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Tenths as Decimals

Introduction

The word decimal comes from the Latin word “Decem” meaning 10. . In algebra, a decimal number can be defined as a number whose complete part and the fractional part are separated by a decimal point. Before we learn what we mean by a tenth of decimal it is important to recall the place value system of decimals that defines the position of a tenth in a decimal number.

Place Value System of Decimals

We know that each place in the place value table has a value ten times the value of the next place on its right. In other words, the value of a place is one-tenth of the value of the next place on its left. We observe that if one digit moves one place left to right its value becomes one-tenth ($\frac{1}{10}$ ) of its previous value and when it moves two places left to right its value comes one-hundredth ( $\frac{1}{100}$ ) of its previous term and so on. Therefore, if we wish to move beyond ones place which is the case of decimals, we will have to extend the place value table by introducing the places of tenths ($\frac{1}{10}$ ), hundredths ($\frac{1}{100}$ ), thousandths ( $\frac{1}{1000}$ ) and so on.

Therefore, the place value table in case of a decimal number will be of the form –

For example, the decimal number 257.32 in the place value system will be written as –

A decimal or a decimal number may contain a whole number part and a decimal part. The following table shows the whole number part and the decimal part of some decimals –

Now, how do we read the decimals using the place value system? Let us find out.

Reading the Decimal Numbers using the Place Value System

In order to read decimals, the following steps are used –

1. Read the whole number part
2. Read the decimal point as point
3. Read the number to the right of the decimal point. For example, 14.35 will be read as Fourteen point three five. Alternatively, the number to the right of the decimal point can also be read by reading the number to the right of the decimal point and naming the place value of the last digit. For instance, the number 8.527 can also be read as eight and five hundred twenty seven thousandths.

What are Tenths in a Decimal?

Consider the following figure. It is divided into ten equal parts and one part is shaded. The shaded part represents one-tenth of the whole figure. It is written as $\frac{1}{10}$. $\frac{1}{10}$ is also written as 0.1 which is read as “ point one “ or “ decimal one “.

Thus the fraction $\frac{1}{10}$ is called one-tenth and is written as 0.1.

Also, 1 ones = 10 tenths.

Consider another figure. The below figure is divided into ten equal parts and three parts are shaded. The shaded parts represent three-tenths of the whole figure. It is written as $\frac{3}{10}$. $\frac{3}{10}$ is also written as 0.3 which is read as “ point three “ or “ decimal three “.

Thus the fraction $\frac{3}{10}$ is called three tenth and is written as 0.3.

Also, consider the below figure. The below figure is divided into ten equal parts and six parts are shaded. The shaded parts represent six-tenths of the whole figure. It is written as $\frac{6}{10}$. $\frac{6}{10}$ is also written as 0.6 which is read as “ point six “ or “ decimal six “.

Thus the fraction $\frac{6}{10}$ is called six-tenth and is written as 0.6.

Similarly, $\frac{2}{10}$ , $\frac{4}{10}$ , $\frac{5}{10}$ , $\frac{7}{10}$ , $\frac{8}{10}$ and $\frac{9}{10}$ are called 2-tenths, 4-tenths, 7-tenths, 8-tenths and 9-tenths respectively and are denoted by 0.2 , 0.3 , 0.4 , 0.5 , 0.7 , 0.8 and 0.9 respectively.

Thus we have,

$\frac{1}{10}$ = 0.1 and is called one-tenths or 1 tenths

$\frac{2}{10}$ = 0.2 and is called two-tenths or 2 tenths

$\frac{3}{10}$ = 0.3 and is called three-tenths or 3 tenths

$\frac{4}{10}$ = 0.4 and is called four-tenths or 4 tenths

$\frac{5}{10}$ = 0.5 and is called five-tenths or 5 tenths

$\frac{6}{10}$ = 0.6 and is called six-tenths or 6 tenths

$\frac{7}{10}$ = 0.7 and is called seven-tenths or 7 tenths

$\frac{8}{10}$ = 0.8 and is called eight-tenths or 8 tenths

$\frac{9}{10}$ = 0.9 and is called nine-tenths or 9 tenths

$\frac{10}{10}$ = 1 and is called ten-tenths or 10 tenths

Also, $\frac{11}{10}$ = 11 tenths = 10 tenths + 1 tenths = 1 + $\frac{1}{10}$ = 1 + 0.1 = 1.1

