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Fractions and decimals are two important ways to represent parts of a whole. Imagine you have a chocolate bar divided into equal pieces; if you eat some of those pieces, fractions and decimals help you describe exactly how much you ate.
In daily life, measurements such as lengths in meters or weights in kilograms often are not whole numbers. For example, a piece of fabric might be 2.5 meters long. Similarly, in financial transactions, prices might include paise and rupees, like Rs.150.75, which is seventy-five paise more than Rs.150.
Understanding fractions and decimals lets you work accurately with these parts, whether you are measuring distances, sharing things, or handling money.
A fraction represents a part of a whole or a collection. It is written as numerator over denominator, separated by a line, like this: \(\frac{a}{b}\).
Numerator (the top number) shows how many parts we have. Denominator (the bottom number) shows into how many equal parts the whole is divided.
For example, the fraction \(\frac{3}{4}\) means we have 3 parts out of 4 equal parts of a whole.
Fractions come in three types:
A decimal number is another way to represent parts of a whole using place values to the right of the decimal point. Decimals use digits 0-9 and a decimal point "." to separate whole numbers from fractions of a whole.
The first digit after the decimal point is called the tenths place, the second digit is the hundredths place, the third is thousandths, and so on. Each place value is 10 times smaller than the previous one.
Decimals are closely related to fractions. For instance, 0.5 means \(\frac{5}{10}\) which simplifies to \(\frac{1}{2}\), and 0.25 means \(\frac{25}{100} = \frac{1}{4}\).
The four basic arithmetic operations-addition, subtraction, multiplication, and division-work slightly differently for fractions and decimals. Let's learn the step-by-step rules and procedures for each.
graph TD A[Adding Fractions] --> B[Find LCM of denominators] B --> C[Adjust numerators to common denominator] C --> D[Add numerators] D --> E[Write sum over common denominator] E --> F[Simplify fraction if possible] G[Adding Decimals] --> H[Align decimal points vertically] H --> I[Add digits column-wise from right to left] I --> J[Place decimal point in the answer directly below the others] J --> K[Write the sum] subgraph Fractions Addition A --> F end subgraph Decimals Addition G --> K end
Operations with fractions always depend on denominators, while decimals require careful alignment of decimal points.
Step 1: Find the Least Common Multiple (LCM) of the denominators 3 and 4.
Multiples of 3: 3, 6, 9, 12, 15...
Multiples of 4: 4, 8, 12, 16...
LCM = 12
Step 2: Convert both fractions to have denominator 12.
\(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)
\(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)
Step 3: Add adjusted numerators.
\(\frac{8}{12} + \frac{9}{12} = \frac{8 + 9}{12} = \frac{17}{12}\)
Step 4: Convert to mixed fraction (optional).
\(\frac{17}{12} = 1 \frac{5}{12}\)
Answer: \(1 \frac{5}{12}\)
Step 1: Ignore decimals and multiply 5 by 2.
\(5 \times 2 = 10\)
Step 2: Count total decimal places in factors.
0.5 has 1 decimal place; 0.2 has 1 decimal place; total = 2.
Step 3: Place decimal point in product so it has 2 decimal places.
Starting with 10, place decimal 2 places from right: 0.10
Answer: \(0.10 = 0.1\)
Step 1: Write decimal as fraction with denominator 10 raised to number of decimal digits.
Decimal \(0.625\) has 3 digits after decimal.
\(0.625 = \frac{625}{1000}\)
Step 2: Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD).
GCD of 625 and 1000 is 125.
\(\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}\)
Answer: \(0.625 = \frac{5}{8}\)
Step 1: Calculate profit.
Profit = Selling Price - Cost Price = Rs.500 - Rs.400 = Rs.100
Step 2: Use percentage profit formula.
\[ \text{Profit \%} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100 = \left( \frac{100}{400} \right) \times 100 \]
Step 3: Calculate.
\(\frac{100}{400} = 0.25\), so, \(0.25 \times 100 = 25\%\)
Answer: Percentage profit is 25%
Step 1: Use the simple interest formula:
\[ SI = \frac{P \times R \times T}{100} \]
Step 2: Substitute the known values:
\(P = 10000\), \(R = 7.5\), \(T = 1.5\)
\[ SI = \frac{10000 \times 7.5 \times 1.5}{100} \]
Step 3: Calculate step-by-step:
\(7.5 \times 1.5 = 11.25\)
\[ SI = \frac{10000 \times 11.25}{100} = \frac{112500}{100} = 1125 \]
Answer: Simple Interest = Rs.1125
When to use: When denominators are unlike and LCM is complex or time-consuming.
When to use: For any decimal addition or subtraction problem to avoid misplacement of digits.
When to use: To simplify fraction addition or subtraction by matching denominators.
When to use: When solving profit/loss or discount problems quickly and efficiently.
When to use: To avoid mistakes in decimal point placement during multiplication.
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