Mastering Fractions in SHS Mathematics: A Comprehensive Guide

Fractions are fundamental to understanding mathematics and solving complex problems in various fields. For SHS students, grasping the concept of fractions is essential for excelling in both Core and Elective Mathematics. In this blog post, we’ll break down fractions, explore their applications, and provide tips and resources to help you master them effectively.

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What Are Fractions?

Fractions represent parts of a whole. They are written as two numbers separated by a line: the numerator (top number) indicates how many parts are considered, and the denominator (bottom number) shows the total number of equal parts.

Types of Fractions

1. Proper Fraction

• Definition: A fraction where the numerator is smaller than the denominator.
• Examples:
  $\frac{2}{5}, \frac{3}{7}, \frac{4}{9}$

2. Improper Fraction

• Definition: A fraction where the numerator is equal to or greater than the denominator.
• Examples:
  $\frac{7}{4}, \frac{9}{9}, \frac{11}{5}$

3. Mixed Fraction

• Definition: A combination of a whole number and a proper fraction.
• Examples:
  $1\frac{1}{2}, 3\frac{2}{5}, 5\frac{3}{4}$

4. Like Fractions

• Definition: Fractions with the same denominator.
• Examples:
  $\frac{2}{7}, \frac{5}{7}, \frac{6}{7}$

5. Unlike Fractions

• Definition: Fractions with different denominators.
• Examples:
  $\frac{3}{8}, \frac{5}{6}, \frac{7}{12}$

6. Equivalent Fractions

• Definition: Fractions that represent the same value but have different numerators and denominators.
• Examples:
  $\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$
  $\frac{3}{5} = \frac{6}{10} = \frac{9}{15}$

How to Check If Two Fractions Are Equivalent

1. Simplify both fractions:
Reduce them to their simplest forms. If they are the same, the fractions are equivalent.
Example:
$\frac{6}{9}$ and $\frac{2}{3}$

Simplify $\frac{6}{9}$: $\frac{6÷3}{9÷3}=\frac{2}{3}$
So $\frac{6}{9}=\frac{2}{3}$.

2. Cross-multiply:
If the cross-products are equal, the fractions are equivalent.
Example:
$\frac{4}{8}$ and $\frac{1}{2}$

Cross-multiply:
$4\times 2=8$, $8\times 1=8$.
Since $8=8$, the fractions are equivalent.

Sample Questions and Answers on Equivalent fractions

1. Determine which of the following pairs of fractions are equivalent:
   (a) $\frac{2}{5}$ and $\frac{4}{10}$
   (b) $\frac{8}{18}$ and $\frac{20}{45}$
   (c) $\frac{2}{11}$ and $\frac{13}{48}$

Solution:

(a) $\frac{2}{5}$ and $\frac{4}{10}$

1. Simplify $\frac{4}{10}$:
   $\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}$.
   Since $\frac{4}{10} = \frac{2}{5}$, they are equivalent.

(b) $\frac{8}{18}$ and $\frac{20}{45}$

1. Simplify $\frac{8}{18}$:
   $\frac{8}{18} = \frac{8 \div 2}{18 \div 2} = \frac{4}{9}$.

2. Simplify $\frac{20}{45}$:
   $\frac{20}{45} = \frac{20 \div 5}{45 \div 5} = \frac{4}{9}$
   Since both fractions simplify to $\frac{4}{9}$, they are equivalent.

(c) $\frac{2}{11}$ and $\frac{13}{48}$

1. Check using cross-multiplication:
   $2 \times 48 = 96$
   $11 \times 13 = 143$
   Since $96 \neq 143$, they are not equivalent.

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Converting Mixed Fractions to Improper Fractions

A mixed fraction consists of a whole number and a proper fraction (e.g., $2\frac{3}{4}$). To convert it into an improper fraction, follow these steps:

Steps for Conversion

1. Multiply the whole number by the denominator of the fraction.

2. Add the result to the numerator of the fraction.

3. Write the sum as the numerator, keeping the original denominator unchanged.

Formula

If a mixed fraction is $a\frac{b}{c}$, then:

$$\text{Improper Fraction}=\frac{(a\times c)+b}{\mathrm{c}}$$

Sample Questions and Answers

Example 1

Convert $3\frac{2}{5}$ into an improper fraction.

Solution:

1. Multiply the whole number $3$ by the denominator $5$:
   $3 \times 5 = 15$
   
2. Add the numerator $2$:
   $15 + 2 = 17$
   
3. Write the result as the numerator over the denominator $5$:
   $\frac{17}{5}$

Answer: $\frac{17}{5}$

Example 2

Convert $5\frac{4}{7}$ into an improper fraction.