$\frac{12}{10}$ = 12 tenths = 10 tenths + 2 tenths = 1 + $\frac{2}{10}$ = 1 + 0.2 = 1.2

$\frac{13}{10}$ = 13 tenths = 10 tenths + 3 tenths = 1 + $\frac{3}{10}$ = 1 + 0.3 = 1.3

Similarly, we have

$\frac{20}{10}$ = 20 tenths = 10 tenths + 10 tenths = 1 + 1 = 2

$\frac{21}{10}$ = 21 tenths = 20 tenths + 1 tenths = 2 + $\frac{1}{10}$ = 2 + 0.1 = 2.1

$\frac{22}{10}$ = 22 tenths = 20 tenths + 2 tenths = 2 + $\frac{2}{10}$ = 2 + 0.2 = 2.2

Thus a fraction of the form $\frac{Number}{10}$ is written as decimal obtained by putting decimal point by leaving one right-most digit.

For example, $\frac{325}{10}$ = 32.5 while $\frac{5894}{10}$ =  589.4

Let us understand it through an example.

Example Write each of the following as decimals

1. Five ones and four tenths
2. Twenty and one tenths

Solution We have been given the following and we need to write them as decimals. Let us do them one by one

1. Five ones and four tenths

Note that the whole value of the given decimal is 5 and the decimal part is four tenths. Therefore, we will proceed in the same manner as we defined different tenths above.

We will get,

Five ones and four tenths = 5 ones + 4 tenths = 5 + $\frac{4}{10}$ = 5 + 0.4 = 5.4

Hence, Five ones and four tenths In decimal form will be 5.4.

1. Twenty and one tenths

Note that the whole value of the given decimal is 20 and the decimal part is one-tenths. Therefore, we will proceed in the same manner as we defined different tenths above.

We will get,

Twenty ones and one tenths = 20 ones + 1 tenths = 20 + $\frac{1}{10}$ = 20 + 0.1 = 20.1

Hence, Twenty ones and one tenths in decimal form will be 20.1.

Let us take another example.

Example Write each of the following as decimals

1. 20 + 7 + + $\frac{3}{10}$
2. 500 + 3 + $\frac{7}{10}$

Solution We have been given an expanded form of two numbers and we are required to find the corresponding decimal number. Let us do them one by one.

1. 20 + 7 + + $\frac{3}{10}$

We can see that there are two whole numbers and one fractional number.

Note that the whole values of the given decimal are 20 and 7 and the decimal part is three tenths. Therefore, we will proceed in the same manner as we defined different tenths above.

We will get,

20 + 7 +  $\frac{3}{10}$ = 20 + 7 + 0.3 = 27.3

1. 500 + 3 + $\frac{7}{10}$

We can see that there are two whole numbers and one fractional number.

Note that the whole values of the given decimal are 500 and 3 and the decimal part is seven tenths. Therefore, we will proceed in the same manner as we defined different tenths above.

We will get,

500 + 3 + $\frac{7}{10}$ = 500 + 3 + 0.7 = 503.7

Let us now see how to plot the tenths of a decimal on a number line.

Representation of tenths of a decimal on a Number Line

Before we learn how to represent a tenth on a number let us recall what we understand by the term number line.

What is a number line?

A number line is a straight horizontal line with numbers placed at even intervals that provides a visual representation of numbers. Primary operations such as addition, subtraction, multiplication, and division can all be performed on a number line. The numbers increase as we move towards the right side of a number line while they decrease as we move left.

Representation on a Number Line

Above is a visual representation of a standard number line. As is clearly visible, as we move from left to right, there is an increase in the value of numbers while it decreases when we move from right to left.

We already know how to represent fractions on a number line. Let us now represent tenths of a decimal on a number line. We can understand this by an example.

Let us represent 0.4 on a number line. We can clearly see that there are 4 tenths in 0.4. Therefore in order to represent 0.4 on a number line we will divide the unit length between 0 and 1 into 10 equal parts and take 4 parts as shown below –

Now, we can know that 0.4 in fraction form is equal to 4/10. Hence we will mark 4/10 as 0.4 which is our desired mark on the number line.

