Decimals | Quantitative Techniques for CLAT PDF Download

Decimal Fractions and Decimals

 The fractions in which the denominators are 10, 100, 1000, etc., are known as decimal fractions. Numbers written in decimal form are called decimals.

• A decimal has two parts: the whole number part (to the left of the decimal point) and the decimal part (to the right of the decimal point).
• The number of digits contained in the decimal part of a decimal is called the number of its decimal places.

Examples: 0.1 = 0.10 = 0.100 and 0.2 = 0.20 = 0.200. Different (unlike) decimals may be converted into like decimals by annexing the requisite number of zeros at the end of the decimal part.

Addition of Decimals

1. Write down the decimal numbers one under the other and line up the decimal points.
2. Convert the given decimals to like decimals (same number of decimal places).
3. Arrange the addends so that digits of the same place value are in the same column.
4. Add the numbers starting from the rightmost column.
5. Place the decimal point in the sum directly under the decimal points of the addends.

Add the decimals:

(i) 1.83, 21.105, 236.8 and 0.9

Solution:

Write each addend as a like decimal with three decimal places (highest number of places among the addends is 3).

1.830 + 21.105 + 236.800 + 0.900

Arrange them in columns and add the digits column-wise from the right.

Subtraction of Decimals

1. Write down the two decimal numbers one under the other and line up the decimal points.
2. Convert the given decimals to like decimals.
3. Write the smaller decimal number under the larger decimal number in the columns.
4. Arrange digits of the same place value in the same column.
5. Subtract the numbers column-wise from the right. Place the decimal point in the difference directly under the decimal points of the numbers above.

Subtract the decimals:

(i) Subtract 27.59 from 31.4

Solution:

Convert both numbers to like decimals (two decimal places).

31.40 - 27.59

Arrange and subtract column by column.

Converting a Decimal into a Fraction

1. Write the given decimal without the decimal point as the numerator.
2. In the denominator, write 1 followed by as many zeros as there are decimal places in the given decimal.
3. Reduce the fraction to its simplest form (if possible).

Convert the following into fractions.

(i) 3.91

Solution:

Write the decimal without the point as numerator.

Numerator = 391

There are two decimal places, so denominator = 100

Thus the fraction is

=391/100

Converting a Fraction into a Decimal

a) When the denominator is 10 or a power of 10

• To convert a fraction with denominator 10, put the decimal point one place to the left of the first digit of the numerator.
• To convert a fraction with denominator 100, put the decimal point two places to the left of the first digit of the numerator.
• To convert a fraction with denominator 1000, put the decimal point three places to the left of the first digit of the numerator.

b) When the denominator is not a power of 10

1. Divide the numerator by the denominator as in whole-number division until a non-zero remainder appears or until remainder becomes zero.
2. Place a decimal point in the dividend and in the quotient when required (after the integer part is obtained).
3. Add a zero to the right of the decimal point in the dividend and also to the remainder.
4. Continue dividing as in whole numbers and bring down zeros as required.
5. Repeat until the remainder becomes zero (terminating decimal) or a remainder repeats (recurring decimal).

Example 1 Convert 29/4 into a decimal fraction

Solution:

Perform the division 29 ÷ 4 using long division.

The result of the division is 7.25, so 29/4 = 7.25 (a terminating decimal).

Example 2 Convert the fraction shown below into a decimal fraction.

The division in the examples above terminates after a finite number of decimal places; such fractions give terminating decimals.

In other cases (for example when dividing certain fractions), the division never terminates but repeats a block of digits; such decimals are non-terminating repeating decimals.

Examples of repeating decimal representation:

Decimal representation of 8/9 is 0.888... and is written as 0.

is 0.888...   or  is

Also,

Every positive rational number is either a terminating decimal or a non-terminating repeating decimal.

To search whether a rational number is a terminating or non-terminating repeating decimal.

If a rational number is expressed in lowest terms and its denominator has no prime factors other than 2 or 5 (i.e., denominator = 2^a × 5^b), then the rational number is a terminating decimal. Otherwise the rational number is a non-terminating repeating decimal.

Examples:

The fractions shown in the images below are terminating decimals:

Whereas the fractions shown below are non-terminating repeating decimals:

Multiplication of Decimals

a) Multiplication of a decimal by a whole number

Multiply the decimal number and the whole number as if both were whole numbers. In the product place the decimal point so that the number of decimal places equals the number of decimal places in the decimal multiplicand.

b) Multiplication of a decimal by 10, 100, 1000, ...

To multiply a decimal by 10, move the decimal point one place to the right. To multiply by 100, move it two places to the right, and so on.

Examples:

2.96 × 10 = 29.6

2.96 × 100 = 296

2.96 × 1000 = 2960

c) Multiplication of a decimal by a decimal

Multiply the numbers ignoring the decimal points as if they were whole numbers. Then place the decimal point in the product so that the total number of decimal places equals the sum of decimal places in the multiplicand and the multiplier.

Example: Find 34.17 × 3.2

Solution:

Multiply 34.17 and 3.2 as whole numbers by ignoring the decimal points.

3417 × 32 = 109344

The multiplicand has 2 decimal places and the multiplier has 1 decimal place, so the product must have 3 decimal places in total.

Therefore 34.17 × 3.2 = 109.344

Division of Decimals

a) Division of a decimal by a whole number

Division proceeds as with whole numbers. Place the decimal point in the quotient directly above the decimal point in the dividend when needed.

Example:

From the working shown, 442.4 ÷ 7 = 63.2

b) Division of a decimal by 10, 100, 1000, ...

To divide a decimal by 10 move the decimal point one place to the left; to divide by 100 move it two places to the left, and so on.

c) Division of a decimal by a decimal

To divide a decimal by another decimal, convert the divisor to a whole number by multiplying both divisor and dividend by the same power of 10, then divide.

Example: Divide 6.25 by 2.5

Solution:

Since the divisor 2.5 has one decimal place, multiply both dividend and divisor by 10 to make the divisor a whole number.

6.25 × 10 = 62.5

2.5 × 10 = 25

Now divide 62.5 by 25 as division of a decimal by a whole number.

Now the problem reduces to the division of a decimal by a whole number.

Similarly, other examples follow the same idea:

d) Division of a whole number by a decimal

Convert the divisor (decimal) into a whole number by multiplying both dividend and divisor by the same power of 10, then carry out the division as usual.