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You Know � fractions � units used in measurement

You will Learn � reading and writing decimals � expanded form of decimals � to convert fractions to decimals � to convert decimals to fractions � use of decimals in money and metric measure � like and unlike decimals � comparison of decimals � equivalent decimals � addition and subtraction of decimals � multiplication of decimals � division of decimals

Kick Start

2 shaded part out of ten equal parts can be expressed in the form of fraction as 2

10.31 shaded parts out of hundred equal parts can be expressed as 31

100 .

The fractions with denominators 10, 100 and 1000 and so on are called decimal fractions or simply decimals.210 can be written as 0.2. It is read as point two or zero point two.

31100 can be written as 0.31. It is read as point three one or zero point three one.

Similarly, 5491000 can be written as 0.549. It is read as zero point five four nine

7. Decimals

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Reading and writing decimalsDecimal fraction with 10 as the denominator is known as tenths, with 100 as hundredths and with 1000 as thousandths.

Tenths

The shaded part is 110 (one

tenths).110 = 0.1 (in decimal form)

It is read as zero point one.

The shaded part is 410 (four

tenths).410 = 0.4 (in decimal form)

It is read as zero point four.

The shaded part is 910 (nine

tenths).9

10 = 0.9 (in decimal form)

It is read as zero point nine.

'Since there are one zero in the denominator, there are one digit after the decimal point.'

Hundredths

The shaded part is 1100

(one hundredths).1

100 = 0.01 (in decimal form)

It is read as zero point zero one.

The shaded part is 28100

(twenty eight hundredths).28100 = 0.28 (in decimal form)

It is read as zero point two eight.

The shaded part is 47100

(forty seven hundredths).47

100 = 0.47 (in decimal form)

It is read as zero point four seven.

'Since there are two zeros in the denominator, there are two digits after the decimal point.'

Thousandths

The shaded part is 11000

(one thousandths).The shaded part is 6

1000 (six thousandths).

The shaded part is 161000

(sixteen thousandths).

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11000 = 0.001 in decimal

form.

It is read as zero point zero zero one.

61000 = 0.006 in decimal

form.

It is read as zero point zero zero six.

161000 = 0.016 in decimal

form.

It is read as zero point zero one six.

'Since there are three zeros in the denominator, there are three digits after the decimal point.'

Whole numbers and decimal partsA decimal number consists of two parts:

a. Whole part b. Decimal partDecimal point (dot) is used to separate these two parts. The places occupied by the digits after the decimal point are called as decimal places.

Decimal point

Whole number part Decimal part

1 . 3 7

7 out of 100 7 hundredths

3 out of 10 3 tenths

1 Whole

Let’s see some more examples:

Coloured parts Decimal form Read as

2.35 Two point three five or two and thirty-five hundredths

1.4 One point four or One and four tenths

1.53 One point five three or One and fifty-three hundreadths

2.007Two point zero zero seven or Two and seven thousandths

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Let’s ExerciseExercise 7.1

1. Identify the decimal fractions from the following fractions.

a. 45 b. 2

10 c. 71100 d. 10

1000 e. 1017 f. 1

8 g. 670 h. 111

1000 i. 27100

2. Write the following as decimals.

a. One point seven two b. Three point one fourc. One point three eight zero d. Twenty one point zero one twoe. Six point three one f. Three point zero zero three

3. Write the following decimals in words.

a. 3.102 b. 11.34 c. 2.532 d. 2.413

e. 10.11 f. 2.5 g. 7.31 h. 1.324

Place value and Expanded form of Decimal numbersPlace value chart for the decimal number up to three decimal places.

Hundreds Tens Ones Decimal point Tenths Hundredths Thousandths

100 10 1 . 110

1100

11000

Let’s write the decimal 132.154 using the place value chart.

