Learning Resources
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: Master the multiplication and division of proper fractions, improper fractions, and mixed numbers through visual area models and the principle of reciprocals.
The Chapati Model: Think of a fraction as sharing slices of a fresh chapati. If one whole chapati is cut into 4 equal quarters, each slice is \(\frac{1}{4}\). If you take 3 slices, you have \(\frac{3}{4}\). The numerator (top) counts your slices, and the denominator (bottom) tells how many equal slices make one whole.
1. Multiplication: Finding a "Fraction of a Fraction"
Suppose you have \(\frac{1}{2}\) of a pawpaw and you decide to share \(\frac{1}{3}\) of your half with your friend. In mathematics, the word "of" means multiplication. What fraction of the original whole pawpaw does your friend get?
To see this visually, divide a whole rectangular unit into 2 columns (left column shaded = \(\frac{1}{2}\)). Then divide the entire unit horizontally into 3 rows. The overlapping section is 1 small cell out of 6 total cells, which is \(\frac{1}{6}\):
\[ \frac{1}{2} \times \frac{1}{3} = \frac{1 \times 1}{2 \times 3} = \frac{1}{6} \]2. Division: "How Many Fit Inside?"
When you ask \(6 \div 2\), you are asking: "How many 2s fit inside 6?" (Answer: 3). Similarly, \(\frac{3}{4} \div \frac{1}{8}\) asks: "How many eighths fit inside three-quarters?" Since each quarter contains two eighths, three quarters contain \(3 \times 2 = 6\) eighths.
Dividing by a fraction is identical to multiplying by its reciprocal (flipping the divisor):
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \]Key Formulas
Multiplication Rule:
\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \] Multiply the numerators together and the denominators together.
\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \] Multiply the numerators together and the denominators together.
Division Rule (Keep, Change, Flip):
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \] Keep the first fraction, change the division sign to multiplication, and flip the second fraction to its reciprocal.
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \] Keep the first fraction, change the division sign to multiplication, and flip the second fraction to its reciprocal.
Converting Mixed Numbers to Improper Fractions:
\[ A\frac{b}{c} = \frac{(A \times c) + b}{c} \] Always convert all mixed numbers to improper fractions before multiplying or dividing!
\[ A\frac{b}{c} = \frac{(A \times c) + b}{c} \] Always convert all mixed numbers to improper fractions before multiplying or dividing!
Simplification & Cross-Cancellation:
\[ \frac{\cancel{a}^1}{b} \times \frac{c}{\cancel{d}_2} \] Divide out common factors between any numerator and any denominator before multiplying to work with smaller, simpler numbers.
\[ \frac{\cancel{a}^1}{b} \times \frac{c}{\cancel{d}_2} \] Divide out common factors between any numerator and any denominator before multiplying to work with smaller, simpler numbers.
Worked Examples
Example 1 (Easy): Basic Fraction Multiplication
Find the value of \(\frac{2}{5} \times \frac{3}{4}\) in simplest form.
Find the value of \(\frac{2}{5} \times \frac{3}{4}\) in simplest form.
- Multiply numerators and denominators:
\[ \frac{2 \times 3}{5 \times 4} = \frac{6}{20} \] - Simplify the fraction:
Find the greatest common divisor of 6 and 20, which is 2.
\[ \frac{6 \div 2}{20 \div 2} = \frac{3}{10} \] - Final Answer: \(\mathbf{\frac{3}{10}}\)
Example 2 (Medium): Multiplying Mixed Numbers
Calculate \(1\frac{1}{2} \times 2\frac{1}{3}\). Give your answer as a mixed number in simplest form.
Calculate \(1\frac{1}{2} \times 2\frac{1}{3}\). Give your answer as a mixed number in simplest form.
- Convert mixed numbers to improper fractions:
\[ 1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2} \] \[ 2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3} \] - Multiply and cross-cancel common factors:
\[ \frac{3}{2} \times \frac{7}{3} = \frac{\cancel{3}^1}{2} \times \frac{7}{\cancel{3}_1} = \frac{1 \times 7}{2 \times 1} = \frac{7}{2} \] - Convert back to a mixed number:
\[ 7 \div 2 = 3 \text{ remainder } 1 \implies 3\frac{1}{2} \] - Final Answer: \(\mathbf{3\frac{1}{2}}\) (or \(\mathbf{\frac{7}{2}}\))
Example 3 (Hard): Multi-step Mixed Number Division in Context
A tailor in Nairobi has a roll of kitenge fabric measuring \(7\frac{1}{2}\) metres. Each traditional skirt requires \(\frac{3}{4}\) metre of fabric. How many complete skirts can the tailor make from this roll?
A tailor in Nairobi has a roll of kitenge fabric measuring \(7\frac{1}{2}\) metres. Each traditional skirt requires \(\frac{3}{4}\) metre of fabric. How many complete skirts can the tailor make from this roll?
