Fractions are an important part of math, representing parts of a whole. To work effectively with fractions, it’s essential to know how to perform basic operations: addition, subtraction, multiplication, and division. Each operation has its own steps and rules to follow, making it easier to find accurate answers.
Adding Fractions
To add fractions:
With the same denominator (Like Fractions): Add the numerators and keep the denominator the same. For example,
$/frac{2}{7}$ + $/frac{3}{7}$ = $/frac{5}{7}$
With different denominators (Unlike Fractions): First, find a common denominator (usually the least common multiple), then rewrite each fraction with this common denominator. Finally, add the numerators while keeping the new denominator. For example,
Aman eats 1/2 of a pizza and Deepa eats 1/3 of a pizza. What portion of the pizza did they both eat together?
The first pizza is divided into 2 parts and the second one into 3 parts. It is easier to add when the two pizzas are divided into equal number of parts. Let’s find a number that can be divided into 2 and 3 parts. 6 is the smallest number that is divisible by 2 and 3. Let’s cut each pizza into 6 parts. The fractions $\frac{1}{2}$ and $\frac{1}{3}$ become their equivalent fractions $\frac{3}{6}$ and $\frac{2}{6}$.
$\frac{3}{6}$ + $\frac{2}{6}$ = $\frac{5}{6}$
Aman and Deepa have together eaten 5/6 parts of the pizza.
Subtracting Fractions
Subtracting fractions is similar to addition:
With the same denominator (Like Fractions): Subtract the numerators and keep the denominator the same. For example,
$/frac{8}{11}$ + $/frac{6}{11}$ = $/frac{2}{11}$
With different denominators (Unlike Fractions): First, find a common denominator (usually the least common multiple), then rewrite each fraction with this common denominator. Finally, add the numerators while keeping the new denominator. For example,
Aman has $\frac{3}{4}$ of a chocolate bar, and he gives $\frac{ of a chocolate bar to his friend, Riya. How much of the chocolate bar does Aman have left?
To solve this, we need to subtract $\frac{1}{3}$$\frac{3}{4}$
Finding a Common Denominator:
- The chocolate bars are divided into different-sized pieces: one into 4 parts and the other into 3 parts. To make subtraction easier, we need to find a common denominator.
- The smallest number that can be divided by both 4 and 3 is 12. So, let’s convert each fraction to have 12 as the denominator.
Convert to Equivalent Fractions:
- Fo $\frac{3}{4}$, we multiply both the numerator and denominator by 3 to get $\frac{9}{12}$.
- For $\frac{1}{3}$, we multiply both the numerator and denominator by 4 to get $\frac{4}{12}$.
Subtract the Fractions:
- Now that the fractions have the same denominator, we can subtract them
$\frac{9}{12}$ – $\frac{4}{12}$ = $\frac{5}{12}$
Aman has $\frac{5}{12}$ of the chocolate bar left after sharing with Riya.
Multiplying Fractions
To multiply fractions:
- Multiply the numerators together to get the new numerator.
- Multiply the denominators together to get the new denominator.
- Simplify the result if possible.
For example,
$\frac{2}{3}$ x $\frac{4}{5}$ = $\frac{8}{15}$
Visual Explanation
Let’s say you have of a chocolate bar, and you want to find out what $\frac{1}{4}$ of that amount would be. In other words, we’re calculating
$\frac{2}{3}$ x $\frac{1}{4}$
Draw the First Fraction:
- Start by drawing a rectangle or square to represent one whole chocolate bar.
- Divide this chocolate bar into 3 equal parts (representing thirds) by drawing vertical lines, and shade 2 out of these 3 parts to represent $\frac{2}{3}$.
Represent the Second Fraction:
- Now, take the same chocolate bar and divide it into 4 equal parts (representing fourths) by drawing horizontal lines.
- You’ll see that the chocolate bar is now divided into a 3×4 grid, giving us 12 smaller sections.
Identify the Overlap:
- Since you have shaded vertically and need $\frac{1}{4}$ of this amount, shade 1 out of every 4 rows of the chocolate bar horizontally.
- The overlapping shaded area (where both the vertical and horizontal shading meet) represents the fraction $\frac{2}{3}$ x $\frac{1}{4}$.
Count the Overlapping Parts:
- Out of the 12 sections, 2 sections are shaded both vertically and horizontally.
- So, $\frac{2}{3}$ x $\frac{1}{4}$ = $\frac{2}{12}$ = $\frac{1}{6}$
The result of $\frac{2}{3}$ x $\frac{1}{4}$ is $\frac{1}{6}$. This visually shows that one-sixth of the entire chocolate bar represents one-fourth of two-thirds.
Dividing Fractions
Dividing fractions involves a few simple steps:
- Keep the first fraction, as such.
- Flip (find the reciprocal of) the second fraction.
- Change the division sign to multiplication.
- Multiply the fractions as in multiplication.
Dividing Whole Numbers by Fractions
Tips for Working with Fractions
- Simplify when possible: After any operation, check if you can simplify the fraction by dividing the numerator and denominator by a common factor.
- Convert improper fractions: If your result is an improper fraction, you might want to convert it to a mixed number for easier interpretation.
- Use visual aids: Drawing models or fraction bars can help visualize addition and subtraction, making operations clearer.
FAQs on Operations on Fractions
We can easily remember the dividing fraction by the KCF method. Which means:
K- Keep the first fraction as it is
C- Change the division symbol to multiplication
F- Flip the second fraction
Now multiply the fractions to get the answer
Example : 1/3 ÷ 2/5
- 1/3 x 5/2 = 5/6
No, this method only applies for addition and subtraction of mixed fractions. To multiply mixed fractions, convert it to improper fractions first and then multiply the numerator with numerator and denominator with denominator. The same applies for division of mixed fractions.
- Cross-multiplication is a technique used to compare fractions or solve proportion equations, not directly for addition, subtraction, multiplication, or division.
- Yes, you can cross-cancel any common factors before multiplying. This simplifies the process and reduces the result right away.
- Dividing by a fraction essentially means determining how many times that fraction fits into another number. Multiplying by the reciprocal achieves this because it reverses the numerator and denominator.