Understanding Decimals: Definitions, Rules and Applications

Decimals

Decimals

Decimals are another way of writing part of a whole number. In a place value table on moving a digit from the right to the left by one place, its place value becomes ten times. Similarly, on moving a digit from the left to the right by one place, its place value becomes one tenth.

1.0Decimal Fractions

A decimal fraction is a fraction whose denominator is 10 or 100 or 1000 etc. Thus, $\frac{1}{10}, \frac{17}{100}, \frac{37}{1000}$ are all decimal fractions. There is another special way of writing such fractions.

Tenths

The 'dec' in decimal means ten. The fraction $\frac{1}{10}$ names the number one-tenth. If we divide a square into ten equal parts, then each part of the square represents one-tenth. Besides writing as $\frac{1}{10}$ , one-tenth can also written as $10%$ .

Tenths

One tenth or $\frac{1}{10}$ or 0.1 The dot written immediately to the right of units place is called the decimal point. The number written in decimal form are called decimal numbers or simply decimals.

Hundredths

The number one hundredth can be named by the decimal fraction 0.01 . The digit 1 is in the hundredths place shows one hundredth and 0 in the tenths place shows 0 tenths. 10 hundredths means 1 tenths and so is written as 0.10 .

You know that $\frac{46}{100} = \frac{40 + 6}{100} = \frac{40}{100} + \frac{6}{100} = \frac{4}{10} + \frac{6}{100}$ Think of $\frac{46}{100}$ as 4 tenth 6 hundredths The decimal number for 4 tenths is 0.4 The decimal number for 6 hundredths is 0.06 The decimal number for 4 tenths 6 hundredths or 46 hundredth is 0.46 $\frac{46}{100} = \frac{4}{10} + \frac{6}{100} = 0.4 + 0.06 = 0.46$

Thousandths

The decimal number for one thousandth is 0.001 . The digit 1 is in the thousandth place shows one thousandth, 0 in the tenth place shows 0 tenths, 0 in the hundredth place shows 0 hundredths. Similarly, 5 thousandths will be written as 0.005 . $\frac{29}{1000} = \frac{20 + 9}{1000} = \frac{20}{1000} + \frac{9}{1000} = \frac{0}{10} + \frac{2}{100} + \frac{9}{1000}$ Think of $\frac{29}{1000}$ as 0 tenth, 2 hundredths and 9 thousandths and written as 0.029. Similarly, $\frac{250}{1000} = \frac{200 + 50 + 6}{1000} = \frac{200}{1000} + \frac{50}{1000} + \frac{6}{1000} = \frac{2}{10} + \frac{5}{100} + \frac{6}{1000}$ $\frac{256}{1000}$ means 2 tenths, 5 hundredths and 6 thousandths and written as 0.256 . (i) Any number of zeros may be put to the extreme right of the decimal part of a decimal.

Eg. $0.75 = \frac{75}{100} = \frac{75 \times 10}{100 \times 10} = \frac{750}{1000} = 0.750$ . Hence, numbers $0.75, 0.750, 0.7500, 0.75000$ are called equivalent decimals. (ii) A decimal like 28.356 has two parts-whole number part and decimal part. These parts are separated by a dot called the decimal point. The whole number part to the left of the decimal point and the decimal part is to its right.

Thousandths

(iii) The number of digits contained in the decimal part of a decimal gives the number of its decimal places. Eg. the number 7.987 has three decimal places, 8.6252 has four decimal places. (iv) Decimals having the same number of decimal places are called like decimals. (v) Decimals having different number of decimal places are called unlike decimals.

Every fraction is a decimal.

2.0Place Value

The fractional part in a decimal number is usually written to the right of the decimal point.

