When we share a pizza among friends or divide a bar of chocolate, we need numbers that can represent parts of a whole. These numbers are called fractions. A fraction is a number that represents a part of a whole. For example, if a pizza is cut into 8 equal slices and you eat 3 slices, you have eaten 3 out of 8 parts, which is written as the fraction 3/8.
A fraction has two parts: the numerator, which is the number above the line, and the denominator, which is the number below the line. In the fraction 3/8, the numerator is 3 and the denominator is 8. The denominator tells us into how many equal parts the whole is divided, and the numerator tells us how many of those parts we are considering. Fractions are used in cooking (half a cup of sugar), measurement (half a metre of cloth) and time (a quarter of an hour).
In this chapter, we learn about proper, improper and mixed fractions, how to compare fractions, how to convert between improper and mixed fractions, and how to add, subtract, multiply and divide fractions. We also learn about equivalent fractions, which are fractions that represent the same part of a whole even though their numerators and denominators look different.
A fraction in which the numerator is less than the denominator is called a proper fraction. Proper fractions are always less than 1. Examples are 1/2, 3/4 and 5/8.
A fraction in which the numerator is greater than or equal to the denominator is called an improper fraction. Improper fractions are always greater than or equal to 1. Examples are 5/4, 7/3 and 9/9.
A mixed fraction is a combination of a whole number and a proper fraction. For example, 2 and 3/5 means 2 whole parts plus 3/5 of a part. A mixed fraction is also called a mixed number.
A fraction whose numerator is 1 is called a unit fraction. Examples are 1/2, 1/3 and 1/10.
Fractions with the same denominators are called like fractions. Fractions with different denominators are called unlike fractions. For example, 2/5 and 3/5 are like fractions, while 2/5 and 3/7 are unlike fractions.
To convert an improper fraction to a mixed fraction, divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the numerator of the fractional part, and the divisor becomes the denominator.
For example, to convert 7/3: 7 divided by 3 gives quotient 2 and remainder 1. So 7/3 = 2 and 1/3.
To convert a mixed fraction to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator.
For example, to convert 2 and 3/5: (2 x 5) + 3 = 10 + 3 = 13, so 2 and 3/5 = 13/5.
Fractions that represent the same part of a whole are called equivalent fractions. To obtain equivalent fractions, we multiply or divide the numerator and denominator by the same non-zero number.
A fraction is in its lowest terms (simplest form) when the numerator and denominator have no common factor other than 1. For example, 3/4 is in lowest terms, but 6/8 is not because both can be divided by 2 to get 3/4.
If the denominators are the same, the fraction with the greater numerator is greater. For example, 5/8 > 3/8.
If the denominators are different, we first make the denominators the same by finding the LCM of the denominators, then compare the numerators.
For example, to compare 2/3 and 3/4: LCM of 3 and 4 is 12. 2/3 = 8/12 and 3/4 = 9/12. Since 9/12 > 8/12, we have 3/4 > 2/3.
To compare mixed fractions, first compare the whole number parts. The mixed fraction with the greater whole number is greater. If the whole numbers are equal, compare the fractional parts.
Among unit fractions, the one with the smaller denominator is greater. For example, 1/2 > 1/3 > 1/6.
To add or subtract like fractions, add or subtract the numerators and keep the denominator the same.
To add or subtract unlike fractions, first convert them to like fractions using the LCM of the denominators, then add or subtract.
Convert the mixed fractions to improper fractions, then add or subtract. Alternatively, add the whole numbers and fractional parts separately, carrying over when needed.
To multiply fractions, multiply the numerators together and multiply the denominators together.
A fraction as an operator: To find a part of a quantity, we multiply the quantity by the fraction. For example, 1/2 of 10 = 1/2 x 10 = 5.
To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
Fractions appear in many everyday situations: half a dozen eggs (6 eggs), a quarter of an hour (15 minutes), half a litre of milk, and three-quarters of a metre of cloth. Understanding fractions helps us divide, share and measure accurately.
| Type of Fraction | Definition | Examples |
|---|---|---|
| Proper fraction | Numerator less than denominator | 1/2, 3/4 |
| Improper fraction | Numerator greater than or equal to denominator | 5/4, 7/3 |
| Mixed fraction | Whole number plus proper fraction | 2 and 1/3 |
| Unit fraction | Numerator is 1 | 1/2, 1/10 |
| Equivalent fractions | Different fractions, same value | 1/2, 2/4, 3/6 |
| Operation | Rule | Example |
|---|---|---|
| Addition of like fractions | Add numerators, keep denominator | 3/8 + 2/8 = 5/8 |
| Subtraction of like fractions | Subtract numerators, keep denominator | 7/9 - 4/9 = 3/9 |
| Multiplication | Multiply numerators and denominators | 2/3 x 4/5 = 8/15 |
| Division | Multiply by reciprocal | 3/4 divided by 2/5 = 15/8 |
Fractions help us represent parts of a whole and handle everyday situations involving sharing and measurement. We learnt about proper, improper, mixed and unit fractions, how to find equivalent fractions and compare them, and how to add, subtract, multiply and divide fractions. Mastery of fractions is essential for decimals, ratios and algebra in the coming chapters.