Hundreds Tens Ones Decimal point Tenths Hundredths Thousandths 1 3 2 . 1 5 4

In the expanded form, 132.154 is written as,

132.154 = 1 hundreds + 3 tens + 2 ones + 1 tenths + 5 hundredths + 4 thousandths

132.154 = 100 + 30 + 2 + 110 + 5

100 + 41000 (fractional expansion)

132.154 = 100 + 30 + 2 + 0.1 + 0.05 + 0.004 (decimal expansion)

Decimal Fractional expansion Decimal expansion

2.234 2 + 210 + 3

100 + 41000 2 + 0.2 + 0.03 + 0.004

1.045 1 + 4100 + 5

1000 1 + 0.04 + 0.005

32.789 30 + 2 + 710 + 8

100 + 91000 30 + 2 + 0.7 + 0.08 + 0.009

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Exercise 7.21. Write the fractional and decimal expansion for the following decimals.

a. 34.5 b. 32.01 c. 11.567 d. 10.45

e. 112.345 f. 19.01 g. 3.342 h. 90.213

2. Write the decimal number for the following expanded form.

a. 60 + 710 b. 40 + 5 + 8

10 + 1100 c. 200 + 4 + 3

100

d. 1 + 2100 + 6

1000 e. 7 + 210 + 1

1000 f. 4 + 510 + 2

100

Converting Fractions into DecimalsTo convert a fraction into a decimal number.

� Count the number of zeros in the denominator.

� Write the numerator and put the decimal point by moving towards left as many places as the number of zeros.

Example 1 Convert 14310 into decimal.

Solution:

Number of zeros in the denominator is one.

Thus, 14310 = 14.3

Example 2 Convert 37100 into decimal.

Solution:

Number of zeros in the denominator is two.

Thus, 37100 = 0.37

Example 3 Convert 71000 into decimal.

Solution:

Number of zeros in the denominator is three.

Thus, 71000 = 0.007

Example 4 Convert 325100 into decimal.

Solution:

Number of zeros in the denominator is two.

Thus, 325100 = 3.25

Converting decimals into fractionsTo convert a decimal number to a fraction.

� Count the number of digits after decimal points.

� Write the number as numerator omitting the decimal point.

� Write in the denominator, as many zero or zeros to the right of 1 as there are number of digit or digits after the decimal point.

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Example 5 Write 1.7 as a fraction.

Solution:

Number of digits after the decimal point is one.

Therefore, denominator = 10

Thus, 1.7 = 1710

Example 6 Write 2.05 as a fraction.

Solution:

Number of digits after the decimal point is two.

Therefore, denominator = 100

Thus, 2.05 = 205100

Example 7 Write 1.12 as a fraction.

Solution:

Number of digits after the decimal point is two.

Therefore, denominator = 100

Thus, 1.12 = 112100

Example 8 Write 7.325 as a fraction.

Solution:

Number of digits after the decimal point is three.

Therefore, denominator = 1000

Thus, 7.325 = 73251000

Use of decimals in money and metric measures

Example 9 Express 565 paise in rupees and paise.

Solution:

100 paise = 1 rupee (`)

565 paise = 565100 = ` 5.65

= ` 5 and 65 paise.

Example 10 Express 5 cm in m.

Solution:

100 cm = 1 m

5 cm = 5100 = 0.05 m

Example 11 Express 108 m in km.

Solution:

1000 m = 1 km

108 m = 1081000 = 0.108 km

Example 12 Express 1045 ml in l.

Solution:

1000 ml = 1 l

1045 ml = 10451000 = 1.045 l

Exercise 7.31. Write the following fractions as decimals.

a. 410 b. 11

1000 c. 23910 d. 208

100 e. 1781000 f. 13

102. Write the following decimals as fractions.

a. 1.9 b. 6.20 c. 3.56 d. 0.609 e. 3.7 f. 0.007

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3. Write the following in the decimal form.a. 45 rupees 30 paise b. 3 l and 300 mlc. 4 m and 24 cm d. 9 kg and 235 g

Like and Unlike DecimalsLike DecimalsDecimals with the same number of decimal places are called like decimals.

0.2, 0.5, 0.7 are like decimals as each one has a single digit after the decimal point.

Unlike DecimalsDecimals with different decimal places are called unlike decimals.

0.2, 0.05, 0.007, etc. are unlike decimals as the number of digits after the decimal point are not the same.

