Learning Resources
Rates, Ratio, Proportions and 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 Objective: Master the interconnected mathematical language of rates, ratios, proportions, and percentages to solve real-life scaling and comparative problems.
Interactive Scaler: The Master Recipe & Proportion Lab
Adjust the servings slider below. Notice how every ingredient scales by the exact same multiplier (the scale factor), while the percentage share of each ingredient remains perfectly constant!
Flour (g)
500
63.7% of total
Water (ml)
250
31.8% of total
Oil (ml)
30
3.8% of total
Salt (g)
5
0.6% of total
Scale Factor (Multiplier): 1.00× (since \(\frac{\text{New Servings}}{4} = \frac{\text{Servings}}{4}\))
(a) Concrete Scenario: The Village Feast
When cooking chapati for a family of 4, you use 500 g of flour and 250 ml of water. When hosting a party for 30 guests, adding just more flour without water will ruin the dough! You must scale every ingredient by the exact same factor: \(\frac{30}{4} = 7.5\). This preservation of proportion is the foundation of proportional reasoning.
(b) Geometric & Relational Insight
A ratio compares two quantities of the same kind (e.g. 500 g flour to 250 ml water, or \(2:1\)). A proportion states that two ratios are equal: \(\frac{a}{b} = \frac{c}{d}\). Geometrically, this means scaling preserves shape and texture, stretching all dimensions uniformly.
(c) The Unified Algebraic Model
Rates, ratios, proportions, and percentages are all aspects of the same concept:
- Ratio: Dimensionless comparison \(a:b\) or \(\frac{a}{b}\).
- Rate: Comparison between two different physical units (e.g., speed = \(\frac{\text{km}}{\text{h}}\), water flow = \(\frac{\text{litres}}{\text{minute}}\)).
- Percentage: A standardized ratio with a denominator of 100: \(\frac{P}{100} = \frac{\text{Part}}{\text{Whole}}\).
- Cross-Multiplication: Whenever \(\frac{a}{b} = \frac{c}{d}\), the fundamental truth is \(a \times d = b \times c\).
Master Key: To check if two ratios are proportional, cross-multiply. If the products are equal, the ratios are directly proportional!
Key Formulas
\(\displaystyle\text{Ratio} = \frac{a}{b} = a:b\)
Direct comparison of two quantities in identical units. Always simplify to simplest terms by dividing by the GCD.
Direct comparison of two quantities in identical units. Always simplify to simplest terms by dividing by the GCD.
\(\displaystyle\text{Rate} = \frac{\text{Quantity}_1 \text{ (unit 1)}}{\text{Quantity}_2 \text{ (unit 2)}}\)
Comparison of quantities with different units (e.g., \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\), \(\text{Unit Price} = \frac{\text{Cost}}{\text{Items}}\)).
Comparison of quantities with different units (e.g., \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\), \(\text{Unit Price} = \frac{\text{Cost}}{\text{Items}}\)).
\(\displaystyle\frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c\)
The fundamental law of proportions (cross-multiplication) used to solve for an unknown value.
The fundamental law of proportions (cross-multiplication) used to solve for an unknown value.
\(\displaystyle\text{Part} = \left(\frac{P}{100}\right) \times \text{Whole} \quad \Longleftrightarrow \quad P\% = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100\)
Standard percentage relationship between part, whole, and rate per hundred.
Standard percentage relationship between part, whole, and rate per hundred.
\(\displaystyle\text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Original Value}} \times 100\%\)
Measures percentage increase (positive) or percentage discount/loss (negative) always relative to the original base.
Measures percentage increase (positive) or percentage discount/loss (negative) always relative to the original base.
\(\displaystyle\text{Unitary Method: } \text{Total} = \left(\frac{\text{Known Amount}}{\text{Number of Units}}\right) \times \text{Desired Units}\)
Find the value of a single unit first, then multiply by the desired target quantity.
Find the value of a single unit first, then multiply by the desired target quantity.
Worked Examples
Example 1 (Easy — Percentage of a Whole):
Edwin's school requires a \(20\%\) deposit before booking a tour to Nairobi National Park that costs KES 2,500 per learner. How much is the deposit?
Edwin's school requires a \(20\%\) deposit before booking a tour to Nairobi National Park that costs KES 2,500 per learner. How much is the deposit?
- Identify the formula: \(\text{Part} = \frac{P}{100} \times \text{Whole}\).
- Substitute values: \(\text{Deposit} = \frac{20}{100} \times 2500\).
- Simplify: \(\frac{1}{5} \times 2500 = 500\).
Example 2 (Medium — Rate and Time):
A solar water pump in Isiolo fills a community tank at a steady rate of \(30\text{ litres per minute}\). How many minutes will it take to completely fill a tank with a capacity of \(1,800\text{ litres}\)?
A solar water pump in Isiolo fills a community tank at a steady rate of \(30\text{ litres per minute}\). How many minutes will it take to completely fill a tank with a capacity of \(1,800\text{ litres}\)?