Solution:

1. Multiply the whole number $\mathrm{5\; by\; the\; denominator }$$7$:
   $5 \times 7 = 35$
   
2. Add the numerator $4$:
   $35 + 4 = 39$
   
3. Write the result as the numerator over the denominator $7$:
   $\frac{39}{7}$

Answer: $\frac{39}{7}$

Practice Questions

1. Convert $4\frac{5}{8}$ into an improper fraction.
2. Convert $6\frac{2}{9}$ into an improper fraction.
3. Convert $7\frac{3}{4}$ into an improper fraction.

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Key Operations with Fractions

1. Addition and Subtraction

Fractions can be added or subtracted depending on whether they have the same denominator (like fractions) or different denominators (unlike fractions). Below are the rules and examples for each case.

Case 1: Like Fractions (Same Denominator)

• Rule: Add or subtract the numerators, keeping the denominator the same.

Example 1: Addition

Question: $\frac{3}{7} + \frac{2}{7}$

Solution:
$\frac{3 + 2}{7} = \frac{5}{7}$

Example 2: Subtraction

Question: $\frac{5}{9} - \frac{2}{9}$

Solution:
$\frac{5 - 2}{9} = \frac{3}{9}$
Simplify: $\frac{3}{9} = \frac{1}{3}$

Case 2: Unlike Fractions (Different Denominators)

• Rule:
  • Find the least common denominator (LCD) of the fractions.
  • Convert the fractions to equivalent fractions with the LCD.
  • Add or subtract the numerators.

Example 1: Addition

Question: $\frac{1}{4} + \frac{2}{3}$

Solution:

1. Find the LCD of 4 and 3: $12$.
2. Convert fractions:
   $\frac{1}{4} = \frac{3}{12}$, $\frac{2}{3} = \frac{8}{12}$.
3. Add the numerators:
   $\frac{3}{12} + \frac{8}{12} = \frac{11}{12}$.

Example 2: Subtraction

Question: $\frac{5}{6} - \frac{1}{4}$
Solution:

1. Find the LCD of 6 and 4: 12
2. Convert fractions:

$\frac{5}{6}=\frac{10}{12}$, $\frac{1}{4}=\frac{3}{12}$

3. Subtract the numerators:

$\frac{10}{12}-\frac{3}{12}=\frac{7}{12}$

Case 3: Mixed Fractions

• Rule: Convert mixed fractions to improper fractions, perform the operation, and simplify if necessary.

Example: Addition

Question: $2\frac{1}{3} + 1\frac{2}{5}$

Solution:

Convert to improper fractions:
$2\frac{1}{3}=\frac{7}{3},1\frac{2}{5}=\frac{7}{5}$

Find the LCD of 3 and 5: $15$.

Convert fractions:
$\frac{7}{3}=\frac{35}{15},\frac{7}{5}=\frac{21}{15}$

Add:
$\frac{35}{15}+\frac{21}{15}=\frac{56}{15}$

Simplify if necessary: $3\frac{11}{15}$

Example: Subtraction

Question: $3\frac{1}{2} - 1\frac{3}{4}$
Solution:

Convert to improper fractions:
$3\frac{1}{2}=\frac{7}{2},1\frac{3}{4}=\frac{7}{4}$

Find the LCD of 2 and 4: $4$.
Convert fractions:
$\frac{7}{2}=\frac{14}{4},\frac{7}{4}=\frac{7}{4}$

Subtract:
$\frac{14}{4}-\frac{7}{4}=\frac{7}{4}$

Simplify if necessary: $1\frac{3}{4}$

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2. Multiplication

Multiplying fractions involves multiplying the numerators together and the denominators together. The result is a new fraction. The basic steps are as follows:

1. Multiply the numerators (top numbers) to get the new numerator.
2. Multiply the denominators (bottom numbers) to get the new denominator.
3. Simplify the fraction, if necessary, by dividing both the numerator and denominator by their greatest common divisor (GCD).

