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Numbers are the foundation of mathematics, and beyond whole numbers, two important ways to express parts of a whole are through fractions and decimals. Fractions and decimals allow us to represent quantities that are not whole, like half a pizza or 0.75 liters of milk.
Understanding these concepts deeply is essential not only for daily life-such as handling money, measuring lengths, or calculating discounts-but also for competitive exams where quick and accurate number manipulation is tested.
In this section, we'll explore what fractions and decimals are, how to convert between them, perform operations like addition, subtraction, multiplication, and division, and apply these skills to practical problems involving percentages, simple interest, and ratios.
Let's begin our journey into the fascinating world of fractions and decimals!
A fraction represents a part of a whole. It consists of two numbers separated by a line. The number on top is called the numerator and tells how many parts we have, while the number on the bottom is the denominator and tells into how many equal parts the whole is divided.
Fractions can be:
Visualizing fractions helps understand them better. Imagine a circle divided into equal parts. The shaded portion represents the fraction.
On the left, the circle is divided into 4 parts with 1 part shaded, representing the fraction \( \frac{1}{4} \). On the right, 3 parts out of 4 are shaded representing \( \frac{3}{4} \). This visual shows how the numerator relates to the portion shaded and the denominator to total parts.
Decimal numbers show parts of a whole using a decimal point. Unlike fractions, which use two numbers, decimals write numbers with digits after a dot to indicate smaller values.
The position of each digit after the decimal point has a specific place value:
Understanding place values helps convert decimals to fractions and perform operations correctly.
In the number 3.47:
This means \( 3.47 = 3 + \frac{4}{10} + \frac{7}{100} \).
Since fractions and decimals both represent parts of a whole, it is important to know how to switch between these forms.
graph TD A[Fraction \(\frac{a}{b}\)] --> B[Divide numerator by denominator \(a \div b\)] B --> C[Result is the Decimal] D[Decimal] --> E[Write decimal number without point over \(10^{\text{number of decimal places}}\)] E --> F[Simplify the fraction]For example, to convert \( \frac{3}{4} \) to decimal, divide 3 by 4 to get 0.75. To convert 0.75 to fraction, write \( \frac{75}{100} \) and simplify to \( \frac{3}{4} \).
Basic arithmetic operations on fractions follow specific rules. When adding or subtracting, denominators need to be the same.
| Operation | Rule / Formula | Explanation |
|---|---|---|
| Addition | \( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \) | Find LCM of denominators \(b\) and \(d\), rewrite fractions, add numerators |
| Subtraction | \( \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \) | Same method as addition; subtract numerators after equivalent denominators |
| Multiplication | \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \) | Multiply numerators and denominators directly |
| Division | \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \) | Invert second fraction and multiply |
Always simplify the result by dividing numerator and denominator by their greatest common divisor (GCD).
Decimal operations are similar to whole numbers but require careful alignment of the decimal point.
Estimating results helps check your answers quickly.
To compare fractions and decimals, convert both to a common format.
| Pair | Comparison Strategy | Result |
|---|---|---|
| \( \frac{3}{5} \) and 0.6 | Convert \( \frac{3}{5} = 0.6 \) | Equal |
| \( \frac{7}{8} \) and 0.85 | Convert \( \frac{7}{8} = 0.875 \), compare decimals | \( \frac{7}{8} > 0.85 \) |
| \( \frac{2}{3} \) and \( \frac{3}{4} \) | Cross multiply: \(2 \times 4 = 8\), \(3 \times 3 = 9\) | \( \frac{3}{4} > \frac{2}{3} \) |
A percentage is a fraction out of 100. It expresses how many parts per hundred something is.
For example, 25% means \( \frac{25}{100} \) or one-quarter.
To find the percentage of a quantity, use:
Applying discounts in shopping involves reducing the price by a certain percentage.
Example: A shirt priced at INR 1500 is discounted by INR 300. Find discount percent.
Discount percent \( = \frac{300}{1500} \times 100 = 20\% \).
Simple Interest (SI) is the interest calculated on the original principal amount over time.
Key components are:
Example: Calculate the interest on INR 10,000 at 5% per annum for 3 years.
A ratio compares two quantities, showing their relative sizes. Ratios are often written as \( a:b \).
A proportion states that two ratios are equal, like \( \frac{a}{b} = \frac{c}{d} \).
Mixture problems involve combining quantities with different properties and finding the resulting ratio or value. These problems are common in competitive exams.
Example: Mixing 2 liters of milk with 3 liters of water gives the ratio milk:water = 2:3.
Step 1: Find the least common denominator (LCD) of 3 and 4, which is 12.
Step 2: Convert each fraction to an equivalent fraction with 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 the numerators: \(8 + 9 = 17\), so sum is \( \frac{17}{12} \).
Step 4: Since \( \frac{17}{12} \) is improper, convert it to mixed number:
\( \frac{17}{12} = 1 \frac{5}{12} \) (because \(12 \times 1 = 12\) and \(17 - 12 = 5\)).
Answer: \( 1 \frac{5}{12} \).
Step 1: Note that 0.375 has 3 decimal places, so write as:
\( \frac{375}{1000} \)
Step 2: Find the greatest common divisor (GCD) of 375 and 1000, which is 125.
Step 3: Divide numerator and denominator by 125:
\( \frac{375 \div 125}{1000 \div 125} = \frac{3}{8} \).
Answer: \( \frac{3}{8} \).
Step 1: Use formula for percentage:
\( \text{Discount \%} = \frac{\text{Discount}}{\text{Marked Price}} \times 100 \)
Step 2: Substitute values:
\( = \frac{2250}{15000} \times 100 = 0.15 \times 100 = 15\% \)
Answer: Discount percentage is 15%.
Step 1: Use formula \( SI = \frac{P \times R \times T}{100} \).
Step 2: Substitute values \(P=10000\), \(R=5\), \(T=3\):
\( SI = \frac{10000 \times 5 \times 3}{100} = \frac{150000}{100} = 1500 \)
Answer: The simple interest is INR 1,500.
Step 1: Write down quantities: Milk = 2 liters, Water = 3 liters.
Step 2: Form ratio milk:water = 2:3.
Answer: The ratio of milk to water is 2:3.
When to use: Compare two fractions like \( \frac{3}{7} \) and \( \frac{4}{9} \) during time-limited exams.
When to use: To get exact fractions from decimals with 3 or fewer digits after the point.
When to use: Adding or subtracting fractions with unlike denominators.
When to use: To avoid errors in decimal operations, especially in competitive tests.
When to use: Calculations involving principal, rate, and time for interest.
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