The following chart shows the place value of each digit in 3675.476:

Digit Place Value	Value of the Digit
3 Thousands	3000
6 Hundreds	600
7 Tens	70
5 Ones	5
4 Tenths	$(\frac{4}{10})$
7 Hundredths	$(\frac{7}{100})$
6 Thousandths	$(\frac{6}{1000})$

Some common mistakes to avoid. $0.35 \neq = \frac{35}{10}; 0.35 = \frac{35}{100}$

3.0Comparison of Decimals

While comparing two decimal numbers: (i) The decimal fraction with the greater integral part will be greater. (ii) If the integral part of two decimal fractions are the same, then beginning from the tenth place, the decimal fraction with the greater digit in the same place is greater.

Conversion of Decimal Fractions to Fractions

To convert a decimal fraction into a fraction, follow the steps given below. Step-1: Count the number of decimal places in the decimal fraction. Step-2: Ignore the decimal point. Write all the digits as the numerator of the fraction. Step-3: Write as many zeroes after 1 in the denominator, as there were decimal places in the decimal fraction. Step-4 : Reduce the fraction to its simplest form.

Some decimal expansions go on forever. For example, $\frac{1}{3} = 0.333 \dots$ ... where the '...' means that the 3's never end.

Conversion by Long Division Method

Step-1: Convert the dividend to a suitable equivalent decimal. Step-2: When the digit to the right of decimal point is brought down, a decimal point is to be placed in the quotient.

4.0Addition of Decimals

While adding two or more decimals, follow the steps given below. Step-1: Convert the decimal fractions into like decimal fractions. Step-2: Arrange the decimal fractions vertically with all the digits in the correct place according to the place value of the digits in the decimal fraction. This will result the decimal points of all the decimal fractions to fall in one vertical line. Then add as the case may be.

5.0Subtraction of Decimals

While subtracting two or more decimals, follow the steps given below. Step-1: Convert the decimal fractions into like decimal fractions. Step-2 : Arrange the decimal fractions vertically with all the digits in the correct place according to the place value of the digits in the decimal fraction. This will result the decimal points of all the decimal fractions to fall in one vertical line. Then subtract, as the case may be.

While doing subtraction the smaller number is written under the bigger number.
While doing addition and subtraction, the numbers are written in such a way that the decimals are in the same vertical line.

6.0Conversion of Units

Rs. $1 = 100$ paisa 1 Paisa $=$ Rs. $\frac{1}{100}$ $1 Km = 1000 m$ $1 m = \frac{1}{1000} Km$ 10 mm (millimeters) $= 1 cm$ (centimeter) 10 centimeters $(cm) = 1$ decimeter $(dm)$ 10 decimeters $(dm) = 1$ meter $(m) = 100 cm = 1000 mm$ 10 meters ( m ) = 1 decameter (dam) 10 decameters $($ dam $) = 1$ hectometer $(hm) = 100 m$ 10 hectometer $= 1$ Kilometer (Km) = 1000 m

Unit Ladder

Kilometer↓ (×10)Hectometer↓ (×10)Decameter↓ (×10)Meter↑ (÷10)Decimeter↑ (÷10)Centimeter↑ (÷10)Millimeter

10 decimeters $= 1$ meter 1 decimeter $= \frac{1}{10}$ of a meter or 0.1 meter 1000 milligrams $= 1$ gram 1000 grams $= 1$ Kilogram 1 Milligram $= \frac{1}{1000}$ of a gram 1 gram $= \frac{1}{1000}$ Kilogram 1000 millilitres $(m /) = 1$ litre ( $l$ ) $1 ml = \frac{1}{1000}$ litre $= 0.001 l$

7.0Word Problems

When you get a word problem that involves adding or subtracting decimals, it's usually a good idea to rewrite all the numbers with the same number of decimal places, so you don't get confused.

KEYWORDS AND PHRASES

Addition	Subtraction
Sum	Difference
Added to	Minus
Increased by	Decreased by
Total	Less
All together	Less than
More than
Combine

8.0Number Line

Draw number line and divide the distance between any two divisions in 10 equal parts as shown below:

Division of Number Line in 10 Parts

Let us locate a point corresponding to 1.6 on the number line. We know that 1.6 is more than 1 and less than 2 . There is one complete whole number and six tenths in it. From 0 , we move one complete step towards right and then six small parts towards right and shade the point which represent 1.6.

