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A decimal is another way of expressing a fraction.
1. Tenths
The fraction 1 / 10, as a decimal, is written as 0.1 and read as zero point one. The following shaded parts can be read as:
The dot or the point between the two digits is called the decimal point.
Whole numbers and decimal numbers can also be combined as shown below: The picture given alongside shows, on a ruler, a length of 1 cm + 7 parts out of 1 cm. Since each cm is further divided into 10 equal parts (as can be seen), the above given number can be written as 1.7.
1.7 is also a decimal number and is read as one point seven.
2. Hundredths
Let us now divide a whole into 100 equal parts.
We read
 0.01 as zero point zero one,
 0.08 as zero point zero eight,
 0.75 as zero point seven five.
0.41 is 41 hundredths or 4 tenths 1 hundredth.
Similarly,
So, a decimal number has two parts—the whole number part and the decimal part separated by a decimal point. Number of digits after the decimal point is called the number of decimal places.
In the above number, there are 2 decimal places after the whole number part.
Edurev Tips: Whenever there is no whole number part, do not forget to write 0 before the decimal point, e.g., .65 is always written as 0.65.
3. Thousandths
If we divide one whole into thousand equal parts, then each part of the whole represents onethousandth.
Thus, 1 / 1000 = 0.001 or onethousandth read as zero point zero zero one.
The decimal 0.716 = 716 / 1000 (716 thousandths) represents 716 parts out of 1000 parts.
Sumit and Ritu are both reading the number 0.716 in different ways. Tick the correct one?In expanded form, we write 0.716 as:
Similarly, 49.107 is fortynine and one hundred seven thousandths and is read as fortynine point one zero seven.
In expanded form,
Notice the following pattern.
The number of digits after the decimal point in the decimal numeral is equal to the number of zeros after 1 in the denominator of the corresponding common fraction.
The place value chart shown below can also be used to understand decimals. The place value of each digit in the number 2222.222 is shown below.
The first place to the right of the decimal point is the tenths place, the second place to the right is the hundredths place and so on. The last place after the decimal point tells us how to name the decimal part.
Edurev Tips: Digit at each place has a place value onetenth (1 / 10) that of the place to its left.
Here it is 222 thousandths.
Using the place value chart, we can write the expanded form of any decimal number. Take the number 1279.364.
1279.364 = 1 thousand + 2 hundreds + 7 tens + 9 ones + 3 tenths + 6 hundredths + 4 thousandths
= 1 × 1000 + 2 × 100 + 7 × 10 + 9 × 1 + 3 × 1 / 10 + 6 × 1 / 100 + 4 × 10 / 100
Or
= 1000 + 200 + 70 + 9 + 3 / 10 + 6 / 100 + 4 / 1000 (Fractional form)
= 1000 + 200 + 70 + 9 + 0.3 + 0.06 + 0.004 (Decimal form)
Decimals that name the same amount are called equivalent decimals.
Observe the following.
As can be seen, these two pictures name the same amount, so 0.8 = 0.80.
8 tenths = 80 hundredths
8 / 10 = 80 / 100
Thus, 0.8 = 0.80 = 0.800 = 0.8000 = ______
Similarly, 1.8 = 1.80 = 1.800 ; 37.41 = 37.410 = 37.4100, etc.
From the above it is clear that writing zeros at the right hand end of a decimal number does not change its value.
1. Like Decimals and Unlike Decimals
Decimals having the same number of decimal places are called like decimals.
Decimals having different number of decimal places are called unlike decimals.
Thus, 1.2, 4.03, 0.895 are all unlike decimals.
2. Converting Unlike Decimals to Like Decimals
Unlike decimals can be converted to like decimals by finding their equivalent decimals.
Example 1: Convert 6.8, 7.83 and 12.040 to like decimals.
The greatest number of decimal places is 3, so we convert all of them to equivalent decimals with 3 decimal places.
6.8 → 6.800, 7.83 → 7.830, 12.040
Hence, 6.8, 7.83 and 12.040 when converted to like decimals becomes 6.800, 7.830, 12.040.
1. Comparing Decimals
To compare decimal numbers, we follow the following steps.
Step 1: Convert the decimals into like decimals.
Step 2: First, compare the whole number parts. The number with the greater whole number part is greater.
Step 3: If the whole number parts are the same, compare the tenths digits. The decimal number having greater tenths digit names a greater number.
Step 4: If the tenths digits are the same, compare the hundredths digits and so on.
Example 2: Which number is bigger: 3.612 or 3.621?
