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Fractions, Percents, Decimals, Ratios

Fractions

The word fraction means something is broken from a whole thing. Thus, a fraction represents a part of something. Suppose that a company consists of 200 employees. The number of male employees is 100. Then,  represents the part of male employees in the company. Consider a chocolate bar, which is cut into 5 equal pieces. Two pieces are eaten and three pieces are remaining. Let us sketch this.

The shaded region represents the remaining pieces. Out of five parts, three parts are remaining. We write this as . Thus,  represents the remaining parts of the chocolate. Here, 3 is called the numerator and 5 is called the denominator. In general, the digit that lies above the vertical line of a fraction is called the numerator and the digit that lies below it is called the denominator.

A fraction is a number of the form , where a is the numerator and b is the denominator. We can also write a whole number in the form of a fraction. For example, 5 can be written as .

Percents

Consider a shopkeeper who sells fruits. If he says that “25% of the fruits are sold”, what does it mean? It means that 25 fruits out of every 100 fruits are sold.

The word percent means “per hundred.” The notation for percent is %. Thus, 25% means “25 per hundred” or “25 out of 100”. We write this as . Note that this is a fraction. From this, we can see that percent can be written as a fraction. Also, we can see that when we write the percent as fraction, the numerator is the number that appears along with the % symbol and the denominator is 100.

Similarly, a fraction can be written as percent also. Let us check how this works.

Example 1

Question: Convert the fraction  to percent notation.

Solution:

We know that when we write the fraction equivalent of a percent, the denominator is 100. So, first let us make the denominator of the given fraction 100. For this, we multiply the denominator by 5.

But this will change the value of the fraction. To avoid this, we multiply the numerator also with the same number.

We know that  means “15 per hundred”. This is equivalent to 15%. The percent is obtained by writing the numerator and putting the % symbol to the right of the number. Therefore, the fraction  is equivalent to 15%.

Decimals

If the cost of a pencil is \$2.75, we say that the cost of the pencil is marked in decimal notation. A decimal notation has an integer part, a decimal point, and a decimal part.

Every decimal notation contains a decimal point. The number of digits after the decimal point denotes the number of decimal places in the notation. Then, 2.75 contains two decimal places.

The part of the notation that lies to the left of the decimal place is called the integer part and the part of the notation that lies to the right of the decimal point is called the decimal part.

The first position to the left of the decimal place is the units place, the second position to the left of the decimal place is the tens place, and so on. Then, the value of 2 in 2.75 is equal to  or 2. The first position to the right of the decimal point is the  place, the second position after the decimal point is the  place, and so on. Then, .75 is . When we convert a decimal to a fraction, we need not follow this method.

Consider the decimal notation 0.75. We can see that there are two decimal places after the decimal point. Then, the denominator of the equivalent fraction will be 100. Remove the decimal point of .75 and write 75 as the numerator. We get . Thus for the decimal notation 0.75, the decimal part .75 is equivalent to .

Again, we ended up with a fraction. This means that a number in decimal notation can be written as a fraction. Now, let us check whether a fraction can be written in decimal notation.

Example: 2

Question: Write  in decimal notation.

Solution:

First, check whether the denominator of the given fraction is a multiple of 10. If yes, count the number of 0’s in the denominator. We can see that for the given fraction, the denominator is 100. There are two 0’s in the denominator. Then, our decimal notation must contain two decimal places. Now, start from the right end of the numerator and count two digits to the left. After the second digit, we put a decimal point.

The decimal notation of the fraction  is .30. The decimal notation has two decimal places and no integer part. The 0’s that lie after the last nonzero digit to the right of the decimal point are of no value. So, we can omit those 0’s. Now, the answer is .3. But sometimes we may miss the decimal point in “.3” and read it as 3. This happens because there is no integer part in the notation. To avoid this confusion, we place a 0 to the left of the decimal point. Then the answer is 0.3. It is important to note that we put this 0 only for the decimal notations that have no integer part. Therefore, the decimal notation of  is 0.3.

For the fraction , there are two 0’s in the denominator. But when we look at the numerator, there is only one digit. This means that we cannot count two digits from the right and put the decimal point. In such situations, we should put a 0 on the left of 7 and put the decimal point on the left of that 0. Then, we get .07.

How will we convert a fraction whose denominator is not a multiple of 10? In such cases, we have to divide the numerator by the denominator to obtain the decimal notation.

Ratios

In a class, there are 21 boys and 20 girls. Then, we say that the ratio of the number of boys to the number of girls is 21:20. We can express the ratio of 21 to 20 as 21 : 20 or . Thus, a ratio can be represented as a fraction.

A ratio is the relative magnitudes of two quantities. It can be expressed as a quotient. This is the reason why we are able to write a ratio as a fraction. Generally, a : b is read as “the ratio of a to b.”

Practice Problems

1. Express the shaded portion as a fraction. Convert the fraction to decimal notation.

2. Convert  to percent notation.

3. Express 0.171 as fraction.

4. Write 33% as a fraction.

5. Write the statement “the ratio of 4 to 13” as ratio and as fraction.