GCSE Maths

XYZ-myexam

Edexcel · Algebra
Introduction Operations with Fractions Addition of Fractions Subtraction of Fractions Multiplication of Fractions Division of Fractions Solved Examples

Introduction

Operations are the basic processes used to manipulate numbers and expressions. The four fundamental operations are:

  • Addition (+)
  • Subtraction (-)
  • Multiplication (x)
  • Division (÷)

Operations with Fractions

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:

1
Parentheses
2
Exponents
3
Multiplication / Division
4
Addition / Subtraction

Addition of Fractions

1. Same Denominator

If the denominators (bottom numbers) are the same, just add the numerators (top numbers):

Example
Evaluate: $\frac{3}{4}+\frac{2}{4}$
SOLUTION
$$\frac{3}{4}+\frac{2}{4}=\frac{6}{4}$$

2. Different Denominators

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)
Example
Evaluate: $\frac{3}{4}+\frac{7}{8}$
SOLUTION
$$\frac{3}{4}+\frac{7}{8}=\frac{3(8)+7(4)}{32}=\frac{52}{32}$$

Final Answer: $\frac{13}{8}$

Solved Example
Convert $3/4+7/2$ as a fraction
SOLUTION
1
Make the bottom numbers the same
$$=\frac{3}{4}+\frac{7}{2}$$
$$=\frac{3+14}{4}$$
2
Add the top numbers
$$=\frac{17}{4}$$
3
Convert back to a Mixed Fraction
$$=3\frac{5}{4}$$

Final Answer: $3\frac{5}{4}$

Subtraction of Fractions

1. Same Denominator

If the denominators (bottom numbers) are the same, subtract the numerators (top numbers):

Example
Evaluate: $\frac{9}{4}-\frac{2}{4}$
SOLUTION
$$\frac{9}{4}-\frac{2}{4}=\frac{7}{4}$$

2. Different Denominators

If the denominators are different, follow these steps:

1
Find the Least Common Denominator (LCD). The smallest number that both denominators divide into.
2
Convert fractions to have the same denominator
3
Subtract the numerators
4
Simplify the result (if possible)
Example
Evaluate: $\frac{3}{8}-\frac{1}{3}$
SOLUTION
$$\frac{3}{8}-\frac{1}{3}=\frac{3(3)-8(1)}{24}$$

Final Answer: $\frac{1}{24}$

Solved Example
Convert $\frac{9}{4} – \frac{5}{3}$ as a fraction
SOLUTION
1
Make the bottom numbers the same
$$=\frac{9}{4}-\frac{5}{3}=\frac{9(3)-5(4)}{12}$$
2
Subtract the top numbers
$$=\frac{7}{12}$$

Final Answer: $\frac{7}{12}$

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Multiplication of Fractions

Multiplying fractions is straightforward-just multiply numerators together and denominators together, then simplify if possible.

1. Basic Rule: For a problem, such as

$$\frac{a}{b}\times\frac{c}{d}=\frac{a\times c}{b\times d}$$
  • Numerator of the product $=a\times c$
  • Denominator of the product $=b\times d$

2. Steps with Simplification

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).
Example
Multiply: $\frac{2}{3}\times\frac{6}{5}$
SOLUTION
$$\frac{2}{3}\times\frac{6}{5}=\frac{2\times6}{3\times5}$$
$$=\frac{12}{15}$$

Final Answer: $\frac{4}{5}$

Solved Example
Convert $8/3\times6/5$ as a fraction
SOLUTION
1
Multiply top numbers together.
$$=\frac{8}{3}\times\frac{6}{5}$$
2
Multiply bottom numbers together.
$$=\frac{8\times6}{3\times5}$$
3
Simplify the result
$$=\frac{48}{15}$$

Final Answer: $\frac{48}{15}$

Division of Fractions

Dividing fractions involves multiplying by the reciprocal (or “flip”) of the divisor.

1. Basic Rule: For a problem, such as

$$\frac{a}{b}\div\frac{c}{d}=\frac{a\times d}{b\times c}$$
  • Numerator of the product $=a\times d$
  • Denominator of the product $=b\times c$

2. Steps with Simplification

1
Write the problem:
$$\frac{a}{b}\div\frac{c}{d}$$
2
Reciprocal, Change to
$$\frac{a}{b}\div\frac{c}{d}=\frac{a\times d}{b\times c}$$
3
Multiply numerators and denominators. Simplify the result by dividing numerator and denominator by their greatest common divisor (GCD).
Example
Divide: $\frac{3}{4}\div\frac{2}{7}$
SOLUTION
$$\frac{3}{4}\times\frac{7}{2}=\frac{21}{8}$$

Final Answer: $\frac{21}{8}$

Solved Example
Convert $3/4\div7/2$ as a fraction
SOLUTION
1
Keep the first fraction same and change the divide sign to multiplication sign and reciprocate the second fraction.
$$=\frac{3}{4}\times\frac{2}{7}$$
2
Multiply bottom and top numbers together.
$$=\frac{6}{28}$$

Final Answer: $\frac{6}{28}$

Solved Examples

Solved Example
Emma baked $2/3$ of a tray of cookies in the morning and $1/4$ 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:
$$\frac{2\times4}{3\times4}=\frac{8}{12}$$
$$\frac{1\times3}{4\times3}=\frac{3}{12}$$
3
Calculate the final result by applying favorable operations. The total amount of baking that has been completed: $=(\frac{8}{12}+\frac{3}{12})$
$$=\frac{8+3}{12}=\frac{11}{12}$$

Final Answer: $\frac{11}{12}$

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:- $2\frac{3}{4}=\frac{11}{4}$
  • Pieces of length :- $\frac{1}{6}$
2
Divide total length by piece length. We know that:
$$=\frac{\frac{11}{4}}{\frac{1}{6}}$$
$$=\frac{11\times6}{4\times1}$$
3
Calculate the final result by applying favorable operations. Pieces that we can cut from $11/4$m length of ribbon: $\frac{11\times6}{4\times1}=\frac{66}{4}$
$$=\frac{66}{4}=\frac{33}{2}=16.5$$

Final Answer: Therefore, we can cut 16 full pieces from that ribbon.

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