To represent a decimal on a number line, divide each segment of the number line into ten equal parts. E.g. To represent 8.4 on a number line, divide the segment between 8 and 9 into ten equal parts.

Dividing Segments 8 and 9 into 10 Parts

The arrow is four parts to the right of 8 where it points at 8.4. Likewise, to represent 8.45 on a number line, divide the segment between 8.4 and 8.5 into ten equal parts.

Dividing Segments 8.4 and 8.5 in 10 Parts

The arrow is five parts to the right of 8.4 where it points at 8.45 . Similarly, we can represent 8.456 on a number line by dividing the segment between 8.45 and 8.46 into ten equal parts.

Dividing 8.45 and 8.46 in 10 Equal Parts

9.0Numerical Ability

Write each of the following as decimals: (i) Seven-tenths (ii) Two tens and nine - tenths (iii) One hundred and two ones

Explanation (i) $\frac{7}{10} = 0.7$ (ii) $20 + \frac{9}{10} = 20.9$ (iii) $100 + 2 = 102.0$

Write the following in decimal forms: (i) $\frac{125}{100}$ (ii) $\frac{15}{10}$ (iii) $\frac{315625}{1000}$

Solution (i) $\frac{125}{100} = 1.25$ (ii) $\frac{15}{10} = 1.5$ (iii) $\frac{315625}{1000} = 315.625$

Write the following in the standard form as decimals. Also write them in words in both the ways. (i) $1 ml = \frac{1}{1000}$ (ii) $46000 + 900 + 40 + 6 + \frac{3}{10} + \frac{6}{100} + \frac{5}{1000}$ (iii) $900 + 0 + 5 + \frac{1}{10} + \frac{0}{100} + \frac{0}{1000} + \frac{8}{10000}$

Explanation (i) $4000 + 700 + 30 + 0 + \frac{4}{10} + \frac{9}{100} + \frac{0}{1000}$ $= 4730.490$ In words :- Four thousand seven hundred thirty point four nine zero. or Four thousand seven hundred thirty and four hundred ninety thousandths. (ii) $46000 + 900 + 40 + 6 + \frac{3}{10} + \frac{6}{100} + \frac{5}{1000}$ $= 46946.365$ In words:- Forty six thousand nine hundred forty six point three six five. or Forty six thousand nine hundred forty six and three hundred sixty five thousandths. (iii) $900 + 0 + 5 + \frac{1}{10} + \frac{0}{100} + \frac{0}{1000} + \frac{8}{10000}$ $= 905.1008$ In words:- Nine hundred five point one zero zero eight. or Nine hundred five and one thousand eight ten thousandths.

Write the following in decimal notation and expanded form. (i) Thirty-three hundredths (ii) Five hundred eighty -three thousandths

Solution (i) Thirty - three hundredths $= \frac{33}{100} = 0.33$ Expanded form $= \frac{3}{10} + \frac{3}{100}$ (ii) Five hundred eighty - three thousandths $= \frac{583}{1000} = 0.583$ Expanded form $= \frac{5}{10} + \frac{8}{100} + \frac{3}{1000}$

In each of the following pairs of decimal numbers, state which number is greater. (i) 539.2, 97.654 (ii) 238.587, 238.589

Solution (i) The whole number part of 539.2 is 539 and that of 97.654 is 97 . Since, $539 > 97$ , so $539.2 > 97.654$ (ii) In the two numbers (a) Whole number parts are equal. (b) In both numbers, digit at the tenths place is 5 . (c) In both numbers, digit at the hundredths place is 8. (d) We compare the thousandths digits. Since, $7 < 9$ , so $238.587 < 238.589$

Write the decimal numbers $3.05, 3.5, 4.8, 3.079, 6$ in order of size starting with the largest.

