Decimals
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 four fundamental operations on decimals and apply them to solve multi-step real-life financial and measurement problems.
Core Concept: A decimal represents a base-10 fractional value where each place to the right of the decimal point represents a division by a power of 10: tenths (\(0.1 = \frac{1}{10}\)), hundredths (\(0.01 = \frac{1}{100}\)), and thousandths (\(0.001 = \frac{1}{1000}\)).
(a) Concrete Scenario: The Dukawalla's Ledger
Imagine shopping at a local kiosk in Nairobi. You purchase \(1.5\text{ kg}\) of sugar at \(\text{Ksh }140.50\text{ per kg}\) and \(0.75\text{ kg}\) of tea leaves at \(\text{Ksh }260.00\text{ per kg}\). To calculate your exact bill and determine the change from a \(\text{Ksh }500\) note, you must combine multiplication, addition, and subtraction of decimal values with absolute precision.
(b) Geometric & Place Value Insight
Think of decimals on a place-value grid. When we add or subtract decimals, we are stacking quantities of identical size (units with units, tenths with tenths, hundredths with hundredths). Misaligning decimal points is equivalent to adding 5 ten-shilling coins to 5 ten-cent coins and calling it 10 whole shillings!
(c) Interactive Place-Value Explorer
Interactive: Decimal Multiplier & Place Shifter
Adjust the sliders to see how place values shift during multiplication.
Whole Number Product: 15 × 24 = 360
Total Decimal Places: 1 + 1 = 2 places
Final Decimal Result: 1.5 × 2.4 = 3.60
Key Formulas
Review these governing mathematical principles for operating on decimals efficiently and error-free:
Pad shorter numbers with trailing zeroes so that digits in the same place-value column line up vertically over the decimal point.
Multiply as integers ignoring points, then place the decimal point so the answer has total decimal places equal to the sum of decimal places in the factors.
Multiply both the dividend \(N\) and the divisor \(D\) by \(10^k\) (where \(k\) is the number of decimal places in \(D\)) to make the divisor a whole integer before dividing.
Worked Examples
Example 1 (Easy: Direct Addition & Subtraction)
A farmer in Eldoret collects \(14.75\text{ litres}\) of milk in the morning and \(12.8\text{ litres}\) in the evening. If \(3.25\text{ litres}\) are consumed by the family, how many litres remain for sale?
Step 1: Find the total milk collected by aligning decimals:
\[14.75 + 12.80 = 27.55\text{ litres}\]Step 2: Subtract the consumed milk:
\[27.55 - 3.25 = 24.30\text{ litres}\]Final Answer: \(24.3\text{ litres}\) (or \(24.30\text{ L}\)).
Example 2 (Medium: Multi-Step Multiplication and Division)
A transport lorry consumes \(0.18\text{ litres}\) of diesel per kilometre. Diesel costs \(\text{Ksh }180.50\text{ per litre}\). Find the total fuel cost for a journey from Nairobi to Nakuru covering \(160\text{ km}\).
Step 1: Calculate the total litres of diesel required:
\[\text{Litres} = 160 \times 0.18\]Multiply \(160 \times 18 = 2880\). With two decimal places: \(28.80 = 28.8\text{ litres}\).
Step 2: Calculate the total cost:
\[\text{Cost} = 28.8 \times 180.50\]Multiply whole numbers: \(288 \times 1805 = 519840\). The factors have \(1 + 1 = 2\) decimal places in reduced form (\(28.8 \times 180.5\)):
\[519840 \div 100 = 5198.40\]Final Answer: \(\text{Ksh }5,198.40\).
Example 3 (Hard: Real-World Multi-Step Problem Solving)
Wanjiku owns a retail shop. She buys a \(50\text{ kg}\) sack of sugar for \(\text{Ksh }6,200.00\). She repacks the entire sack into smaller packets of \(0.75\text{ kg}\) each. She sells each small packet for \(\text{Ksh }115.50\) and gives the remaining fraction of sugar away for free. Calculate her net profit.
Step 1: Determine how many complete \(0.75\text{ kg}\) packets she can make:
\[50 \div 0.75 = \frac{50 \times 100}{0.75 \times 100} = \frac{5000}{75} = \frac{200}{3} = 66.666...\text{ packets}\]She makes exactly \(66\) complete packets.
Step 2: Calculate total sales revenue from the \(66\) packets:
\[\text{Revenue} = 66 \times 115.50 = 66 \times \frac{231}{2} = 33 \times 231 = \text{Ksh }7,623.00\]Step 3: Calculate net profit (\(\text{Revenue} - \text{Cost}\)):
\[\text{Profit} = 7,623.00 - 6,200.00 = \text{Ksh }1,423.00\]Final Answer: \(\text{Ksh }1,423.00\).
Common Mistakes
Misconception 1: Right-aligning numbers regardless of the decimal point
The Mistake: Adding \(4.25 + 12.3\) by writing:
4.25 + 12.3 ------ 4.48 (WRONG! Adding hundredths to tenths)
Correction: Always pad decimals so place values match: \(4.25 + 12.30 = 16.55\). The decimal points must form a vertical line.
Why it feels right: Whole number addition is always aligned along the right edge, so our brain tends to push all numbers rightwards instinctively.
Misconception 2: "Multiplying always makes a number bigger"
The Mistake: Assuming that \(20 \times 0.4\) must be greater than \(20\).
Correction: Multiplying by a decimal between \(0\) and \(1\) calculates a fractional part of the number, which results in a smaller value: \(20 \times 0.4 = 8\).
Why it feels right: Throughout early primary school with whole integers (\(2, 3, 4\dots\)), multiplication always scaled values upwards.
Misconception 3: Shifting only the divisor when dividing decimals
The Mistake: Calculating \(1.25 \div 0.5\) as \(1.25 \div 5 = 0.25\).
Correction: Whatever power of 10 you multiply the divisor by, you MUST multiply the dividend by the exact same amount: \[\frac{1.25 \times 10}{0.5 \times 10} = \frac{12.5}{5} = 2.5\]
Real World
📱 Mobile Money & Banking
M-PESA transaction charges, excise duties (e.g. \(15\% = 0.15\)), and daily interest on savings (e.g., M-Shwari \(0.025\%\)) depend on multi-step decimal arithmetic to ensure accurate balance deductions to the exact cent.
🌾 Agriculture & Crop Yield
Kenyan coffee and tea factories weigh deliveries on electronic weighbridges down to hundredths of a kilogram (e.g., \(48.65\text{ kg}\)). Rates per kg (e.g., \(\text{Ksh }82.40\)) require decimal multiplication for exact farmer payout computation.
🏥 Medicine & Pharmacy
Nurses calculate medicine dosages per kilogram of body weight (e.g., \(0.25\text{ mg/kg}\) for a child weighing \(14.8\text{ kg}\)). Exact decimal calculations are critical for patient safety.
⛽ Fuel Economy & Logistics
Matatu SACCOs track fleet efficiency in kilometres per litre (e.g., \(7.45\text{ km/L}\)) across fluctuating pump prices (e.g., \(\text{Ksh }201.75\text{/L}\)) to set accurate passenger fares.
Practice