# Decimals

Fractions in which denominators are powers of 10 are called decimal fractions.

1/10, 1/100, 1/100 etc. are respectively the tenth, the hundredth and the thousandth part of 1.

7/10 is 7 tenths, written as 7 = 0.7, 13/100 = 0.13, 9/100 = 0.09 and so on.

To convert decimals into fractions, we remove the decimal and divide by the multiple of 10 and then cancel out the numbers.

**Illustration 3:**

(*i*) 0.56 = 56/100 = 28/50
If numerator and denominator of a fraction contain the same number of decimal places, then we may remove the decimal sign.

**Illustration 4:**

5.87/6.93 = 587/693

To add decimals, note that the decimal point should lie in one column. The numbers so arranged can now be added or subtracted in a usual way.

**Illustration 5:**

713 + 7.013 + .713 + 7.0013 = ?

Write all decimals in one vertical line.

To multiply by any power of 10, shift the decimal point to the right by as many places of decimal, as is the power of 10.

**Illustration 6:**

0.3456 ×10000. We just have to shift the decimal point by 4 places = 3456.

To multiply the two decimal numbers, simply ignore the decimal and multiply them. In the product, the decimal point is placed to as many places of decimal as is the sum of the decimal places in the given number.

**Illustration 7:**1.71 × 1.2. First multiply the numbers: 171 × 12 = 2052. Since there are three decimals, the answer becomes 2.052.

To divide decimal numbers, first divide without considering the decimal point. In the quotient, put the decimal point to give as many places of decimal as are there in the dividend.

**Illustration 8:**

To divide 0.49 by 7, first we get 49/7 = 7. Then add two decimals, which was in the given number: 0.07.

To compare fractions, convert each one of the given fractions in the decimal form. Arrange them in ascending or descending order, as per requirement.

**Illustration 9:**

Arrange the fractions 3/8, 7/12, 2/3, 14/19, 16/25 and 1⁄2 in ascending order. Convert each of the given fractions into decimal form, we get:

3/8 = 0.375,

7/12 = 0.583,

2/3 = 0.666,

14/18 = 0.736,

16/25 = 0.64 and

1⁄2 =0.5.

Clearly, 0.375 < 0.5 < 0.583 < 0.64 < 0.666 < 0.736. Hence 3/8 < 1⁄2 < 7/12 < 16/25 < 2/3 < 14/19

# Recurring Decimals

If in a decimal fraction, a figure or set of figures is repeated continuously, then such a number is called a recurring decimal. If a single figures is repeated, then it is expressed by putting a dot on it If a set of figures is repeated, it is expressed by putting a bar on the set.

**Illustration 10:**

2/3 = 0.666 .......... = 0.Å. where Å = 6.

1/7 = 0.142857 142857....... = 0.142857

To convert a pure recurring decimal into a vulgar fraction, write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.

**Illustration 11:**

0.7777... = 0.7; 0.057 = 57/999.

A decimal fraction in which some figures do not repeated and some of them repeat, is called a *mixed recurring decimal.*

To convert a mixed recurring decimal into a vulgar fraction, in the numerator, take the difference between the number formed by all the digits after decimal point and divide that by as many nines as repeating digits, followed by as many zeros as is the of non-repeating digits.

# Conversion of decimals into fractions and vice versa

It is useful to learn the conversion as given in the table as it helps acquire speed.

This is a very useful table, and one can easily find decimals and percentages related to any fraction.

Note that from a basic fraction of 1⁄2 or 1/3, we can derive subsequent fractions by successively dividing by 2. So if 1/6 = 0.166, then 1/12 = 0.166/2 = 0.0833.

Similarly, other fractions can also be converted if we know the base.

For instance, 1/7 = 0.1428, then

1/14 = 0.1428/2 = 0.0714

Also 3/7 = 0.1428 ́ 3 = 0.4285.

Applying the above, we can solve the following sums quite easily.

**Illustration 12:**

Calculate 0.0627 × 0.285 × 8
The above sum can be written as (by remembering values from the above table): 1/16 × 2/7 × 8, which on solving gives us: 1/7 or 0.1428.

Hence 0.0627 × 0.285 × 8 = 0.1428 approx.

**Illustration 13:**

Calculate 2.77% of 7216
We see from the above table that 2.77% corresponds to the value of 1/36. Hence the sum becomes 1/36 ́ 7216 = 200 approx.

The student will appreciate that the above table is very useful in solving quants and data calculations quickly. A sum that may look difficult can be quickly done by conversions.