Mass
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 the measurement of mass and convert smoothly between grams (\(\text{g}\)) and kilograms (\(\text{kg}\)).
Concrete Market Scenario: In Gikomba market, Mama Njeri sells dry beans using a traditional balance scale. When a customer orders 2 kilograms of yellow beans, she places two 1 kg metal weights on one pan and scoops beans onto the other until both sides balance perfectly at zero level.
The Metric Foundation: The prefix "kilo-" comes from the Greek word for one thousand (\(1{,}000\)). Therefore, \(1\text{ kilogram}\) is literally \(1{,}000\text{ individual grams}\).
Visualizer: The Metric Balance Scale
Click or tap on standard weights to place them on the measuring pan to balance Mama Njeri's sack of maize (Target: \(1{,}850\text{ g}\)).
Key Formulas
1. Fundamental Unit Equivalence:
\[ 1\text{ kg} = 1{,}000\text{ g} \] \[ 1\text{ g} = \frac{1}{1{,}000}\text{ kg} = 0.001\text{ kg} \]2. Kilograms to Grams (Larger to Smaller Unit):
Multiply the number of kilograms by \(1{,}000\) (shift decimal point 3 places right):
\[ \text{Mass (g)} = \text{Mass (kg)} \times 1{,}000 \]3. Grams to Kilograms (Smaller to Larger Unit):
Divide the number of grams by \(1{,}000\) (shift decimal point 3 places left):
\[ \text{Mass (kg)} = \frac{\text{Mass (g)}}{1{,}000} \]• kg \(\rightarrow\) g : Going to a smaller unit gives a bigger number \((\times 1{,}000)\).
• g \(\rightarrow\) kg : Going to a larger unit gives a smaller number \((\div 1{,}000)\).
Worked Examples
Example 1 (Easy): Converting Grams to Kilograms
Problem: A packet of tea leaves from Kericho weighs \(750\text{ g}\). Express this mass in kilograms.
Step-by-Step Solution:
- Identify the conversion rule: To convert from grams to kilograms, divide by \(1{,}000\). \[ \text{Mass in kg} = \frac{\text{Mass in g}}{1{,}000} \]
- Substitute the given mass: \[ \text{Mass in kg} = \frac{750}{1{,}000} \]
- Compute the quotient: \[ \frac{750}{1{,}000} = 0.75\text{ kg} \]
Final Answer: The packet of tea leaves has a mass of \(0.75\text{ kg}\).
Example 2 (Medium): Combining Multiple Units & Subtraction
Problem: A posho mill worker has a sack containing \(5\text{ kg } 350\text{ g}\) of maize flour. A customer purchases \(1{,}800\text{ g}\). What is the remaining mass of flour in kilograms?
Step-by-Step Solution:
- Convert the initial mass into uniform units (grams): \[ 5\text{ kg } 350\text{ g} = (5 \times 1{,}000\text{ g}) + 350\text{ g} = 5{,}000\text{ g} + 350\text{ g} = 5{,}350\text{ g} \]
- Subtract the purchased quantity: \[ 5{,}350\text{ g} - 1{,}800\text{ g} = 3{,}550\text{ g} \]
- Convert the remaining mass to kilograms: \[ \text{Mass in kg} = \frac{3{,}550}{1{,}000} = 3.55\text{ kg} \]
Final Answer: The remaining flour weighs \(3.55\text{ kg}\).
Example 3 (Hard): Multi-Item Batch Calculation
Problem: A school caterer in Nakuru prepares breakfast by mixing \(12\) packets of cocoa powder, each weighing \(250\text{ g}\), and \(3.5\text{ kg}\) of sugar into a large storage drum. What is the total combined mass in kilograms?
Step-by-Step Solution:
- Find total mass of cocoa powder: \[ 12 \times 250\text{ g} = 3{,}000\text{ g} \]
- Convert cocoa powder mass to kilograms: \[ \frac{3{,}000\text{ g}}{1{,}000} = 3\text{ kg} \]
- Add the sugar mass: \[ \text{Total mass} = 3\text{ kg (cocoa)} + 3.5\text{ kg (sugar)} = 6.5\text{ kg} \]
Final Answer: The total combined mass is \(6.5\text{ kg}\) (or \(6{,}500\text{ g}\)).
Common Mistakes
Misconception 1: Dividing by 100 Instead of 1,000
The Mistake: Converting \(2{,}500\text{ g}\) to \(25\text{ kg}\) by dividing by \(100\).
Why it feels right: Learners often remember that centimetres convert to metres by dividing by \(100\) (\(1\text{ m} = 100\text{ cm}\)) and incorrectly assume all metric conversions use \(100\).
Correction: The prefix "kilo-" means \(1{,}000\). Always divide or multiply by \(1{,}000\): \[ 2{,}500\text{ g} \div 1{,}000 = 2.5\text{ kg} \]
Misconception 2: Decimal Point Alignment Errors with Leading Zeros
The Mistake: Writing \(4\text{ kg } 50\text{ g}\) as \(4.5\text{ kg}\).
Why it feels right: The learner directly attaches \(50\) after the decimal point without considering the 3-decimal place value required for thousandths.
Correction: \(50\text{ g} = \frac{50}{1{,}000}\text{ kg} = 0.050\text{ kg} = 0.05\text{ kg}\). Therefore: \[ 4\text{ kg } 50\text{ g} = 4.05\text{ kg} \] (Note that \(4.5\text{ kg} = 4\text{ kg } 500\text{ g}\), which is \(450\text{ g}\) heavier!)
Misconception 3: Adding Numbers in Different Units Directly
The Mistake: Adding \(2\text{ kg} + 400\text{ g} = 402\text{ kg}\) or \(402\text{ g}\).
Why it feels right: Simply adding the visible numbers (\(2 + 400 = 402\)) without checking whether their units match.
Correction: Units must be identical before performing addition or subtraction: \[ 2\text{ kg} = 2{,}000\text{ g} \implies 2{,}000\text{ g} + 400\text{ g} = 2{,}400\text{ g} = 2.4\text{ kg} \]
Real World
1. Fresh Produce & Cereal Markets
At Wakulima Market in Nairobi, onions and Irish potatoes are weighed in 50 kg sacks. If a retailer sells 500 g bags to domestic buyers, one 50 kg sack produces exactly \(50{,}000 \div 500 = 100\) individual bags.
2. Postal & Courier Shipping
Courier services charge freight based on mass tiers. A parcel weighing \(1{,}450\text{ g}\) is billed in the \(1.5\text{ kg}\) category because \(1{,}450\text{ g} = 1.45\text{ kg}\).
3. Baking & Commercial Kitchens
A bakery recipe requires \(3.75\text{ kg}\) of wheat flour. Standard kitchen scales measure in grams, so the baker measures \(3.75 \times 1{,}000 = 3{,}750\text{ g}\) into the mixing bowl.
4. Farm Harvest & SACCO Deliveries
Dairy cooperatives weigh raw milk deliveries (where 1 litre of water/milk is approximately 1 kg). Farmers tracking crop yield record harvests in kilograms and convert to metric tonnes or grams for packaging seeds.
Practice