Solution The numbers starting with the largest are: $6 \leftarrow$ Put the largest whole number first 4.8 $3.5 \leftarrow$ Put the largest 'tenth' first $3.079 \leftarrow$ Put the largest 'hundredth' first 3.05 $6 > 4.8 > 3.5 > 3.079 > 3.05$ largest to smallest.

Convert the following into a fraction. (i) 7.258 (ii) 94.62

Explanation (i) Decimal fraction $= 7.258$ Decimal places $= 3$ Thus numerator $= 7258$ Denominator $= 1000$ Fraction $= \frac{7258}{1000} = \frac{3629}{500}$ (ii) Decimal fraction $= 94.62$ Decimal places $= 2$ Thus numerator $= 9462$ Denominator $= 100$ Fraction $= \frac{9462}{100} = \frac{4731}{50}$

Which decimal fraction is greater? 0.293 or 0.027

Explanation The integral parts of both numbers are 0 . Let us now compare the digit in the tenths place. The digits in the tenths place are 2 and 0 . As $2 > 0$ $∴ 0.293 > 0.027$ .

Which decimal fraction is smaller? 7.692 or 7.648

Explanation The integral part of both the numbers are 7. Again, the digit in the tenths places are equal. Now, compare the digits in the hundredths place. The digits are 9 and 4 . As $9 > 4$ $∴ 7.692 > 7.648$ .

10. Arrange 448.7, 4.4487, 4.4587, 4.444 and 44.87 in descending order.

Explanation First write the decimals in a place-value chart. The greatest integral part is 448, followed by 44, after which there are three decimals with the same integral part. Here, 4.444 is the smallest decimal. Moving left to right after decimal point, we find 4.444 $< 4.4487$ < 4.4587. Thus, $539 > 97$

11. Convert $\frac{8}{25}$ into a decimal fraction.

Solution (i) 8 unit $= 800$ hundredths or $8 = 8.00$ $25 8.00 ⟵ .32 8$ wholes $\frac{- 0}{80 ⟵} 80$ tenths $\frac{- 75}{50 ⟵} 50$ hundredths $\frac{- 50}{0}$ The number of digits in the decimal part of a decimal fraction is referred to as the number of decimal places. $∴ \frac{8}{25} = 0.32$

12. Find the sum in each of the following: (i) $= 94.62$ (ii) $15 + 0.632 + 13.8$ Solution

(i)

+ 7.283 + 592.732 600.015

(ii)

+ 15.000 + 0.632 + 13.800 29.432

13. Find the sum of (i) 0.71 and 0.86 (ii) 15.1, 315.23, 53.036 Solution

(i)

+ 0.71 + 0.86 1.57

(ii)

+ 15.100 315.230 + 53.036 383.366

14. Subtract the following (i) 67.46-43.21 (ii) 600.431-100.00 (iii) 117.462-19.364 Explanation

(i)

- 67.46 - 43.21 - 24.25

(ii)

- 600.431 - 100.00 - 500.431

(iii)

- 117.462 - 19.364 - 98.098

15. Subtract and Solve (i) Subtract: 483.742-48.732 (ii) Solve: $7.983 + 4.6429 - 5.7893$ Solution

(i)

- 483.742 - 48.732 - 435.010

(ii)

7.983 + 4.6429 - 5.7893

First add:

- 7.9830 + 4.6429 12.6259

Now subtract:

12.6259 - 5.7893 - 6.8366

16. Subtract: (i) 49.497 from 637.652 Solution

(i)

- 637.652 - 049.497 - 588.155

(ii)

- 95.20 - 78.34 - 16.86

17. Express (i) 8 paise and Rs.15.36 in Rs. (ii) $1033 m - 428 cm$ in meter (iii) $778 g + 1.939 kg$ in kg (iv) $4169 g - 0.798 kg$ in kg

