Quantitative Aptitude
Decimals and Fractions
Master decimals and fractions with this comprehensive guide! Learn core concepts, conversions, comparisons, and operations with clear examples and practical tips to boost your aptitude skills.
Decimals and fractions are fundamental concepts in mathematics that appear in various competitive exams, daily life, and technical fields. In this guide, we’ll cover the essentials of decimals and fractions, their inter-conversions, comparisons, operations, and practical applications. By the end, you’ll feel confident in handling aptitude questions involving these concepts.
Simplification
What is decimals & fractions
Decimals and fractions are essential mathematical tools for representing parts of a whole. Decimals use a point to divide whole numbers from fractional parts (e.g., 0.75), while fractions use a numerator and denominator separated by a line (e.g., ¾). Decimals are based on powers of ten, making them easy for calculations in finance and science. Fractions, however, show exact parts of a whole and come in forms like proper and improper fractions. Both formats are interchangeable, allowing us to compare, measure, and solve problems accurately across various applications.”
Understanding Decimals
What is a Decimal?
A decimal represents a part of a whole, expressed in a format with a decimal point, e.g., 0.75. The decimal system is based on powers of 10, making it simple to read and write.
Types of Decimals
- Terminating Decimals: These decimals come to an end, e.g., 0.5, 2.75.
- Recurring/Repeating Decimals: These have one or more digits repeating indefinitely, e.g., 0.333… or 1.666….
- Non-terminating Non-repeating Decimals: Decimals that go on forever without repeating, e.g., π = 3.14159…
Place Value in Decimals
The digits to the right of the decimal point hold values of tenths, hundredths, thousandths, etc. For example, in 3.456:
- 6 is in the thousandths place (0.006).
- 4 is in the tenths place (0.4),
- 5 is in the hundredths place (0.05),
Converting Decimals to Fractions
To convert a decimal to a fraction:
- Identify the place value of the last digit.
- Write the decimal as a fraction with the respective place value as the denominator. for Example: 0.75 = 75/100 = 3/4 after simplifying.
Understanding Fractions
What is a Fraction?
A fraction represents a part of a whole, expressed as two numbers separated by a line. For instance, in 3/4, 3 is the numerator, and 4 is the denominator.
Types of Fractions
Fractions are categorized based on the relationship between their numerator and denominator into three main types—proper fractions (numerator smaller than the denominator), improper fractions (numerator equal to or larger than the denominator), and mixed fractions (a whole number combined with a proper fraction).
- Proper Fractions: The numerator is smaller than the denominator (e.g., 3/4).
- Improper Fractions: The numerator is larger than the denominator (e.g., 7/4).
- Mixed Fractions: A whole number combined with a proper fraction [e.g., 1x(1/2)].
Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number). This division gives a decimal representation of the fraction.
- Example: 3/4 = 3 ÷ 4 = 0.75.
Simplifying Fractions
Reduce fractions to their simplest form by dividing both the numerator and the denominator by their Greatest Common Divisor (GCD).
- Example: (15/20) = (3/4).
Comparing
Comparing Decimals and Fractions
Understanding how to compare decimals and fractions is essential in solving mathematical problems, especially in aptitude tests and everyday applications. Decimals and fractions both represent parts of a whole but do so differently, which can make comparisons slightly challenging. However, with the right approach, comparing these values becomes straightforward.
Which is Greater?
To determine which of two numbers is larger, first identify if both numbers are in decimal or fraction form. If both are decimals, comparing them is simple—begin by looking at the digits from left to right. For example, to compare 0.75 and 0.8, observe that in the tenths place, 8 is greater than 7, so 0.8 is larger than 0.75.
When comparing fractions, you have two options:
Convert Fractions to Decimals: One way to compare fractions is to convert them into decimal form. This can be done by dividing the numerator by the denominator. For example, to compare 3/4 and 5/8, first divide: 3/4=0.75 and 5/8=0.625, making it clear that 3/4 is greater than 5/8.
Use Cross-Multiplication: This technique avoids conversion to decimals. Cross-multiply the fractions, multiplying the numerator of one fraction by the denominator of the other and vice versa. The larger product indicates the larger fraction. For instance, comparing 3/4 and 2/3:
- 3×3=9
- 4×2=8 Since 9 is greater than 8, 3/4 is greater than 2/3.
Common Conversion Techniques
In some cases, it may be helpful to convert both numbers to the same form for an easier comparison. Here are a few techniques:
- Converting Fractions to Decimals: To convert a fraction into a decimal, divide the numerator by the denominator.
- Converting Decimals to Fractions: For decimals, write the number over its place value. For example, 0.25 becomes 25/100, which can be simplified to 1/4.
- Using a Common Denominator: For fractions with different denominators, finding a common denominator allows for direct comparison. For instance, to compare 1/2 and 3/4, convert 1/2 to 2/4 so both fractions have the same denominator, making it easy to see that 3/4 is larger.