The numbers are
On comparing the digits from the left, you find that the hundredths digits differ. Since 2 hundredths > 1 hundredths, therefore, 3.621 > 3.612.
2. Ordering Decimals
Example 3: Arrange 21.012, 21.002, 24.102 in increasing order.
On comparing the whole number parts, you find that 24.102 is the greatest. Now, compare 210.12 and 21.002.
The whole number parts being the same, we start comparing from the tenths digit.
The digits at the hundredths differ. Since 1 hundredths > 0 hundredth, so 21.012 > 21.002.
Thus, the given numbers in increasing order are: 21.002, 21.012, 24.102.
1. Addition
To add two or more decimals, we follow these steps.
Step 1: Convert the decimal to like decimals.
Step 2: Line up the decimal points, i.e., the addends are placed one below the other such that the tens digit is below tens, ones is below ones, decimal points below the decimal point, tenths below the tenths and so on.
Step 3: Add, as in case of whole numbers. Carry wherever necessary.
Step 4: Place the decimal point in the sum directly below the decimal point in the addends.
Example 4: Add: 4.83, 312.9 and 23.031.
First, we convert the numbers to be added to like decimals.
We have: 4.83 → 4.830, 312.9 → 312.900, 23.031.
Now, line up the decimals and add.
Example 5: Add: 6.9, 3.405 and 9.46.
First, we convert the numbers to be added to like decimals.
Converting into like decimals 6.9 → 6.900, 3.405, 9.46 → 9.460.
Now, line up the decimals and add.
Edurev Tips: A whole number can be expressed in the decimal form as: 12 = 12.0 or 12.00 or 12.000 and so on.
2. Subtraction
To subtract one decimal number from the other, we follow the steps given below.
Step 1: Convert the decimal to like decimals.
Step 2: Line up the decimal points.
Step 3: Subtract in each place value.
Step 4: Regroup, wherever necessary.
Example 6: Subtract 6.253 from 16.67.
 Convert into like decimals as 16.67 → 16.670, 6.253 and line up the decimal points.
 Subtract the thousandths.
You cannot subtract 3 thousandths from 0 thousandths, so borrow 1 hundredths from 7 hundredths leaving behind 6 hundredths.
1 hundredths = 10 thousandths
0 thousandths → 10 thousandths
Now, subtract thousandths, 10 – 3 = 7 thousandths. Subtract the hundredths.
6 – 5 = 1 hundredth Subtract the tenths.
6 – 2 = 4 tenths Subtract the ones.
6 – 6 = 0 ones Subtract the tens.
1 – 0 = 1 ten
So, 16.67 – 6.253 = 10.417.
Example 7: Find 312.8 – 59.98.
1. Multiplication by 10, 100, 1000
Observe the following examples.
(i)
5 tenths × 10 = 5 ones
0.5 × 10 = 5
(ii)
2 ones 7 tenths × 10 = 2 tens 7 ones
2.7 × 10 = 27
We observe that:
Multiplying a decimal by 10 moves the decimal point 1 place to the right.
(iii)
1 ones 7 tenths 8 hundredths × 100
= 1 hundreds 7 tens 8 ones
(iv)
3 tenths × 100 = 3 tens
We observe that:
Multiplying a decimal by 100 moves the decimal point 2 places to the right.
(v)
2 thousandths × 1000 = 2 ones
Edurev Tips: A number without a decimal point is considered to have the decimal point to the extreme right. 317 means 317.0
(vi)
1 one 8 tenths = 1 thousand 8 hundreds
(vii)
We observe that:
Multiplying a decimal by 1000 moves the decimal point 3 places to the right.
2. Multiplication by a Counting Number
Example 8: If one litre of petrol costs ₹ 74.03, what will be the cost of 12 litres of petrol?
Cost of 12 L petrol = ₹ 74.03 × 12
Step 1: Multiply, as you would multiply counting numbers.
Step 2: Place the decimal point in the product at a place so that the number of decimal places in the product is equal to the total number of decimal places of the factors.
Example 9: Multiply 6.095 by 45.
3. Multiplication of a Decimal Number by a Decimal Number
Lets us find the product of 12.7 and 0.4.
The above work can also be set as:
Working
Multiply, as you would multiply the counting numbers.
The total number of decimal places in the product is equal to the sum of the decimal places of the factors.
Example 10: Multiply:
(a) 0.8 × 0.3
(b) 0.007 × 0.03
(c) 0.009 × 1.2
(a)
(b)
(c)
Example 11: Multiply 23.02 by 0.12.
