Ratio, Proportion & Financial Mathematics
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Master direct proportion, ratio sharing, scaling, and financial calculations including currency exchange and simple interest.
Fundamental Principle of Ratio & Proportion:
- Ratio \(a : b\): A multiplicative comparison of two quantities. Multiplying or dividing both parts by the same non-zero number leaves the ratio unchanged: \(a : b = ka : kb\).
- Direct Proportion: Two quantities \(x\) and \(y\) are directly proportional (\(y \propto x\)) if their quotient is constant: \(\frac{y}{x} = k\), which means \(y = kx\).
- Unitary Method: Find the value of 1 unit first, then multiply by the desired number of units.
Key Formulas
1. Ratio Sharing
\[ \text{Value of 1 part} = \frac{\text{Total Amount}}{\text{Sum of parts}} \] \[ \text{Share of } a = a \times \text{Value of 1 part} \]2. Direct Proportion & Cross-Multiplication
\[ \frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c \]3. Simple Interest
\[ I = \frac{P \times R \times T}{100} \] \[ \text{Total Amount } A = P + I = P\left(1 + \frac{RT}{100}\right) \]Where \(P =\) Principal, \(R =\) Annual interest rate (%), and \(T =\) Time in years.
4. Currency Conversion
\[ \text{Foreign Currency} = \frac{\text{Local Currency}}{\text{Exchange Rate (Local per 1 Foreign)}} \] \[ \text{Local Currency} = \text{Foreign Currency} \times \text{Exchange Rate} \]5. Percentage Profit & Loss
\[ \text{Percentage Profit} = \left(\frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}}\right) \times 100\% \]Worked Examples
Problem: Wanjiku and Otieno share KES 15,000 from a joint market stall in the ratio \(3 : 2\). How much does Wanjiku receive?
- Total parts: \(3 + 2 = 5\) parts.
- Value of 1 part: \[ \frac{\text{KES } 15{,}000}{5} = \text{KES } 3{,}000 \]
- Wanjiku's share (3 parts): \[ 3 \times \text{KES } 3{,}000 = \text{KES } 9{,}000 \]
Problem: Kwame deposited money into a saccos account earning \(6\%\) simple interest per annum. After \(4\) years, he earned \(\text{KES } 7{,}200\) in interest. Find the original principal deposited.
- Identify given values: \(I = 7{,}200\), \(R = 6\), \(T = 4\).
- Apply simple interest formula: \[ I = \frac{P \times R \times T}{100} \implies 7{,}200 = \frac{P \times 6 \times 4}{100} \]
- Solve for \(P\): \[ 7{,}200 = \frac{24P}{100} = 0.24P \implies P = \frac{7{,}200}{0.24} = \text{KES } 30{,}000 \]
Problem: An electronics trader in Nairobi imports 50 solar lamps from the US costing \($20\) each. The exchange rate is \(1\text{ USD} = 140\text{ KES}\). Shipping and customs clearance cost \(\text{KES } 30{,}000\) in total. If he sells all lamps for \(\text{KES } 216{,}000\), calculate his percentage profit.
- Total purchase cost in USD: \[ 50 \times 20 = \$1{,}000 \]
- Convert purchase cost to KES: \[ 1{,}000 \times 140 = \text{KES } 140{,}000 \]
- Total cost price (CP): \[ \text{CP} = 140{,}000 + 30{,}000 = \text{KES } 170{,}000 \]
- Calculate profit: \[ \text{Profit} = \text{SP} - \text{CP} = 216{,}000 - 170{,}000 = \text{KES } 46{,}000 \]
- Calculate percentage profit: \[ \text{Percentage Profit} = \frac{46{,}000}{170{,}000} \times 100\% \approx 27.06\% \]
Common Mistakes
Incorrect: A student converting KES 70,000 to USD at \(1\text{ USD} = 140\text{ KES}\) multiplies \(70{,}000 \times 140 = 9{,}800{,}000\text{ USD}\).
Correction: Since \(1\text{ USD}\) is worth multiple shillings, the dollar amount must be smaller than the shilling amount. You must divide: \(\frac{70{,}000}{140} = 500\text{ USD}\).
Incorrect: Giving the interest \(I\) as the final answer when asked for the total amount repaid.
Correction: The final repayment or total accumulated amount is the principal plus the interest: \(A = P + I\).
Incorrect: \(\text{Profit \%} = \frac{\text{Profit}}{\text{Selling Price}} \times 100\%\).
Correction: Unless explicitly specified as margin, percentage profit is always calculated based on the original Cost Price (CP): \(\frac{\text{Profit}}{\text{Cost Price}} \times 100\%\).
Real World
Mobile Money & SACCO Loans
In Kenya, small agribusiness owners borrow capital from Chama groups or SACCOs. Understanding simple interest and flat repayment schedules prevents overpaying hidden fees.
Cross-Border Trading
Merchants trading between Kenya, Uganda, and Tanzania convert currencies daily at border posts like Busia and Namanga, using direct proportion to calculate exchange margins and optimal pricing.
Concrete & Construction Ratios
Civil engineers mix cement, sand, and gravel in strict volumetric ratios (e.g., \(1 : 2 : 4\) for standard reinforced concrete) to guarantee structural safety and calculate exact bag requirements.
Practice