Steps to Multiply Fractions:

1. Multiply the numerators:
$\text{Numerator}=\text{Numerator of Fraction 1}\times \text{Numerator of Fraction 2}$

2. Multiply the denominators:
$\text{Denominator}=\text{Denominator of Fraction 1}\times \text{Denominator of Fraction 2}$

3. Simplify the result, if possible.

Sample Questions and Answers:

Example 1:

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

Solution:

Multiply the numerators:
$2\times 3=6$

Multiply the denominators:
$5\times 4=20$

The result is:
$\frac{6}{20}$

Simplify the fraction:
$\frac{6}{20}=\frac{6÷2}{20÷2}=\frac{3}{10}$

Answer:
$\frac{2}{5} \times \frac{3}{4} = \frac{3}{10}$

Example 2:

Multiply the fractions:
$\frac{7}{9} \times \frac{2}{3}$

Solution:

Multiply the numerators:
$7\times 2=14$

Multiply the denominators:
$9\times 3=27$

The result is:
$\frac{14}{27}$

Since 14 and 27 do not have any common factors other than 1, the fraction is already in its simplest form.

Answer:
$\frac{7}{9} \times \frac{2}{3} = \frac{14}{27}$

Example 3:

Multiply the fractions:
$\frac{5}{8} \times \frac{4}{10}$

Solution:

Multiply the numerators:
$5\times 4=20$

Multiply the denominators:
$8\times 10=80$

The result is:
$\frac{20}{80}$

Simplify the fraction:
$\frac{20}{80}=\frac{20÷20}{80÷20}=\frac{1}{4}$

Answer:
$\frac{5}{8} \times \frac{4}{10} = \frac{1}{4}$

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3. Division 

In mathematics, division of fractions refers to the operation where one fraction is divided by another. Instead of directly dividing fractions, we multiply the first fraction by the reciprocal (or inverse) of the second fraction.

Steps to Divide Fractions:

1. Flip the second fraction (find its reciprocal).
2. Multiply the first fraction by the reciprocal of the second fraction.
3. Simplify the result if possible.

Formula:

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Sample Questions and Answers:

Example 1:

Question:
Divide $\frac{3}{4}$ by $\frac{2}{5}$

Solution:

1. Find the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$
   
2. Multiply $\frac{3}{4}$ by $\frac{5}{2}$: $$\frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8}$$
   
3. $\frac{15}{8}$ is already in its simplest form.

Answer:
$\frac{3}{4} \div \frac{2}{5} = \frac{15}{8}$

Example 2:

Question:
Divide $\frac{7}{10}$ by $\frac{3}{4}$

Solution:

1. Find the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$
   
2. Multiply $\frac{7}{10}$ by $\frac{4}{3}$: $$\frac{7}{10} \times \frac{4}{3} = \frac{7 \times 4}{10 \times 3} = \frac{28}{30}$$
   
3. Simplify $\frac{28}{30}$: $$\frac{28}{30} = \frac{28 \div 2}{30 \div 2} = \frac{14}{15}$$
   

Answer:
$\frac{7}{10} \div \frac{3}{4} = \frac{14}{15}$

Example 3:

Question:
Divide $\frac{5}{6}$ by $\frac{5}{9}$

Solution:

1. Find the reciprocal of $\frac{5}{9}$, which is $\frac{9}{5}$.
2. Multiply $\frac{5}{6}$ by $\frac{9}{5}$: $$\frac{5}{6} \times \frac{9}{5} = \frac{5 \times 9}{6 \times 5} = \frac{45}{30}$$
3. Simplify $\frac{45}{30}$: $$\frac{45}{30} = \frac{45 \div 15}{30 \div 15} = \frac{3}{2}$$

Answer:
$\frac{5}{6} \div \frac{5}{9} = \frac{3}{2}$

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Tips for Mastering Fractions

1. Practice Regularly: Solve fraction problems daily to improve your accuracy and speed.

2. Visualize with Diagrams: Use pie charts or bar models to understand fractions better.

3. Use Fraction Apps: Tools like “Fractions Calculator” and “Mathway” can help.

4. Learn LCM and HCF: These concepts simplify adding and subtracting fractions.

5. Seek Help: Don’t hesitate to ask teachers or peers if you’re stuck.

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Applications of Fractions in Real Life

Fractions are used in various real-world scenarios, including:

• Cooking: Measuring ingredients.

• Finance: Calculating interest rates and discounts.

• Engineering: Designing structures and solving technical problems.

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Recommended Resources for SHS Students

1. Textbooks: GAST SHS Core Mathematics Textbook.

2. Websites: Khan Academy and Math is Fun.

3. Apps: “Fraction Basics” and “My SHS Mate.”

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Conclusion

Fractions are a vital component of SHS Mathematics, and mastering them sets the stage for tackling more advanced topics. By practicing consistently and using the resources recommended here, you’ll develop confidence and proficiency in fractions.

Ready to excel in fractions? Start practicing today with resources like Khan Academy or download the “Mathway” app. Share this guide with your classmates and make mathematics enjoyable for everyone!