Example 12: Multiply 1.035 by 2.95.
Example 13: Multiply:
(a) 2.6 by 30
(b) 13.46 by 600
(a) 2.6 × 30 = 2.6 × 3 × 10 = 7.8 × 10 = 78.
(b) 13.46 × 600 = 13.46 × 6 × 100 = 80.76 × 100 = 8076.
1. Division by 10, 100, 1000
Observe the following examples.
(i)
2 ones ÷ 10 = 2 tenths = 0.2
(ii)
6 tenths ÷ 10 = 6 hundredths = 0.06
We observe that:
Dividing a decimal by 10 moves the decimal point one place to the left.
(iii)
4 ones ÷ 100 = 4 hundredths = 0.04
(iv)
2 ones 7 tenths ÷ 100 = 2 hundredths 7 thousandths = 0.027
Dividing a decimal by 100 moves the decimal point 2 places to the left.
(v)
7 ones ÷ 1000 = 7 thousandths = 0.007
(vi)
(vii)
Dividing a decimal by 1000 moves the decimal point 3 places to the left.
2. Division by a Counting Number
Example 14: Find: 20.22 ÷ 6.
Compare the two workings shown below.
Step 1:
1. 20.22 / 6
Step 2:
2.
Step 3:
3. 2022 / 6 = 337
Step 4:
4. 337 ÷ 100 = 3.37
Actual Working
Method:
1. Divide, as you would divide counting numbers.
2. Place the decimal point in the quotient directly above the decimal point in the dividend.
Short formSo, 20.22 ÷ 6 = 3.37
Example 15: Divide: 2.67 by 3.
3 > 2, so we put a zero in the quotient.
Check
3 × 0.89 = 2.67
∴ 2.67 ÷ 3 = 0.89.
Example 16: Solve: 1414.332 ÷ 17.
∴ 1414.332 ÷ 17 = 83.196.
Example 17: Divide the following.
(a) 14.448 ÷ 14
(b) 262.85 ÷ 35
(a)
∴ 14.448 ÷ 14 = 1.032.
(b)
∴ 262.85 ÷ 35 = 7.51.
Check
35 × 7.51 = 262.85
If the last remainder is not zero, write as many zeros on the right of the dividend as are necessary to make the last remainder zero.
Example 18: Find: 9.2 ÷ 16.
∴ 9.2 ÷ 16 = 0.575.When the divisor is a multiple of 10 or 100 or 1000, etc., then the process of division can be shortened, as shown below.
Example 19: Find: 94.24 ÷ 80.
∴ 94.24 ÷ 80 = 1.178
Short form:
Edurev Tips: 94.24 ÷ 10 = 9.424
3. Division of a Decimal Number by a Decimal Number
You have learnt how to divide a decimal number by a whole number. To divide a decimal number by a decimal number, we make the divisor a whole number.
Look at the following example.
(i) Multiply the divisor by a multiple of 10 to make it a whole number.
Thus, 0.7 × 10 = 7.
(ii) Multiply the dividend by the same multiple of 10.
Thus, 4.41 × 10 = 44.1.
Now, divide as usual:
44.1 ÷ 7 = 44.1 / 7 = 6.3
The work can be set as:
Example 20: Divide : (a) 6.265 ÷ 0.07 (b) 1.8 ÷ 0.003
(a)
(b)
Example 21: Divide: 4.20 by 1.05.
4.20 ÷ 1.05 = 4.20 / 1.05
2 decimal places in divisor, s o shif t de cima l p oint 2 places to the right both in divisor and dividend.
∴ 4.20 ÷ 1.05 = 4.
Example 22: Find 20.169 ÷ 0.83.
Shift decimal point 2 places to the right.
∴ 20.169 ÷ 0.83 = 24.3.
Example 23: Divide 66.468 by 76.4.
∴ 66.468 ÷ 76.4 = 0.87.
Example 24: Divide:
(a) 90 ÷ 0.60
(b) 36 ÷ 0.06
(a)
(b)
Example 25: Change 4 / 5 to a decimal fraction.
To convert 4 / 5 to a decimal, we divide the numerator 4 by the denominator 5, as follows.
Put decimal point after the whole number and add 1 zero to ensure complete division.
∴ 4 / 5 = 0.8.
Example 26: Express 85 / 16 as a decimal.
To convert 85 / 16 to a decimal, divide the numerator 85 by the denominator 16.
Put decimal point after the whole number and add 4 zeros to ensure complete division.
∴ 85 / 16 = 5.3125.
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