Fractions, Decimals, Percentages
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Core Concept: Fractions, decimals, and percentages are three different languages used to describe the exact same quantity: a portion of a whole.
Concrete Kenyan Context: Imagine Mama Wanjiku sharing a rectangular tray of fresh mahamri baked for breakfast. If she divides the tray equally into \(8\) squares and sells \(3\) squares to a customer, the sold portion is \(\frac{3}{8}\) of the batch.
- Fraction form: \(\frac{3}{8}\) (3 out of 8 equal pieces)
- Decimal form: \(3 \div 8 = 0.375\) (value relative to a unit of 1)
- Percentage form: \(0.375 \times 100\% = 37.5\%\) (parts per 100)
1. The Geometry of Equivalence
Consider a standard strip of length \(1\):
- A Fraction \(\frac{a}{b}\) partitions the strip into \(b\) equal segments and selects \(a\) of them.
- A Decimal measures the strip using base-10 subdivisions (tenths, hundredths, thousandths).
- A Percentage rescales the entire length to \(100\) units. The physical size of the strip does not change; only the ruler changes!
Interactive Equivalence Explorer
Adjust the slider to see how fraction, decimal, and percentage represent the exact same filled bar.
Key Formulas
Converting between representations relies on identity transformations (multiplying or dividing by \(1\) or powers of \(10\)).
Worked Examples
Level 1: Easy (Direct Conversion)
Problem: Express \(\frac{7}{8}\) as a decimal and as a percentage.
- Step 1 (Fraction to Decimal): Divide numerator by denominator: \(7 \div 8\). \[7.000 \div 8 = 0.875\]
- Step 2 (Decimal to Percentage): Multiply the decimal by \(100\%\): \[0.875 \times 100\% = 87.5\%\]
Final Answer: \(\frac{7}{8} = 0.875 = 87.5\%\)
Level 2: Medium (Mixed Computation & Comparison)
Problem: In a Nakuru agribusiness cooperative, farmer Otieno allocated \(\frac{2}{5}\) of his farm to maize, \(35\%\) to French beans, and the remaining \(0.25\) portion to passion fruits. Check if the total allocation equals 1 whole, and determine which crop occupies the largest area.
- Step 1: Convert all allocations into percentages:
- Maize: \(\frac{2}{5} \times 100\% = 40\%\)
- French beans: \(35\%\)
- Passion fruits: \(0.25 \times 100\% = 25\%\)
- Step 2: Check the sum: \[40\% + 35\% + 25\% = 100\% = 1.0\quad (\text{Consistent Whole})\]
- Step 3: Compare sizes: \(40\% > 35\% > 25\%\).
Final Answer: The sum equals \(100\%\) (1 whole). Maize occupies the largest portion (\(40\%\)).
Level 3: Hard (Multi-step Financial Application)
Problem: An electronics distributor in Nairobi lists a smartphone at KSh \(24{,}000\). During a promotional sale, a discount of \(15\%\) is offered. Value Added Tax (VAT) of \(16\%\) is subsequently added to the discounted price. Calculate the final price paid by the customer.
- Step 1: Calculate the discounted price: \[\text{Discount} = 15\% \text{ of } 24{,}000 = 0.15 \times 24{,}000 = \text{KSh } 3{,}600\] \[\text{Discounted Price} = 24{,}000 - 3{,}600 = \text{KSh } 20{,}400\] Alternative: \(24{,}000 \times (1 - 0.15) = 24{,}000 \times 0.85 = \text{KSh } 20{,}400\)
- Step 2: Compute VAT on the discounted amount: \[\text{VAT} = 16\% \text{ of } 20{,}400 = 0.16 \times 20{,}400 = \text{KSh } 3{,}264\]
- Step 3: Compute final payable amount: \[\text{Final Price} = 20{,}400 + 3{,}264 = \text{KSh } 23{,}664\]
Final Answer: KSh \(23{,}664\)
Common Mistakes
The Mistake: Writing \(4.5\%\) as \(0.45\) or \(0.05\) as \(50\%\).
Why it feels right: Learners see one non-zero digit and move the decimal only one place or guess based on digit appearances.
The Correction: The percent symbol \(\%\) means "per hundred" (\(\div 100\)). Therefore, you must always shift exactly two decimal places: \[4.5\% = 4.5 \div 100 = 0.045\] \[0.05 = 0.05 \times 100\% = 5\%\]
The Mistake: Assuming that increasing a price by \(20\%\) and then decreasing the new price by \(20\%\) returns you to the original price.
Why it feels right: \(+20 - 20 = 0\), so intuitively learners expect zero net change.
The Correction: The second percentage is calculated on a larger base! For example, starting with KSh \(100\): \[100 + 20\% = 120\] \[120 - 20\% \text{ of } 120 = 120 - 24 = 96 \neq 100\]
The Mistake: Thinking you can only find percentages when the denominator divides \(100\) evenly (like 2, 4, 5, 10, 20, 25, 50).
The Correction: Any fraction \(\frac{a}{b}\) can be converted to a percentage by multiplying by \(100\%\). For \(\frac{3}{7}\): \[\frac{3}{7} \times 100\% = \frac{300}{7}\% = 42.86\%\]
Real World
1. Mobile Money & Bank Charges
When sending KSh \(2{,}500\) via mobile money, a service fee of \(1.2\%\) is charged. Knowing how to convert \(1.2\% = 0.012\) allows instant computation: \[2{,}500 \times 0.012 = \text{KSh } 30\]
2. KCSE Examination Grading
If a student scores \(48\) marks out of \(60\) in Paper 1 and \(65\) marks out of \(80\) in Paper 2: \[\text{Paper 1: } \frac{48}{60} = 0.80 = 80\%\] \[\text{Paper 2: } \frac{65}{80} = 0.8125 = 81.25\%\] Comparing decimal values allows examiners to standardize performance across papers with unequal maximum marks.
3. Agricultural Fertilizer Blends
A \(50\text{ kg}\) bag of NPK 17:17:17 fertilizer contains \(17\%\) nitrogen by mass. Farmers calculate the actual nitrogen content as: \[50 \times 0.17 = 8.5\text{ kg of active Nitrogen}\]
Practice