Explanation (i) 8 Paise and Rs. $15.36 = 8 + 15.36 \times 100$ ( $∵ 100$ Paisa = 1 Rupee) $= 8 + 1536 = 1544$ Paise or $\frac{1544}{100} =$ Rs. 15.44 (ii) $1033 m - 428 cm$ $7.983 + 4.6429 - 5.7893$ (iii) $778 g + 1.939 kg$ $= \frac{778}{1000} kg + 1.939 kg = 0.778 kg + 1.939 kg$ $= 2.717 kg$ (iv) $4169 g - 0.798 kg = \frac{4169}{1000} kg - 0.798 kg$ $= 4.169 kg - 0.798 kg$ $= 3.371 kg$

18. Express as rupees using decimals. (i) 5 paise (ii) 75 paise

Solution We know that there are 100 paise in 1 rupee. (i) 5 paise $15.36 = 8 + 15.36 \times 100$ rupees $=$ Rs. 0.05 (ii) 75 paise $= \frac{75}{100}$ rupees $=$ Rs. 0.75

19. Express as meters using decimals. (i) 15 cm (ii) 2 m 45 cm

Solution $1 m = 100 cm$ (i) $15 cm = \frac{15}{100} m = 0.15 m$ (ii) $2 m 45 cm = (2 + \frac{45}{100}) m = 2.45 m$

20. Rani had Rs 18.50. She bought one ice-cream for Rs. 11.75. How much money does she have now?

Explanation Money with Rani = Rs. 18.50 Money spent for an ice cream = Rs. 11.75 The money left with Rani will be the difference of these two.

- 18.50 - 11.75 - 6.75

Hence, the money left is Rs 6.75

21. By how much does the sum of 34.07 and 15.239 exceeds the sum of 16.40 and 27.08 ?

Solution Sum of 34.07 and 15.239 $= 34.070 + 15.239 = 49.309$ and sum of 16.40 and $27.08 = 16.40 + 27.08 = 43.48$ $∴$ difference between their sums $= 49.309 - 43.48 = 49.309 - 43.480 = 5.829$

22. Amit bought a Maths book for Rs. 45.60 and a geometry box for Rs. 62.55. What is the total amount spent by Amit?

Solution Money spent on Maths book = Rs. 45.60 Money spent on Geometry box $=$ Rs. 62.55 $∴$ Total amount spent $=$ Rs. $45.60 +$ Rs. $62.55 =$ Rs. 108.15

+ 45.60 + 62.55 108.15

23. Priya travelled 8 km 95 m in the first hour, 6 km 298 m in the second hour and 7 km $9 m$ in the third hour. Find the total distance travelled by her in three hours.

Solution Distance travelled in first hour $= 8 km 95 m = 8.095 km$ Distance travelled in second hour $= 6 km 298 m = 6.298 km$ Distance travelled in third hour $= 7 km 9 m = 7.009 km$ $∴$ Total distance travelled in 3 hours $= 8.095 km + 6.298 km + 7.009 km = 21.402 km$

24. Between which two whole numbers on the number line do the given numbers line? Which of these whole numbers is nearer the number?

Whole Numbers on the Number Line

(i) 0.8 (ii) 5.1 (iii) 2.6

Explanation (i) 0.8 exists between 0 and 1 , and is nearer to 1 . (ii) 5.1 exists between 5 and 6 , and is nearer to 5 . (iii) 2.6 exists between 2 and 3 , and is nearer to 3 .

25. Show the following numbers on the number line. (i) 0.2 (ii) 1.9

Solution (i) Since space between 0 and 1 is divided into 10 equal parts therefore, each part is equal to one - tenth. Now, 0.2 is the second point between 0 and 1

Space Between 0 and 1 Divded in 10 Parts

(ii) Since space between 1 and 2 is divided into 10 equal parts therefore, each part is equal to one - tenth. Now, 1.9 is the ninth point between 1 and 2

Space Between 1 and 2 is divided in 10 Parts

10.0Memory Map

Decimals Memory Map