Decimal to Fraction conversion

Decimal to Fraction Calculator


Decimal Number: 0
Fraction Number: 0

What are fractions and decimals?

A fraction is a number of the form a/b, where b ≠ 0. It represents some part of a whole. Fractions are of three types: Proper fraction, improper fraction and mixed fraction. In a proper fraction, the numerator is lesser than the denominator, whereas it is the opposite for an improper fraction. A mixed fraction (also called a mixed number) is a fraction which has a whole part and a fractional part. For example, 2 ¼ is a mixed fraction, where 2 is the whole number and ¼ is the fractional part. 3/5 is another example of a fraction. Since the numerator is lesser than the denominator, it is a proper fraction.

A decimal number is a non-integer number, and it has a decimal point. It is an alternative representation for fractional numbers. They are represented in the form a.b, where the whole number part and the fractional part are separated by a decimal point. So, ‘a’ is the whole number ‘b’ is the decimal part. For example, 0.75 is a decimal number. As a fraction, this can be written as 75/100, which is equivalent to ¾ in its simplest form.

Consider a pizza cut into 4 slices. Suppose one whole pizza and one half of another pizza have been given as shown here. The one whole pizza represents the whole number, which is 1 in this case, and ½ is the fractional part. Combining these two together gives the mixed fraction 1 ½, which indicates one whole pizza and ½ part of another pizza.

Fraction examples -

If we were to write the mixed fraction 1 ½ as a decimal number, we would need to divide the numerator of the fractional part by the denominator first, and then add the whole number of the fraction to this decimal number. The calculations reveal 1.5 as the decimal value of the fraction 1 ½.

Fractions and their equivalent decimal value

If the fraction has a power of 10 in the denominator, it is quite easy to express it in the form of a decimal.

For example, $$ {1\over10} = 0.1, \; {1\over100}=0.01, \; {1\over1000}=0.001, {1\over10000}=0.0001 \; and \; so \; on $$

Some other examples are: $$ 0.56 = {56\over100}, \; 3.795={3975\over1000}, \; {4\over10}=0.4 $$

It should be noted that the number of digits after the decimal point is equal to the number of zeros in the denominator of the fraction.

Some commonly used decimals and their corresponding fractions are

Decimal form Fraction
0.001 $$ {1\over1000} $$
0.01 $$ {1\over100} $$
0.1 $$ {1\over10} $$
0.125 $$ {1\over8} $$
0.2 $$ {1\over5} $$
0.25 $$ {1\over4} $$
0.3 $$ {3\over10} $$
0.3333 $$ {1\over3} $$
0.375 $$ {3\over8} $$
0.4 $$ {2\over5} $$
0.5 $$ {1\over2} $$
0.625 $$ {5\over8} $$
0.9 $$ {9\over10} $$
1.25 $$ {5\over4} $$
1.5 $$ {3\over2} $$
2.5 $$ {5\over2} $$

Converting a decimal to a fraction

Consider the fraction 1.625. Given below are the steps to convert this decimal to a fraction.

1) Write the decimal number in the form of a fraction. This can be done by writing the numerator as the decimal number itself and the denominator as 1. $$ 1.625 = {1.625 \over 1} $$

2) Remove the decimal point by multiplication. If the given decimal number has 2 places after the decimal point, multiply the numerator and denominator of the fraction obtained in the previous step by 100. In this example, 1.625 has 3 decimal places, so we multiply the numerator and denominator of 1.625/1 by 1000. $$ {1.625 \over 1} = {{1.625 * 1000} \over {1*1000}} = {1625\over1000} $$

3) Now try to reduce the resulting fraction. This can be done by first finding the GCD of the numerator and the denominator (if possible), and then dividing the numerator and denominator by it. In the present example, the GCD of 1625 and 1000 is 125, so we divide these two numbers by 125. $$ {{1625÷125} \over {1000÷125}} = {13\over8} $$

4) Write the fraction thus obtained as an improper/proper fraction or a mixed number. $$ {13\over8} = 1{5\over8} $$

Converting a recurring decimal to a fraction

If the decimal number is a recurring one, then the procedure of converting it into a fraction is slightly different. For this, we consider the recurring decimal 1.666. There are 3 decimal places repeating here.

i) For this, we first set up an equation. Let x equal the recurring decimal number. $$ 𝑥 = 1.666 … (1) $$

ii) Now count the number of places after the decimal point, named y. There is a group of 3 decimal places which are repeating, so y = 3.

iii) Create a second equation by multiplying both sides of the first equation by 10y = 103 = 1000. $$ 1000𝑥 = 1666.666 … (2) $$

iv) Now subtract equation (1) from equation (2). $$ 1000𝑥 = 1666.666 … \\ {{𝑥 = 1.666 …}\over{999x = 1665}} $$

v) Solve for x. $$ x = {1665\over999} $$

vi) Find the GCD of the numerator and the denominator. In this case, it is 333. Divide the numerator and the denominator by the GCD. $$ {{1665 ÷ 333}\over{999÷333}} = {5\over3} $$

vii) Write the resulting fraction as proper/improper fraction or a mixed number. $$ {5\over3} = 1{2\over3} $$

Therefore, $$ 1.666 = 1{2\over3} $$

Areas of application

Decimal to fraction conversions (and vice versa) arise in different situations in science and technology.

1) It is much easier to do arithmetic operations using decimals compared to fractions, as we do not need to find the LCM. Therefore, operations and comparisons of decimal numbers is quite straightforward.

2) In some cases, the fraction notation is much more useful. This is especially true for the case of recurring decimals. It is more convenient to denote the decimal number 1.666 𝑎𝑠 1(2/3). This is easier to enter in calculators as we would not have to write down all the decimal places.

3) In physics problems, while dealing with lots of whole numbers, it is best to express them as fractions.

4) However, when we are working with large numbers and scientific notations, decimals are preferred. For example, if we need to multiply 500 × 4000, and the number of zeros seem overwhelming, then the numbers can be written as $$ 500 × 4000 = 5 × 10^2 × 4 × 10^2 $$

The powers of 10 indicate the number of zeros after 5 and 4. At this point, we can simply multiply the numbers, and add up their powers. $$ 500 × 4000 = 20 × 10^5 = 2 × 10^6. $$

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