Operations Involving Decimals and Fractions
Addition and Subtraction
Decimals:
- Align the Decimal Points: When adding or subtracting decimals, it’s essential to line up the decimal points vertically. This ensures each digit aligns with its corresponding place value (ones, tenths, hundredths, etc.).
- Fill in Zeros (if necessary): If the numbers have different decimal places, you can add trailing zeros to make them the same length, which makes the calculation easier to visualize.
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Perform the Operation: Add or subtract as you would with whole numbers, moving left to right. Place the decimal point in the result directly below the decimal points of the numbers above. Example:
3.25 + 1.7 = 3.25 + 1.70 = 4.95.
Fractions:
- Find a Common Denominator: To add or subtract fractions, both fractions need the same denominator. Multiply the denominators (if necessary) to find the least common denominator (LCD).
- Adjust the Numerators: Once the denominators are the same, adjust the numerators to correspond to the new denominator.
- Add or Subtract: Add or subtract the numerators as instructed, keeping the denominator the same.
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Simplify: If possible, simplify the fraction by dividing the numerator and denominator by their GCD. Example:
{(3/4)+(1/2)} = {(3/4)+(2/4)} = 5/4 or 1*(1/4).
Multiplication
Decimals:
- Multiply as Whole Numbers: Temporarily ignore the decimal points and multiply the numbers as if they were integers.
- Count Decimal Places: After multiplication, count the total number of decimal places in both original numbers.
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Place the Decimal: In the result, place the decimal point so that it has the same total number of decimal places as the sum counted. Example:
2.3×1.5 = 23×15 = 345, with two decimal places in total, so the result is 3.45.
Fractions:
- Multiply Numerators and Denominators: Multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.
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Simplify: If possible, reduce the fraction by dividing both the numerator and denominator by their GCD. Example:
3/4 × 2/5 = 6/20 = (3/10).
Division
Decimals:
- Eliminate the Decimal (if possible): Move the decimal points in both the divisor and the dividend to make the divisor a whole number (shift the decimal to the right in both numbers).
- Divide Normally: Perform the division as you would with whole numbers.
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Place the Decimal: In the result, place the decimal directly above where it appears in the dividend. Example:
4.5÷1.5. Shift the decimal one place in both to get 45÷15=3.
Fractions:
- Invert the Second Fraction: To divide by a fraction, flip the second fraction to find its reciprocal.
- Multiply: After flipping, multiply the first fraction by this reciprocal, as in multiplication.
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Simplify: Simplify the resulting fraction if possible. Example:
3/4 ÷ 2/5 = 3/4 × 5/2 =15/8 = 1*(7/8).
Application
Real-World Applications
Decimals and fractions are deeply integrated into everyday activities and professional fields, making them vital for precise calculations, measurements, and data interpretations.
Financial Calculations (Currency)
In financial transactions, decimals are essential to represent parts of currency, especially in situations that require precision, like banking, trading, and budgeting. For instance:
- Currency: Most currencies are divided into smaller units (e.g., cents in dollars or paise in rupees), expressed as decimals. An item costing $2.50 uses the decimal to show both whole dollars and fractional cents, making it easy to add, subtract, or calculate interest.
- Interest Calculations: In finance, interest rates are often expressed as decimals, like 5.75% or 3.25%, which are critical for calculating savings, loans, and investments.
- Stock Market: Decimal values represent stock prices, returns, and percentages, helping investors make informed trading decisions.
Measurements (Construction, Cooking)
Both fractions and decimals are indispensable in fields that involve precise measurements, such as construction and cooking:
- Cooking and Baking: Recipes often use fractions to specify ingredient amounts, like ½ cup or ¾ teaspoon. Similarly, decimals are used in cooking temperatures or time (e.g., 1.5 hours). Both ensure precise portions and consistent results, which are particularly important in professional cooking or baking.
- Construction: In construction, fractions and decimals are used to represent measurements for building materials, like wood or metal lengths, widths, and thicknesses. For example, 3½ inches or 1.75 feet provide specific values that allow for accurate cutting, aligning, and assembling.
Data Precision (Science & Engineering)
Decimals and fractions are crucial in fields requiring high data accuracy, such as science, engineering, and technology:
- Scientific Measurements: Decimals allow for precise measurements of quantities like mass, volume, or temperature (e.g., 9.81 m/s² for gravitational acceleration). Such accuracy is essential for consistency in experiments, formulas, and results.
- Engineering Calculations: Engineers use fractions and decimals for measurements, tolerances, and parts dimensions to ensure accurate designs. For instance, a tolerance of ±0.005 inches might be required for machine parts to fit perfectly.
- Digital Data: Decimals are used to express values in technology, like measurements in nanometers for microchip manufacturing or data percentages in statistics (e.g., accuracy rates in algorithms).
Decimals & Fractions Mastery Challenge
Strengthen your grip on Decimals and Fractions with this step-by-step three-stage quiz that builds accuracy and confidence!
Conclusion
Mastering decimals and fractions equips you with the aptitude skills for exams and real-world problem-solving. Consistent practice with these principles will solidify your understanding and enhance your efficiency in handling related questions.
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