Converting Unlike Decimal to Like DecimalTo convert unlike decimals to like decimals, first identify the decimal with the largest number of decimal places and then add as many zeroes needed to get equal number of decimal places in each decimal.

Example 1 Convert the decimals 0.2, 0.05, 0.007, 1.234 into like decimals.

Solution:0.007 and 1.234 has the largest number of decimal places, that is, 3.

In order to make like decimals each decimal number should have three decimal places.

So, the set of like decimals will be 0.200, 0.050, 0.007, 1.234.

Comparison of DecimalsDecimals can be compared on the basis of their place value.

We first compare the whole numbers and then the decimal places in the sequence tenths, hundredths, thousandths and so on.

Example 2 Compare 2.48 and 7.34

Solution:First compare the integral parts (whole number).

The decimal with the greater integral part is greater.

Here, 7 > 2

Therefore, 7.34 > 2.48

Example 3 Compare 3.564 and 3.519

Solution:The integral parts (whole number) of the decimals are the same, so compare the digit in the tenths place.

The digits in the tenths place are 5 in both the decimals.

So, compare the digits in the hundredths place.

Here, 6 > 1

Therefore, 3.564 > 3.519

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Equivalent DecimalsThe decimals having the same value are called as equivalent decimals.

Let’s took a look at the model given below to understand equivalent decimals in better way.

Ten hundredths10

100 = 0.10

One tenths110 = 0.1

Here, the shaded parts in both the model shows the same amount of space taken.

Thus, 0.1 or one tenths = 0.10 or ten hundredths

Adding any number of zeros to the write of the last digit is an equivalent decimal to the same decimal number without the zeros.

For example, the equivalent decimals for 3.5 is 3.50 or 3.500

Exercise 7.41. Form the three groups of like decimals.

4.325, 25.01, 3.435, 27.321, 4.88, 6.2, 0.99, 99.9

2. Convert the decimals to like decimals.

a. 7.86, 5.908, 5.5 b. 8.23, 8.416, 1.09 c. 6.638, 1.83, 7.3

3. Put the correct sign '>', '<', '='.

a. 1.263 0.871 b. 5.876 9.876 c. 4.602 4.601

d. 6.04 6.40 e. 1.23 1.230 f. 0.89 0.794

4. Write the equivalent decimals for the following decimals.

a. 4.01 b. 0.1 c. 0.80 d. 9.05 e. 34.06 f. 1.32

Addition of DecimalsStep 1: While adding decimals, if the addends are unlike decimals first convert them to like decimals.

Step 2: Arrange them by placing the decimal points one below the other and then add.

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Example 1 Add 1.354 and 1.246

Solution:

Here, addends are like decimals, so arrange the decimals by placing decimal points one below the other as shown and add.

1 11 . 3 5 4

+ 1 . 2 4 62 . 6 0 0

1.354 + 1.246 = 2.600

Example 2 Add 5.637 and 8.56

Solution:

Here, addends are unlike decimals, so make them like decimals.

Therefore, 5.637 and 8.560 (like decimals)

Now arrange the decimals by placing decimal points one below the other as shown and add.

15 . 6 3 7

+ 8 . 5 6 01 4 . 1 9 7

5.637 + 8.56 = 14.197Subtraction of DecimalsStep 1: While subtracting decimals, if the subtrahend and minuend are unlike decimals first convert them to like decimals.

Step 2: Arrange them by placing the decimal points one below the other and then subtract.

Example 3 Subtract 4.32 from 8.57

Solution:

Here, decimals are like, so arrange the decimals by placing decimal points one below the other as shown and subtract.

8 . 5 7– 4 . 3 2

4 . 2 58.57 – 4.32 = 4.25

Example 4 Subtract 4.875 from 7.9

Solution:

Here, decimals are unlike decimals, so make them like decimals.

4.875 and 7.900

Arrange the decimals as shown and subtract.

