Ratios, Rates, Proportions
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Ratios, Rates and Direct Proportion
In everyday life in Kenya—whether sharing farming yields in Nakuru, mixing feeds for dairy cows, calculating matatu travel times, or converting currencies—we constantly compare quantities.
1. Concrete Understanding: What is a Ratio?
A ratio is a comparison of two or more quantities of the same kind measured in the same units. For example, if a builder mixes \(1\) bag of cement with \(4\) wheelbarrows of sand, the ratio is written as \(1 : 4\). Ratios have no units because the units cancel out.
2. Rates vs. Ratios
While ratios compare quantities with the same units, a rate compares two quantities with different units. Common examples include:
- Speed: Distance per unit time (e.g., \(\text{km/h}\) or \(\text{m/s}\)).
- Price Rate: Cost per item or per kilogram (e.g., \(\text{Ksh/kg}\)).
- Consumption: Fuel usage per unit distance (e.g., \(\text{km/litre}\)).
3. Direct Proportion
Two quantities are in direct proportion if an increase in one leads to a corresponding proportional increase in the other (i.e., their quotient is a constant \(k\)):
\[\frac{y}{x} = k \implies y = kx\]Key Formulas
1. Simplifying and Expressing Ratios
\[a : b = \frac{a}{b}\]A ratio is in its simplest form when \(a\) and \(b\) are whole numbers with a greatest common divisor (GCD) of \(1\).
2. Sharing a Quantity in a Given Ratio
To divide a total quantity \(T\) into parts given by the ratio \(a : b : c\):
\[\text{Total Parts} = a + b + c\] \[\text{Value of 1 Part} = \frac{T}{a + b + c}\] \[\text{Share of } a = \frac{a}{a + b + c} \times T\]3. Rate and Unitary Method
\[\text{Rate} = \frac{\text{Quantity } A}{\text{Quantity } B}\] \[\text{Cost for } n \text{ units} = \left(\frac{\text{Total Cost}}{\text{Given Units}}\right) \times n\]4. Proportion and Cross-Multiplication
\[\frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c\]Worked Examples
Example 1 (Easy): Sharing Harvest in a Given Ratio
Problem: Two farmers, Amani and Baraka, shared a harvest of \(60\) bags of maize in the ratio \(3 : 2\). How many bags did Baraka receive?
Solution:
- Find total parts: \(3 + 2 = 5\) equal parts.
- Find the value of 1 part: \(\frac{60}{5} = 12\) bags.
- Calculate Baraka's share (2 parts): \(2 \times 12 = 24\) bags.
Answer: \(24\) bags
Example 2 (Medium): Direct Proportion Unitary Cost
Problem: At a local posho mill, \(6\) tins of maize cost \(\text{Ksh } 720\). Assuming the rate is directly proportional, how much will \(11\) tins cost?
Solution:
- Find the unit rate (cost of 1 tin): \[\text{Unit Rate} = \frac{720}{6} = \text{Ksh } 120 \text{ per tin}\]
- Multiply the unit rate by the desired quantity (\(11\) tins): \[\text{Total Cost} = 11 \times 120 = \text{Ksh } 1,320\]
Answer: \(\text{Ksh } 1,320\)
Example 3 (Hard): Three-Part Ratio with Difference Given
Problem: Three business partners—Wanjiku, Otieno, and Kiprono—contributed capital in the ratio \(2 : 5 : 7\). Kiprono contributed \(\text{Ksh } 25,000\) more than Wanjiku. Calculate the total capital contributed by all three partners.
Solution:
- Identify the difference in ratio parts between Kiprono and Wanjiku: \[\text{Difference in parts} = 7 - 2 = 5 \text{ parts}\]
- Determine the value of \(1\) part: \[5 \text{ parts} = \text{Ksh } 25,000 \implies 1 \text{ part} = \frac{25,000}{5} = \text{Ksh } 5,000\]
- Calculate the total number of parts for all three: \[\text{Total parts} = 2 + 5 + 7 = 14 \text{ parts}\]
- Calculate the total capital: \[\text{Total Capital} = 14 \times 5,000 = \text{Ksh } 70,000\]
Answer: \(\text{Ksh } 70,000\)
Common Mistakes
Misconception 1: Dividing by the Number of People Instead of Total Parts
Mistake: Dividing a total quantity \(T\) by \(2\) when sharing in the ratio \(3 : 2\).
Why it happens: The word "share" often reminds learners of equal splitting between two people.
Correct Approach: Always sum the parts of the ratio first (\(3 + 2 = 5\)), then divide \(T\) by \(5\) to find the value of a single part.
Misconception 2: Comparing Quantities with Different Units Directly
Mistake: Stating that the ratio of \(50\text{ cents}\) to \(\text{Ksh } 2\) is \(50 : 2 = 25 : 1\).
Why it happens: Ignoring the units before writing down the numbers.
Correct Approach: Convert to the same units first! \(\text{Ksh } 2 = 200\text{ cents}\). Thus, the correct ratio is \(50 : 200 = 1 : 4\).
Misconception 3: Inverting Cross-Multiplication
Mistake: Solving \(\frac{x}{6} = \frac{10}{15}\) by calculating \(x = 6 \times \frac{15}{10}\).
Why it happens: Memorizing division rules without setting up \(a \times d = b \times c\).
Correct Approach: Cross-multiply cleanly: \(15 \times x = 6 \times 10 \implies 15x = 60 \implies x = 4\).
Real World
Agriculture & Fertilizers
A farmer in Eldoret applies DAP fertilizer at a rate of \(50\text{ kg}\) per acre. For a \(4.5\text{-acre}\) farm, direct proportion gives \(4.5 \times 50 = 225\text{ kg}\) (or \(4.5\) fifty-kilo bags).
Matatu Transport
A matatu covers \(160\text{ km}\) from Nairobi to Nakuru in \(2\) hours. Its average rate of travel is \(\frac{160}{2} = 80\text{ km/h}\). Drivers use this rate to estimate arrival times.
Concrete Mixing
Standard building concrete uses a ratio of \(1 : 2 : 4\) (Cement : Sand : Ballast). To prepare \(14\text{ m}^3\) of mix, a contractor knows exactly \(2\text{ m}^3\) of cement is required.
Foreign Exchange
Forex bureaus use exchange rates such as \(1\text{ USD} = 130\text{ KES}\). To convert \(250\text{ USD}\), direct multiplication gives \(250 \times 130 = \text{Ksh } 32,500\).
Practice