Fractions, Decimals and Percentages

Introduction to Fractions, Decimals and Percentages

Fractions, decimals, and percentages are three different but closely related ways of representing parts of a whole. Understanding how to work with each of them, and how to convert between them, is an essential mathematical skill used constantly in everyday life, from sharing food and money to understanding statistics, discounts, and measurements. This topic explores what fractions, decimals, and percentages mean, how they relate to one another, and how to perform basic operations with each.

Meaning of a Fraction

A fraction represents a part of a whole and is written in the form a/b, where a is called the numerator (the number of parts being considered) and b is called the denominator (the total number of equal parts the whole is divided into). For example, in the fraction 3/4, the numerator 3 tells us we have three parts, and the denominator 4 tells us the whole was divided into four equal parts.

Types of Fractions

Fractions can be classified into several types. A proper fraction has a numerator smaller than its denominator (such as 2/5), representing a value less than one whole. An improper fraction has a numerator equal to or larger than its denominator (such as 7/4), representing a value equal to or greater than one whole. A mixed number combines a whole number with a proper fraction (such as 1¾), and can be converted to and from an improper fraction. Equivalent fractions represent the same value even though they look different, such as 1/2, 2/4, and 3/6, all of which represent exactly the same amount.

Simplifying Fractions

A fraction is in its simplest (lowest) form when the numerator and denominator have no common factor other than 1. To simplify a fraction, both the numerator and denominator are divided by their highest common factor (HCF). For example, to simplify 12/18, since the HCF of 12 and 18 is 6, dividing both by 6 gives 2/3, which is the simplest form.

Operations with Fractions

To add or subtract fractions, they must first have the same denominator (a common denominator); if they do not, equivalent fractions must be found first. For example, to add 1/3 + 1/4, the common denominator is 12, so the fractions become 4/12 + 3/12 = 7/12. To multiply fractions, simply multiply the numerators together and the denominators together: 2/3 × 3/5 = 6/15, which simplifies to 2/5. To divide fractions, multiply the first fraction by the reciprocal (upside-down version) of the second fraction: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12, which simplifies to 5/6.

Meaning of a Decimal

A decimal is another way of representing a fraction, using place value based on powers of ten, separated from the whole number part by a decimal point. Digits to the right of the decimal point represent tenths, hundredths, thousandths, and so on. For example, in the decimal 4.56, the digit 5 represents 5 tenths (5/10) and the digit 6 represents 6 hundredths (6/100), so 4.56 means 4 whole units plus 56 hundredths.

Converting Between Fractions and Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75. To convert a decimal to a fraction, write the decimal digits over the appropriate power of ten based on place value, then simplify. For example, 0.6 = 6/10, which simplifies to 3/5.

Operations with Decimals

When adding or subtracting decimals, the decimal points must be lined up vertically to ensure digits of the same place value are combined correctly. For example, 3.45 + 2.7 is calculated by aligning as 3.45 + 2.70 = 6.15. When multiplying decimals, multiply as if there were no decimal points, then count the total number of decimal places in both numbers being multiplied and place the decimal point accordingly in the answer. When dividing decimals, it is often easiest to first convert the divisor into a whole number by multiplying both numbers by an appropriate power of ten.

Meaning of a Percentage

A percentage represents a number as a fraction of 100, using the percent symbol (%). The word "percent" literally means "per hundred." For example, 45% means 45 out of every 100, which can also be written as the fraction 45/100 or the decimal 0.45. Percentages are widely used to express discounts, interest rates, exam scores, statistics, and proportions in a way that is easy to compare and understand.

Converting Between Percentages, Fractions, and Decimals

To convert a percentage to a fraction, write it over 100 and simplify: 60% = 60/100 = 3/5. To convert a percentage to a decimal, divide by 100 (or move the decimal point two places to the left): 60% = 0.6. To convert a fraction to a percentage, first convert it to a decimal, then multiply by 100: 3/5 = 0.6 = 60%. To convert a decimal to a percentage, multiply by 100: 0.6 = 60%.