8 9 107 . 9 0 0

– 4 . 8 7 53 . 0 2 5

7.9 – 4.875 = 3.025Exercise 7.51. Solve the following.

a. 6 . 2 9+ 3 . 3 2

b. 1 7 . 1+ 3 8 . 9

c. 9 . 7 5 1+ 5 . 2 6 1

d. 2 3 . 9– 1 7 . 5

e. 8 . 0 0– 5 . 4 2

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2. Arrange vertically and solve the following.a. 4.7 + 5.2 b. 32.89 + 6.785 c. 9.721 + 4.467d. 9.76 + 5.087 e. 7.608 + 7.364 f. 27.25 + 15.13g. 12.6 – 7.4 h. 9.62 – 8.76 i. 1.498 – 0.99j. 11.89 – 11.21 k. 8.74 – 3.5 l. 5.124 – 1.03

Multiplication of Decimals Multiplication of decimals is done in the same way as multiplication of numbers.

Multiplication of Decimals by a Whole Number

Step 1: Ignore the decimal point and multiply the decimal number as simple multiplication.

Step 2: Count the number of decimal places in the multiplicand and skip that many number of digits from the right and put the decimal point in the product.

Example 1 Multiply 2.41 by 3.

Solution:

Ignore the decimal point and multiply.

2 4 1× 3

7 2 3Since there are two digits after the decimal point in the multiplicand, put the decimal point in the product after two digits from the right.

Thus, 2.41 × 3 = 7.23

Example 2 Multiply 4.617 by 12

Solution:

Ignore the decimal point and multiply.

4 6 1 7× 1 2

9 2 3 4+ 4 6 1 7 0

5 5 4 0 4Since there are three digits after the decimal point in the multiplicand, put the decimal point in the product after three digits from the right.

Thus, 4.617 × 12 = 55.404Multiplication of Decimals by 10, 100 and 1000

When you multiply a decimal number by 10, the decimal point moves to the right by one place. For example, (i) 17.1 × 10 = 171.0 = 171 (ii) 331.45 × 10 = 3314.5 (iii) 14.267 × 10 = 142.67When you multiply a decimal number by 100, the decimal point moves to the right by two places. For example, (i) 51.56 × 100 = 5156 (ii) 24.091 × 100 = 2409.1 (iii) 16.8 × 100 = 1680When you multiply a decimal number by 1000, the decimal point moves to the right by three places. For example, (i) 28.315 × 1000 = 28315 (ii) 57.6 × 1000 = 57600 (iii) 7.92 × 1000 = 7920

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Division of DecimalsDivision of decimals is done in the same way as division of numbers.

Division of Decimals by a Whole Number

Step 1: Divide as in division of numbers ignoring the decimal point.

Step 2: Put the decimal point in the quotient when the division of whole number part of the dividend gets completed.

Example 3 Divide 25.4 by 5.

Solution:5.08

5 25.4 25

40400

−

−

Example 4 Divide 3.21 by 2.

Solution:1.605

2 3.21 21212

101010

0

−

−

−

−

When the number of digits in the dividend is less and the division is not complete, keep adding zeroes at every step till the division is complete as in example 1 and example 2.

Division of Decimals by 10, 100 and 1000

When you divide a decimal number by 10, the decimal point moves to the left by one place. For example,

(i) 21.1 ÷ 10 = 2.11 (ii) 272.45 ÷ 10 = 27.245 (iii) 12.67 ÷ 10 = 1.267

When you divide a decimal number by 100, the decimal point moves to the left by two places. For example,

(i) 13.36 ÷ 100 = 0.1336 (ii) 240.71 ÷ 100 = 2.4071 (iii) 6.7 ÷ 100 = 0.067

When you divide a decimal number by 1000, the decimal point moves to the left by three places. For example,

(i) 38.5 ÷ 1000 = 0.0385 (ii) 157.9 ÷ 1000 = 0.1579 (iii) 3.92 ÷ 1000 = 0.00392

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Exercise 7.61. Solve the following.

a. 24.22 × 3 b. 8.133 × 7 c. 4.21 × 9d. 12.6 × 11 e. 6.089 × 4 f. 7.881 × 5

2. Find the quotient.

a. 36.9 ÷ 3 b. 16.14 ÷ 6 c. 8.76 ÷ 8d. 33.21 ÷ 15 e. 83.4 ÷ 2 f. 5.28 ÷ 5

3. Fill in the blanks.

a. 2.82 × 10 = b. 18.39 × 1000 = c. 3.827 × 100 = d. 23.21 × 10 = e. 6.520 × 1000 = f. 7.082 × 100 = g. 4.8 ÷ 10 = h. 63.48 ÷ 100 = i. 15.87 ÷ 100 = j. 314.3 ÷ 1000 =

Math Lab ActivityObjective: To reinforce the concept of decimals.

