Edexcel Mathematics (4XMAF) -Unit 1 - 1.6 Percentages- Study Notes- New Syllabus
Edexcel Mathematics (4XMAF) -Unit 1 – 1.6 Percentages- Study Notes- New syllabus
Edexcel Mathematics (4XMAF) -Unit 1 – 1.6 Percentages- Study Notes -Edexcel iGCSE Mathematics – per latest Syllabus.
Key Concepts:
A understand that ‛percentage’ means ‛number of parts per 100’
B express a given number as a percentage of another number
C express a percentage as a fraction and as a decimal
Edexcel iGCSE Mathematics -Concise Summary Notes- All Topics
Meaning of Percentage
The word percentage means “per 100” or “out of 100”.
The symbol for percentage is \( \% \).
\( 1\% = \dfrac{1}{100} \)
So a percentage is simply a fraction whose denominator is \( 100 \).
Examples
\( 25\% = \dfrac{25}{100} \)
\( 50\% = \dfrac{50}{100} = \dfrac{1}{2} \)
\( 100\% = \dfrac{100}{100} = 1 \)
Interpreting Percentages
Percentages are often used in everyday situations such as marks, discounts and statistics.
\( 80\% \) means 80 parts out of 100
\( 10\% \) means 10 parts out of 100
Key Idea
Percentages compare a quantity with 100.
Example 1:
Write \( 40\% \) as a fraction.
▶️ Answer/Explanation
\( 40\% = \dfrac{40}{100} \)
\( \dfrac{40}{100} = \dfrac{2}{5} \)
Conclusion: \( \dfrac{2}{5} \).
Example 2:
What does \( 5\% \) mean?
▶️ Answer/Explanation
\( 5\% = \dfrac{5}{100} \)
Conclusion: 5 out of every 100.
Example 3:
A student scored 90 marks out of 100. Write this as a percentage.
▶️ Answer/Explanation
\( \dfrac{90}{100} = 90\% \)
Conclusion: \( 90\% \).
Expressing One Number as a Percentage of Another
To express a number as a percentage of another number, compare the two quantities using a fraction.
Method
\( \text{Percentage} = \dfrac{\text{part}}{\text{whole}} \times 100\% \)
The first number is the part and the second number is the whole.
Example Idea
“What percentage is 20 of 50?”
\( \dfrac{20}{50} \times 100\% = 40\% \)
Important
Part ÷ Whole first, then multiply by \( 100 \).
Example 1:
Express 15 as a percentage of 60.
▶️ Answer/Explanation
\( \dfrac{15}{60} \times 100\% \)
\( \dfrac{1}{4} \times 100\% = 25\% \)
Conclusion: \( 25\% \).
Example 2:
A class has 18 girls out of 24 students. What percentage are girls?
▶️ Answer/Explanation
\( \dfrac{18}{24} \times 100\% \)
\( \dfrac{3}{4} \times 100\% = 75\% \)
Conclusion: \( 75\% \) are girls.
Example 3:
In a test, a student scored 42 marks out of 70. Find the percentage.
▶️ Answer/Explanation
\( \dfrac{42}{70} \times 100\% \)
\( \dfrac{3}{5} \times 100\% = 60\% \)
Conclusion: \( 60\% \).
Expressing a Percentage as a Fraction and as a Decimal
Percentage to Fraction
A percentage means “per 100”, so write the percentage over \( 100 \) and simplify.
\( p\% = \dfrac{p}{100} \)
Examples:
\( 25\% = \dfrac{25}{100} = \dfrac{1}{4} \)
\( 40\% = \dfrac{40}{100} = \dfrac{2}{5} \)
Percentage to Decimal
Divide the percentage by \( 100 \), or move the decimal point two places to the left.
\( p\% = \dfrac{p}{100} \)
\( 25\% = 0.25 \)
\( 6\% = 0.06 \)
Key Idea
Percentage → fraction: write over \( 100 \)
Percentage → decimal: divide by \( 100 \)
Example 1:
Convert \( 60\% \) to a fraction and a decimal.
▶️ Answer/Explanation
Fraction: \( \dfrac{60}{100} = \dfrac{3}{5} \)
Decimal: \( 60\% = 0.60 = 0.6 \)
Conclusion: \( \dfrac{3}{5} \) and \( 0.6 \).
Example 2:
Write \( 12\% \) as a fraction in simplest form.
▶️ Answer/Explanation
\( \dfrac{12}{100} \)
\( \dfrac{12}{100} = \dfrac{3}{25} \)
Conclusion: \( \dfrac{3}{25} \).
Example 3:
Convert \( 7.5\% \) to a decimal.
▶️ Answer/Explanation
\( 7.5\% = \dfrac{7.5}{100} \)
\( = 0.075 \)
Conclusion: \( 0.075 \).