Fractions, Decimals, Percentages
Interactive lessons, worked examples, and problem-solving techniques to help Kenyan students master KJSEA, 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.
Fraction ↔ Decimal ↔ Percentage Explorer
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