Fractions, Percents, Decimals, Ratios
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 .
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.
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.
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.
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.”