Finding a Percentage of a Quantity

To find a percentage of a given quantity, convert the percentage to a decimal or fraction, then multiply by the quantity. For example, to find 25% of 80, convert 25% to 0.25, then calculate 0.25 × 80 = 20. This type of calculation is common when working out discounts in shops, calculating tax, or determining scores on a test.

Percentage Increase and Decrease

To calculate a percentage increase, find the amount of increase, divide it by the original amount, then multiply by 100. For example, if a price rises from ₦200 to ₦250, the increase is ₦50, so the percentage increase is (50/200) × 100 = 25%. Similarly, for a percentage decrease, find the amount of decrease, divide by the original amount, and multiply by 100. These calculations are useful for understanding price changes, population growth or decline, and performance improvements.

Comparing Fractions, Decimals and Percentages

Because fractions, decimals, and percentages can all represent the same value in different forms, it is often useful to convert them to a common form (usually decimals) when comparing them. For example, to compare 3/5, 0.55, and 58%, converting all three to decimals gives 0.6, 0.55, and 0.58, making it clear that 3/5 is the largest and 0.55 is the smallest.

Real-Life Applications

Fractions, decimals, and percentages appear constantly in daily life: sharing food or money among family members (fractions), measuring quantities in cooking or construction (decimals), calculating discounts during sales, understanding interest rates on savings or loans, interpreting statistics in news reports, and understanding exam or match results (percentages). Being confident with all three forms, and knowing how to convert between them, makes it much easier to interpret and solve real-world problems involving quantities and proportions.

Summary

Fractions, decimals, and percentages are three interconnected ways of expressing parts of a whole. A fraction shows a numerator over a denominator, a decimal uses place value based on powers of ten, and a percentage expresses a value out of 100. Knowing how to convert between these forms, and how to add, subtract, multiply, and divide with each, is a fundamental mathematical skill that supports countless everyday tasks, from shopping and cooking to understanding statistics and financial information.

Practising for Mastery

Because fractions, decimals, and percentages appear so frequently in both examinations and daily life, regular practice is the best way to build speed and confidence. Useful strategies include memorising common fraction-decimal-percentage equivalents (such as 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, and 1/10 = 0.1 = 10%), practising mental conversions during everyday activities like shopping, and always checking whether an answer makes sense by estimating first. With consistent practice, students learn to move fluidly between these three representations, which is a skill that continues to be useful throughout secondary school mathematics and beyond.

Worked Examples

Example 1: Add 2/3 and 1/6. The common denominator is 6: 2/3 = 4/6, so 4/6 + 1/6 = 5/6.

Example 2: Convert 7/8 to a decimal and a percentage. 7 ÷ 8 = 0.875. Multiplying by 100 gives 87.5%. So 7/8 = 0.875 = 87.5%.

Example 3: Find 15% of 340. Convert 15% to a decimal: 0.15. Then 0.15 × 340 = 51.

Example 4: A trader's daily sales rose from ₦150,000 to ₦180,000. Find the percentage increase. Increase = 180,000 − 150,000 = 30,000. Percentage increase = (30,000/150,000) × 100 = 20%.

Example 5: Arrange 0.7, 3/4, and 68% in order from smallest to largest. Convert all to decimals: 0.7, 3/4 = 0.75, and 68% = 0.68. In order: 68% (0.68), 0.7, then 3/4 (0.75).

Student Exercise

Solve the following problems, showing all your working:

1. Add 3/5 and 1/4.
2. Simplify 24/36 to its lowest terms.
3. Convert 5/8 to a decimal.
4. Convert 0.45 to a percentage.
5. Convert 72% to a fraction in its lowest terms.
6. Find 30% of 250.
7. A shirt priced ₦1,200 is now sold at ₦1,500. Find the percentage increase in price.
8. Multiply 2/5 by 3/7.
9. Divide 3/4 by 1/2.
10. Arrange 0.6, 5/8, and 62% in ascending order.