Fractions
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: Understand how to compare, add, and subtract fractions with different denominators by finding common unit parts.
The Core Principle: A fraction describes equal parts of a whole. The denominator (bottom) defines the size of each slice, while the numerator (top) counts how many slices you have. You cannot directly combine or compare slices of different sizes (like halves and thirds) without first cutting them into identical pieces using a Common Denominator.
Interactive Visualizer: Fraction Strip Alignment
Select two fractions below to see how finding the Least Common Multiple (LCM) creates matching subdivisions.
Connecting the Steps
- Identify Denominators: Look at the denominators to see if the parts are equal.
- Find the Lowest Common Multiple (LCM): Determine the smallest number both denominators divide into evenly.
- Create Equivalent Fractions: Multiply both the numerator and denominator of each fraction by the factor needed to reach the common denominator.
- Compute and Simplify: Add or subtract the numerators while keeping the common denominator unchanged. Simplify to lowest terms if possible.
Key Formulas
1. Addition with Common Denominators
\[ \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \]When the denominators match, add only the numerators. The denominator remains unchanged.
2. Subtraction with Common Denominators
\[ \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} \]Subtract the second numerator from the first over the common denominator.
3. Converting to Common Denominators
\[ \frac{a}{b} = \frac{a \times k}{b \times k} \]Multiply numerator and denominator by the same non-zero integer \(k\) to create an equivalent fraction where \(b \times k = \text{LCM}\).
4. Comparing Fractions
\[ \text{To compare } \frac{a}{b} \text{ and } \frac{c}{d}, \text{ rewrite as } \frac{a \cdot d}{b \cdot d} \text{ and } \frac{c \cdot b}{b \cdot d} \]Once denominators match, compare the numerators: \(a \cdot d > c \cdot b \implies \frac{a}{b} > \frac{c}{d}\).
Worked Examples
Example 1 (Easy): Same Denominators
Problem: In a community kitchen in Machakos, chefs used \(\frac{3}{8}\) bag of maize flour in the morning and \(\frac{2}{8}\) bag in the afternoon. What fraction of the bag was used in total?
- Check the denominators: Both fractions have a denominator of \(8\).
- Add the numerators: \[ \frac{3}{8} + \frac{2}{8} = \frac{3 + 2}{8} = \frac{5}{8} \]
- Check if simplifiable: \(5\) and \(8\) share no common factor other than \(1\).
Answer: \(\frac{5}{8}\) of a bag
Example 2 (Medium): One Denominator is a Multiple of the Other
Problem: A dairy farmer in Eldoret collected milk in a large churn. In the morning it was \(\frac{3}{4}\) full. After delivery, \(\frac{1}{2}\) of the churn's capacity had been sold. What fraction of the churn remains full?
- Identify the operation: Subtraction: \(\frac{3}{4} - \frac{1}{2}\).
- Find the LCM of \(4\) and \(2\): \(\text{LCM}(4, 2) = 4\).
- Convert \(\frac{1}{2}\) to fourths: \[ \frac{1 \times 2}{2 \times 2} = \frac{2}{4} \]
- Subtract numerators: \[ \frac{3}{4} - \frac{2}{4} = \frac{3 - 2}{4} = \frac{1}{4} \]
Answer: \(\frac{1}{4}\) of the churn
Example 3 (Hard): Different Denominators Requiring LCM
Problem: Wanjiku is preparing her plot in Nyandarua. She plants potatoes on \(\frac{2}{5}\) of the land and cabbages on \(\frac{1}{3}\) of the land. What fraction of the land is left unplanted?
- Find total planted area: \(\frac{2}{5} + \frac{1}{3}\).
- Find the LCM of \(5\) and \(3\): \(\text{LCM}(5, 3) = 15\).
- Convert both fractions to equivalent fractions with denominator \(15\): \[ \frac{2 \times 3}{5 \times 3} = \frac{6}{15}, \quad \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \]
- Add planted portions: \[ \frac{6}{15} + \frac{5}{15} = \frac{11}{15} \]
- Subtract from the whole land (\(1 = \frac{15}{15}\)): \[ \frac{15}{15} - \frac{11}{15} = \frac{4}{15} \]
Answer: \(\frac{4}{15}\) of the land
Common Mistakes
Misconception 1: Adding Across the Denominators
Mistake: \(\frac{1}{3} + \frac{1}{4} = \frac{2}{7}\)
Why it feels right: Whole-number intuition leads students to add the two top numbers and the two bottom numbers together.
Correction: The denominator defines piece size, not the count. You must convert to equal-sized pieces first: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).
Misconception 2: "Bigger Denominator Means Bigger Value"
Mistake: Believing \(\frac{1}{8} > \frac{1}{4}\) because \(8 > 4\).
Why it feels right: In whole numbers, \(8\) is bigger than \(4\).
Correction: In fractions, the denominator is the number of divisions. Cutting a chapati into \(8\) slices produces smaller pieces than cutting it into \(4\) slices. Hence, \(\frac{1}{4} > \frac{1}{8}\).
Misconception 3: Multiplying Only the Numerator
Mistake: Changing \(\frac{2}{3}\) to \(\frac{4}{3}\) when trying to get a common denominator of \(6\).
Why it feels right: Remembering that "we need to multiply by 2" but applying it only to one part of the fraction.
Correction: An equivalent fraction requires multiplying both numerator and denominator by \(2\): \(\frac{2 \times 2}{3 \times 2} = \frac{4}{6}\).
Real World
1. Matatu Fare Collection & Expense Splitting
A matatu stage manager tracks revenue shares. If \(\frac{1}{2}\) of the trip income pays for fuel and \(\frac{1}{5}\) is reserved for county stage fees, finding \(\frac{1}{2} + \frac{1}{5} = \frac{5}{10} + \frac{2}{10} = \frac{7}{10}\) allows the crew to determine that \(\frac{3}{10}\) remains as wages.
2. Shamba Land Allocation
Farmers in the Rift Valley divide their acreage among crops: \(\frac{3}{8}\) for maize, \(\frac{1}{4}\) for napier grass for dairy cows, and the rest for the homestead. Finding the common denominator of \(8\) enables accurate boundary planning.
3. Rainwater Harvesting Tanks
A 5,000-litre school water tank in Kitui has \(\frac{5}{6}\) of its capacity filled after rain. If the school uses \(\frac{1}{3}\) of total capacity over two days, subtracting \(\frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}\) shows exactly half a tank remains.
Practice