200\) are the equivalent decimals or not.
Ans: Converting \(0.20\) we have, \(\frac{{20}}{{100}}.\) Now, converting \(0.200\) we have, \(\frac{{200}}{{1000}} = \frac{{20}}{{100}}.\)
Therefore, \(0.20,0.200\) are the equivalent decimals as they represent the same fractions.

Q.3. Check whether \(0.40\) and \(0.04\) are the equivalent decimals or not.
Ans: Let us convert \(0.40\) into a fraction and we have, \(\frac{40}{{100}}.\) Now, converting \(0.04\) we have, \(\frac{4}{{100}}.\)
Therefore, \(0.40,0.04\) are not equivalent decimals as they are not representing the same fractions.

Q.4. Check whether we get equivalent decimals or not if we convert the fractions \(\frac{5}{4}\) and \(\frac{50}{40}\) in decimals.
Ans: Given, \(\frac{5}{4},\frac{{50}}{{40}}\)
Converting \(\frac{5}{4}\) in decimal we have, \(\frac{{5 \times 25}}{{4 \times 25}} = \frac{{125}}{{100}} = 1.25.\)
Converting \(\frac{50}{40}\) in decimal we have, \(\frac{{50 \times 25}}{{40 \times 25}} = \frac{{1250}}{{1000}} = 1.25.\)
Therefore, we get equivalent decimal numbers from the given fractions as we are getting the same decimal value from the fractions.

Q.5. Write two equivalent decimal numbers of \(2.15.\)
Ans: We know that the equivalent decimals have the same value.
We can form equivalent decimals of the given by adding zeros.
Therefore, the three equivalent decimals of the given are \(2.150,2.1500.\)

Summary

This article covered the definition of decimals, types of decimals, and equivalent decimals. We have learned that equivalent decimals have the same value and the methods to identify the equivalent decimals. The equivalent decimals notes will further help in solving the questions quickly.

FAQs on Equivalent Decimals

Check frequently asked questions related to equivalent decimals below:

Q.1. How do you find equivalent decimals?
Ans: We can find equivalent decimals by adding zero/zeros at the end or the right side of a decimal number.

Q.2. Are the equivalent fractions and the equivalent decimals the same?
Ans: No, the equivalent fractions and the equivalent decimals are not the same.
To find a fraction equivalent to a given fraction, we multiply/divide the numerator and the denominator of the given by the common, same non-zero number.
To find the equivalent decimals, we need to add zero/zeros at the end of the decimal numbers.

Q.3. Explain equivalent decimals with an example.
Ans: Two decimals are said to be equivalent when they have the same value (or the same amount). For example, by showing \(0.4\) and \(0.40\) cover the same space, we can see that these two decimals have the same value. Hence, these can be said as equivalent decimals.

Q.4. How do we verify a decimal number is equivalent to another decimal number?
Ans: Generally, we can verify a decimal number is equivalent to another by using two methods. These are,
a. Checking the place value of the digits of the decimal numbers.
b. Converting the decimal numbers into fractions.

Q.5. What is an equivalent decimal fraction?
Ans: Equivalent decimal fractions are equal in value, and they are unlike fractions. The decimals \(0.5,0.50\) and \(0.500\) are equivalent decimals.

Q.6. Find the equivalent decimals of \(\frac{7}{8}\)?
Ans: The given fraction is \(\frac{7}{8}.\)
To find the equivalent decimals, we need to convert the number into decimal form. Decimal form of \(\frac{7}{8}\) is \(0.875.\)
If we add zero/zeros after the decimal number, we will get its equivalent decimals.
Hence, \(0.8750,0.87500,0.875000,\) etc are the equivalent decimals of \(\frac{7}{8}.\)

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