Fractions and Decimals

Both fractions and decimals are ways of representing a number. Fractions and decimals represent the relationship of part to whole. These concepts are greatly important when precision is needed in any given mathematical value. It is used in various fields like science, engineering, finance, and everyday life.  In this article, you will understand these concepts in a detailed manner.

1.0What are Fractions and Decimals?

Fractions and decimals are two ways of representing numbers. Fractions are written in the form of p/q, where q≠0, while decimals are a combination of the whole number and fraction part. Take, for example, the number 0.5.

2.0Types of Fractions

3.0Simplifying Fractions

Simplifying fractions involves reducing the numerator and denominator to their smallest possible values without changing the fraction's value.

Steps for simplifying fractions:

1. Find the GCD (greatest common divisor) of the numerator and denominator.
2. Divide both the given numerator and denominator by the GCD.

Example: Simplify 8/12:

• GCD of 8 and 12 is 4.
• Divide both by 4: $\frac{(8÷4)}{(12÷4)}=2/3$.

Thus, 8/12 simplifies to 2/3.

4.0Conversion Between Fractions and Decimals

Understanding the conversion between fractions and decimals is crucial because sometimes it’s easier to work with one form over the other, depending on the situation.

Converting Fractions to Decimals

To convert a fraction to a decimal:

• Divide the numerator by the denominator.

Example: Convert 3/4 to decimal:

• 3 ÷ 4 = 0.75

Converting Decimals to Fractions

To convert any given decimal to a fraction:

1. Write the decimal in a fraction form with a denominator being a power of 10 (10, 100, 1000, etc.).
2. Simplify the fraction.

Example: Convert 0.6 to a fraction:

• 0.6 = 6/10
• Simplify 6/10 by dividing by $\frac{2}{\frac{3}{5}}$

Thus, 0.6 = 3/5.

5.0Operations on Fractions and Decimals

Performing operations on fractions and decimals involves addition, subtraction, multiplication, and division. Let's break them down separately for fractions and decimals.

Operations on Fractions

Addition and Subtraction

• Like Fractions: Add or subtract the given numerators and keep the denominator unchanged.
• Unlike Fractions: First, find a common denominator, then add or subtract the numerators.

Example: Add 2/5 and 1/3:

• Common denominator = 15
• $\left(\frac{1}{3}\right)=\frac{1}{15}$
• $\left(\frac{2}{5}\right)=\frac{6}{15}$
• $\frac{6}{15}+\frac{5}{15}=\frac{11}{15}$

Multiplication

• Multiply both the numerators and the denominators separately.

Example: Multiply $\frac{2}{3}\times \frac{4}{5}$:

$\frac{2}{3}\times \frac{4}{5}=\frac{2}{3}\times \frac{8}{15}$

Division

• Multiply the first fraction by the reciprocal (flip the numerator and denominator) of the second fraction.

Example: Divide $\frac{4}{3}÷\frac{2}{5}$:

$\frac{3}{4}\times \frac{5}{2}=\frac{(3\times 5)}{(4\times 2)}=\frac{15}{8}$

Operations on Decimals

Working with decimals involves slightly different techniques.

Adding and Subtracting Decimals

Adding and subtracting decimals is straightforward:

1. Line up the numbers by their decimal points.
2. Add or subtract as with whole numbers.
3. Place the decimal point directly below in the answer.

Example: Add 12.5 and 7.35:

• In the example, the first number has 1 digit after the decimal point, while the second number has 2. To equalise the number of digits in both numbers after the decimals, add a zero in the unit place of the first digit, that is, 12.50
• Now add the numbers vertically, aligning the decimal points; that is, 12.50 + 7.35 = 19.85.

Multiplying and Dividing Decimals

Multiplying decimals

1. Multiply the numbers, ignoring the decimal points.
2. Count the total decimal places in the given numbers.
3. Place the decimal point in the product accordingly.

Example: Multiply 2.4 × 1.5:

• First, let’s multiply the whole numbers themselves without decimals, like this: 24 × 15 = 360
• Now, the decimal places in both numbers are after 1 digit; hence, the total decimal places is two. So, the number will be 3.60.

Dividing decimals:

1. Move the decimal point in the divisor to make it a whole number.
2. Move the decimal point in the dividend the same number of places.
3. Divide as usual.

Example: Divide 6.4 ÷ 0.8:

• In the example, 6.4/0.8
• Here, note that the decimals are after one place in both the numbers; hence, multiply both the numbers by 10, that is, $\frac{6.4\times 10}{0.8\times 10}$
• After further division 64/8 = 8

Understanding adding and subtracting decimals, as well as multiplying and dividing decimals is vital for handling everyday math problems efficiently.

6.0Common Mistakes to Avoid

When working with fractions and decimals, it’s easy to make mistakes if you're not careful. Here are common pitfalls:

• Ignoring simplification: Always simplify fractions when possible.
• Not aligning decimal points: When adding or subtracting decimals, align decimal points precisely.
• Wrong reciprocal: In fraction division, flipping the wrong fraction leads to incorrect results.
• Decimal misplacement: Especially while multiplying or dividing, carefully count the decimal places.

7.0Why Understanding Fractions and Decimals Matters

Being skilled at working with fractions and decimals isn't just for school. It has practical applications in:

• Financial literacy (managing money, interest rates)
• Cooking (ingredient measurements)
• Construction (measurements and scaling)
• Science (data analysis, measurements)

In almost every career, from engineering to business, a solid grasp of these concepts is a valuable asset.

8.0Conclusion

Fractions and decimals form the foundation of many mathematical concepts and daily life applications. By understanding the types of fractions, mastering simplifying fractions, becoming proficient in conversion between fractions and decimals, and practicing adding and subtracting decimals as well as multiplying and dividing decimals, you lay the groundwork for mathematical success.