The steps that we used above to represent a tenth on a number line can be summarised as  –

1. We draw a number line between 0 and 1.
2. We then raw 10 lines dividing the total distance between 0 and 1 into 10 equal parts.
3. Now, one whole divided into 10 parts is equal to  $\frac{1}{10}$.
4. $\frac{1}{10}$ in decimal form is equal to 0.1.
5. At each new line we are adding $\frac{1}{10}$ or 0.1.
6. So, between 0 and 1 we have, 0 . 1 , 0 . 2 , . 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 7 , 0 . 8 and 0 . 9. Similarly, between 1 and 2 we have, 1 . 1 , 1 . 2 , 1 . 3 , 1 . 4 , 1 . 5 , 1 . 6 , 1 . 7 , 1 . 8 and 1 . 9.
7. We can also say that the line representing $\frac{1}{2}$ or 0.5 is the half way mark between 0 and 1. Similarly, the line representing $1\frac{5}{10}$ or 1.5 is the half way mark between 1 and 2.
8. Ten tenths is equal to one whole.

Now let us go through some solved examples on tenth of a decimal.

Solved Examples

Example 1 Label the missing decimal numbers on the number line.

Solution We have been given four numbers marked as A , B , C and D on a number line and we need to find out which decimal numbers they represent. Let us mark them one by one.

We will start with completing the marking of the lines that have not been marked on the given number line. It can be clearly seen that there are 10 lines between two whole numbers on the number line. This means that the lines represent one tenth of the number in the decimal form. Therefore, the lines between 7 and 8 will be marked as  7 . 1 , 7 . 2 , 7 . 3 , 7 . 4 , 7  5 , 7 . 6 , 7 . 7 , 7 . 8 and d 7. 9. Similarly, We between the whole numbers 8 and 9 we have,  8 . 1 , 8 . 2 , 8 . 3 , 8 . 4 , 8 . 5 , 8 . 6 , 8 . 7 , 8 . 8  and 8 . 9. The number line so obtained will be –

Now, we shall check the position of the four points on this number line.

We can see that from the number line above, the point A lies on the decimal number 8.7. Hence A = 8 . 7

Now, let us check the position of point B.

We can see that from the number line above, the point B lies on the decimal number 8.2. Hence B = 8 . 2

Now, let us check the position of point C.

We can see that from the number line above, the point C lies on the decimal number 7 . 1. Hence C = 7 . 1

Now, let us check the position of point D.

We can see that from the number line above, the point D lies on the decimal number 7 . 8 . Hence D =  7 . 8

Therefore, we have,

A = 8 . 7

B = 8 . 2

C = 7 . 1

D =  7 . 8

Example 2 Between what two numbers is does the decimal number 5.4 lie on the number line?

Solution We have been given the decimal number 5.4 and we need to check between which two whole numbers will it lie.

On observing the number 5.4 we can see that the number represents a tenth of a decimal as it has one digit after the decimal point.

Also, we know that 5.4 = 5 + $\frac{4}{10}$

This means that 5.4 is equal to 5 whole parts plus 4 tenths. Et us plot it on the number line. we will have,

We can clealry see that 5.4 will lie between 5 and 6. The point on the number line will be –

Hence, we can say that the number 5.4 will be between the whole numbers 5 and 6.

Example 3 Write the following fraction in decimal form

18  $\frac{5}{10}$

Solution We have been given the fraction 18  510 and we need to write it in decimal form. we can see that the given fraction has one whole number 18 and five tenth. We also know that 510 = 0.5 and is called five-tenths or 5 tenths. Therefore, we have,

18  $\frac{5}{10}$ = 18 + $\frac{5}{10}$ = 18 + 0.5 = 18.5

Hence, 18  $\frac{5}{10}$ = 18.5

Key Facts and Summary

1. A decimal number can be defined as a number whose complete part and the fractional part are separated by a decimal point.
2. The Place Value System is the system in which the position of a digit in a number determines its value. The place value of a digit in a number is the value it holds to be at the place in the number.
3. In order to read decimals, we first
4. Read the whole number part
5. Read the decimal point as point
6. Read the number to the right of the decimal point. For example, 14.35 will be read as Fourteen point three five.
7. The fraction $\frac{1}{10}$ is called one-tenth and is written as 0.1.
8. 1 ones = 10 tenths
9. A number line is a straight horizontal line with numbers placed at even intervals that provides a visual representation of numbers. Primary operations such as addition, subtraction, multiplication, and division can all be performed on a number line.