Introduction
- Operations are the basic processes used to manipulate numbers and expressions. The four fundamental operations are:
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)
- In mathematics, an Operation is a process or action that produces a new value from one or more inputs, such as addition, subtraction, multiplication, or division.
Order of Operations:
- To solve expressions correctly, follow the order:
Parentheses → Exponents → Multiplication/Division → Addition/Subtraction
Watch: Fractions
Addition of Fractions
1. Same Denominator- If the denominators (bottom numbers) are the same, just add the numerators (top numbers):
$$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$
If the denominators are different, follow these steps:
1
Find the Least Common Denominator (LCD), the smallest number that both denominators can divide into.
2
Convert fractions to have the same denominator.
3
Add the numerators.
4
Simplify the result (if needed).
$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$
Subtraction of Fractions
1. Same Denominator- If the denominators (bottom numbers) are the same, just subtract the numerators (top numbers):
$$\frac{a}{c} – \frac{b}{c} = \frac{a – b}{c}$$
If the denominators are different, follow these steps:
1
Find the Least Common Denominator (LCD), the smallest number that both denominators can divide into.
2
Convert fractions to have the same denominator.
3
Subtract the numerators.
4
Simplify the result (if needed).
$$\frac{a}{b} – \frac{c}{d} = \frac{ad – bc}{bd}$$
Multiplication of Fractions
1. Basic Rule- For a problem, such as:$$\frac{a}{b} \times \frac{c}{d}$$
- Numerator of the product = $a \times c$
- Denominator of the product = $b \times d$
1
Multiply the numerators: $a \times c$.
2
Multiply the denominators: $b \times d$.
3
Simplify the resulting fraction by dividing numerator and denominator by their greatest common divisor (GCD).
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1. Basic Rule- For a problem, such as:$$\frac{a}{b} \div \frac{c}{d}$$
- Numerator of the product = $a \times d$
- Denominator of the product = $b \times c$
1
Write the problem:
$$\frac{a}{b} \div \frac{c}{d}$$
2
Reciprocal, Change to:
$$\frac{a}{b} \times \frac{d}{c}$$
3
Multiply numerators and denominators. Simplify the result by dividing numerator and denominator by their greatest common divisor (GCD).
Solved Examples
Solved Example
Convert $\frac{3}{4} + \frac{7}{2}$ into a single fraction.
SOLUTION
1
Make the bottom numbers the same:
$$\frac{3}{4} + \frac{7 \times 2}{2 \times 2} \longrightarrow \frac{3}{4} + \frac{14}{4}$$
2
Add the top numbers:
$$\frac{3 + 14}{4} = \frac{17}{4}$$
3
Convert back to a Mixed Fraction:
$$4\frac{1}{4}$$
Final Answer: $4\frac{1}{4}$
Solved Example
Convert $\frac{9}{4} – \frac{5}{2}$ as a fraction.
SOLUTION
1
Make the bottom numbers the same:
$$\frac{9}{4} – \frac{5 \times 2}{2 \times 2} \longrightarrow \frac{9}{4} – \frac{10}{4}$$
2
Subtract the top numbers:
$$\frac{9 – 10}{4} = -\frac{1}{4}$$
Final Answer: $-\frac{1}{4}$
Solved Example
Convert $\frac{8}{3} \times \frac{6}{5}$ as a fraction.
SOLUTION
1
Multiply top numbers together:
$$8 \times 6 = 48$$
2
Multiply bottom numbers together:
$$3 \times 5 = 15$$
3
Simplify the result:
$$\frac{48}{15} \longrightarrow \frac{16}{5}$$
Final Answer: $\frac{16}{5}$
Solved Example
Convert $\frac{3}{4} \div \frac{7}{2}$ as a fraction.
SOLUTION
1
Keep the first fraction same, change the divide sign to a multiplication sign, and reciprocate the second fraction:
$$\frac{3}{4} \times \frac{2}{7}$$
2
Multiply bottom and top numbers together:
$$\frac{3 \times 2}{4 \times 7} = \frac{6}{28}$$
Simplify the result:
$$\frac{6}{28} = \frac{3}{14}$$
Final Answer: $\frac{3}{14}$
Solved Example
Emma baked $\frac{2}{3}$ of a tray of cookies in the morning and $\frac{1}{4}$ of a tray in the afternoon. How much of a full tray did she bake in total?
SOLUTION
1
Write down the given information:
- In Morning, Emma baked: $\frac{2}{3}$
- At afternoon, fraction of tray gets completed: $\frac{1}{4}$
2
Simplify to make a common denominator. We know that:
$$\frac{2}{3} + \frac{1}{4} \longrightarrow \frac{8}{12} + \frac{3}{12}$$
3
Calculate the final result by applying favorable operations. The total amount of baking that has been completed:
$$\frac{8 + 3}{12} = \frac{11}{12}$$
Final Answer: $\frac{11}{12}$ of a tray
Solved Example
A ribbon is $2\frac{3}{4}$ meter long. You need pieces of length $\frac{1}{6}$ meter. How many full pieces can you cut?
SOLUTION
1
Write down the given information:
- Length of ribbon: $2\frac{3}{4} = \frac{11}{4}$ meter
- Pieces of length: $\frac{1}{6}$ meter
2
Divide total length by piece length. We know that:
$$\frac{11}{4} \div \frac{1}{6}$$
3
Calculate the final result by applying favorable operations:
$$\frac{11}{4} \times \frac{6}{1} = \frac{66}{4} = 16\frac{1}{2}$$
Final Answer: $16$ full pieces
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