Materials required: colours

Procedure:

Observe the decimal number and colour the grids given alongside for each of the given decimals. One example is solved for you.

Decimal Whole number 10th part of decimal

100th part of decimal

3.78

2.91

4.32

3.09

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Math Challenge

Fill in the magic square.

4.8 0.5 2.4

2.9

1.1

1.6 0.2

Math around me

Summer comes and ice cream is a perfect cool treat for all. The largest worldwide consumption of ice cream is in the United States and the most popular flavour is vanilla.

The first ice creams were made by the Chinese somewhere in 3000 BC. In 1843, Nancy Johnson

developed the first hand crank ice cream maker. However, large scale production of ice cream began in 1851 in Boston in the United States.

a. It takes approximately 45.42 litres of milk to make 3.75 litres of ice cream. Express 45.42 and 3.75 as a fraction.

Ice cream cones were invented during 1904. At a fair in St. Louis, a large demand forced an ice cream vendor to ask help from a waffle vendor. He rolled his pastries into horns. Together they made

history.

Hawaii is a home to an “ice cream bean”, a fruit that tastes like vanilla ice cream.

b. The cost of one ice cream bean plant is ` 572.58. How much will 5 such plants cost?

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Let’s Summarize

Revision Exercise1. Write the following in the numeral form.

a. Zero point zero five .

b. Thirty point one zero three .

c. Nine point seven zero eight .

d. Fifty four point seven two .

e. Zero point zero one five three .

f. One hundred and three point seven five .

2. Write the following decimals in words.

a. 3.406 b. 0.137 c. 39.58 d. 420.802e. 23.471 f. 7.045 g. 15.003 h. 12.108

3. Write the following fractions as decimals.

a. 37610 b. 452

1000 c. 51000 d. 52

10

e. 73100 f. 9

100 g. 451100 h. 39

1000

4. Write the following decimals as fractions.

a. 1.508 b. 3.089 c. 30.25 d. 53.50e. 0.860 f. 6.42 g. 46.105 h. 4.306

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5. Write the decimal number for the following expanded form.

a. 23 + 81000 b. 3 + 7

10 + 11000 c. 50 + 2

10 + 91000

d. 1 + 71000 e. 40 + 7 + 4

100 + 11000 f. 10 + 4 + 2

100 + 11000

6. Form groups of like decimals.

19.792, 103.14, 8.609, 0.781, 67.48, 34.797, 94.5, 12.46, 7.8

7. Convert the decimal numbers to like decimals.

a. 10.724, 8.7, 5.13, 51.9, 3.02 b. 9.46, 29, 4.7, 5.008c. 1.67, 51.4, 0.34, 14.2 d. 13.24, 7.9, 14.03, 1.67, 0.3

8. Fill the box with correct sign '>', '<', =.

a. 3.465 3.564 b. 8.201 8.021 c. 54.51 45.25

d. 41.2 41.20 e. 5.368 5.260 f. 0.79 0.079

9. Write the equivalent decimals for the following.

a. 27.15 b. 60.01 c. 0.04 d. 4.36 e. 3.0 f. 22.26

10. Solve.

a. 4.08 + 63.076 b. 2.903 + 6.5 c. 76.9 + 2.46d. 3.805 + 73.05 e. 23.857 – 06.37 f. 82.63 – 57.984g. 34.587 – 28.05 h. 23.45 – 17.378 i. 32.5 × 8j. 21.54 × 9 k. 7.127 × 1000 l. 4.35 × 10m. 31.73 ÷ 2 n. 2.57 ÷ 100 o. 53.26 ÷ 1000

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As on 16.07.2019