- Identify the relationship: \(\text{Rate} = \frac{\text{Volume}}{\text{Time}} \implies \text{Time} = \frac{\text{Volume}}{\text{Rate}}\).
- Substitute values: \(\text{Time} = \frac{1800\text{ L}}{30\text{ L/min}}\).
- Calculate: \(\text{Time} = 60\text{ minutes} = 1\text{ hour}\).
Example 3 (Hard — Percentage Increase and Multi-Step Cost):
A boutique importer in Mombasa orders solar lanterns costing £1,250. An international courier charges a shipping fee of \(7.5\%\) on top of the goods' cost. Calculate the total expenditure in pounds (£) including shipping.
A boutique importer in Mombasa orders solar lanterns costing £1,250. An international courier charges a shipping fee of \(7.5\%\) on top of the goods' cost. Calculate the total expenditure in pounds (£) including shipping.
- Method 1 (Two-step):
Find shipping fee: \(\text{Fee} = \frac{7.5}{100} \times 1250 = 0.075 \times 1250 = 93.75\).
Add to original cost: \(\text{Total} = 1250 + 93.75 = 1343.75\). - Method 2 (Multiplier method):
Total percent = \(100\% + 7.5\% = 107.5\%\).
\(\text{Total} = 1.075 \times 1250 = 1343.75\).
Common Mistakes
Mistake 1: Scaling only one variable instead of all parts.
Example: Tripling a recipe by multiplying flour by 3, but keeping salt and water the same.
Example: Tripling a recipe by multiplying flour by 3, but keeping salt and water the same.
Why it happens Learners focus on the bulk ingredient and forget that a ratio describes the strict relationship between all components.
Correction In any direct proportion, every single component must be multiplied by the exact same scale factor \(k\).
Mistake 2: Dividing by the new amount instead of the original base when calculating % change.
Example: An item rises from KES 100 to KES 120. A learner calculates \(\frac{20}{120} = 16.7\%\) instead of \(\frac{20}{100} = 20\%\).
Example: An item rises from KES 100 to KES 120. A learner calculates \(\frac{20}{120} = 16.7\%\) instead of \(\frac{20}{100} = 20\%\).
Why it happens The larger or final number is mistakenly treated as the reference point.
Correction Always use the original starting value in the denominator: \(\%\text{ change} = \frac{\text{Difference}}{\text{Original Base}} \times 100\).
Mistake 3: Swapping numerator and denominator in unit rates or proportions.
Example: Writing \(\frac{\text{Distance}}{\text{Time}} = \frac{\text{Time}}{\text{Distance}}\).
Example: Writing \(\frac{\text{Distance}}{\text{Time}} = \frac{\text{Time}}{\text{Distance}}\).
Why it happens Rushing through cross-multiplication without checking units.
Correction Keep matching units aligned on corresponding sides of the equals sign: \(\frac{\text{Unit } A_1}{\text{Unit } B_1} = \frac{\text{Unit } A_2}{\text{Unit } B_2}\).
Real World
Agribusiness & Tea Farming in Kericho: Tea collection centers calculate crop yields and delivery targets against seasonal quotas using proportions and percentage achievement: \(\text{Yield Rate} = \frac{\text{Actual Harvest (kg)}}{\text{Target (kg)}} \times 100\%\).
M-Pesa & Mobile Money Fee Structures: Financial transactions apply percentage tariffs or tiered flat rates. Calculating transaction costs and deposits accurately ensures proper business budgeting for local dukas.
Construction & Cement Mixing: Concrete for building school classrooms requires a strict standard ratio of \(1\text{ part cement} : 2\text{ parts sand} : 4\text{ parts ballast}\). Altering these proportions weakens structural integrity.
Community Water Projects: Sizing borehole storage tanks in arid counties like Turkana and Isiolo relies on flow rates (litres/minute) multiplied by daylight pumping hours.
Practice
Edwin's class is going on a field trip that costs KES 2,500 per learner. The school requires a 20% deposit before booking. How much must each learner pay as a deposit?
(Type only the number, e.g., 42)
Review the concepts above.
A water tank in Isiolo is being filled at a steady rate of 30 litres per minute. How many whole minutes will it take to completely fill a tank that holds 1,800 litres?
(Type only the number, e.g., 42)
Review the concepts above.
Kimani harvested 63 kg of tea leaves from his farm in Kericho. The total expected harvest for the season was 105 kg. What percentage of the total harvest did Kimani collect?
(Type only the number, e.g., 85)
Review the concepts above.
At Gikomba market, a trader bought a batch of mangoes for KES 10,500 and later sold them, making a profit of KES 3,150. What percentage profit did the trader earn?
(Type only the number, e.g., 45)
Review the concepts above.
Gabriel is helping the school PTA calculate 31.25% of KES 320 needed for a bulk purchase of water bottles for sports day. What amount in KES should he record?
(Type only the number, e.g., 75)
Review the concepts above.
Maina is calculating the total cost of a shipment. The items cost £1,250 and a shipping fee of 7.5% is added to this price. Calculate the total cost in pounds including the shipping fee.
(Type only the number, e.g., 1450.00)
Review the concepts above.