- Formulate the mathematical expression:
We need to find how many \(\frac{3}{4}\)-metre pieces fit into \(7\frac{1}{2}\) metres:
\[ 7\frac{1}{2} \div \frac{3}{4} \] - Convert \(7\frac{1}{2}\) to an improper fraction:
\[ 7\frac{1}{2} = \frac{(7 \times 2) + 1}{2} = \frac{15}{2} \] - Apply the Keep-Change-Flip rule:
\[ \frac{15}{2} \div \frac{3}{4} = \frac{15}{2} \times \frac{4}{3} \] - Simplify by cross-cancelling:
\[ \frac{\cancel{15}^5}{\cancel{2}_1} \times \frac{\cancel{4}^2}{\cancel{3}_1} = \frac{5 \times 2}{1 \times 1} = 10 \] - Final Answer: The tailor can make 10 skirts.
Common Mistakes
Misconception 1: Multiplying Mixed Numbers Part-by-Part
Mistake Computing \(1\frac{1}{2} \times 2\frac{1}{3}\) as \((1 \times 2) + (\frac{1}{2} \times \frac{1}{3}) = 2\frac{1}{6}\).
Why it is wrong Multiplication distributes over both whole and fractional parts \((1 + \frac{1}{2})(2 + \frac{1}{3}) = 2 + \frac{1}{3} + 1 + \frac{1}{6} = 3\frac{1}{2}\).
Correction Always convert mixed numbers to improper fractions first before multiplying or dividing: \(\frac{3}{2} \times \frac{7}{3} = \frac{7}{2} = 3\frac{1}{2}\).
Misconception 2: Forgetting to Flip the Divisor (Keep-Change-Flip)
Mistake Solving \(\frac{1}{2} \div \frac{3}{4}\) by multiplying straight across to get \(\frac{1 \times 3}{2 \times 4} = \frac{3}{8}\).
Why it happens Students mistake the division sign for multiplication.
Correction Division requires multiplying by the reciprocal of the second fraction: \(\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}\).
Misconception 3: Flipping the First Fraction Instead of the Second
Mistake Writing \(\frac{3}{4} \div \frac{2}{5} = \frac{4}{3} \times \frac{2}{5}\).
Correction Always keep the first fraction unchanged! Only flip the fraction that comes directly after the division sign (the divisor): \(\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}\).
Real World
Cooking Oil Distribution: An entrepreneur in Kisumu buys cooking oil in bulk (a 20-litre jerrycan) and packages it into \(\frac{1}{2}\)-litre and \(\frac{3}{4}\)-litre retail containers. To determine how many \(\frac{3}{4}\)-litre bottles can be filled from 6 litres of oil, they compute \(6 \div \frac{3}{4} = 6 \times \frac{4}{3} = 8\) bottles.
Agriculture & Fertiliser Application: A maize farmer in Eldoret needs to apply \(\frac{2}{5}\) of a bag of fertiliser per acre. For a farm plot measuring \(3\frac{1}{2}\) acres, the total bags needed is \(3\frac{1}{2} \times \frac{2}{5} = \frac{7}{2} \times \frac{2}{5} = \frac{7}{5} = 1\frac{2}{5}\) bags.
Carpentry and Construction: A carpenter cutting timber joists needs to cut a \(4\frac{1}{2}\) m timber plank into pieces that are each \(\frac{3}{8}\) m long. The number of pieces is \(\frac{9}{2} \div \frac{3}{8} = \frac{9}{2} \times \frac{8}{3} = 12\) pieces.
Practice
Grace is making mandazi in Kisumu. Each batch requires \(\frac{2}{3}\) cup of sugar. If she decides to prepare only \(\frac{5}{8}\) of a batch, how many cups of sugar does she need? Give your answer as a simplified fraction.
(Type only the fraction, e.g., 5/12)
Review the concepts above.
A farmer has a piece of sisal rope measuring \(\frac{3}{4}\) metre. He cuts it into smaller pieces of equal length, each measuring \(\frac{1}{8}\) metre. How many pieces of rope does he obtain?
(Type only the number, e.g., 6)
Review the concepts above.
A tailor in Gikomba uses \(1\frac{1}{2}\) metres of kitenge fabric to make one dress. How many metres of fabric will she need to make \(2\frac{1}{3}\) dresses? Give your answer as a simplified improper fraction.
(Type only the fraction, e.g., 7/2)
Review the concepts above.
When mixing animal feed on a farm, a nutritionist multiplies two feed ratios: \(\frac{15}{28} \times \frac{14}{25}\). What is the resulting ratio in simplest form?
(Type only the fraction, e.g., 3/10)
Review the concepts above.
A dairy farmer in Eldoret has \(7\frac{1}{2}\) litres of fresh milk. He pours all the milk into small jugs, each holding \(\frac{3}{4}\) litre. How many full jugs can he fill?
(Type only the number, e.g., 10)
Review the concepts above.
A school store in Nakuru has a sack containing \(22\frac{1}{2}\) kg of sugar. The head cook takes \(\frac{2}{3}\) of this sack and shares it equally among 5 boarding houses. How many kilograms of sugar does each boarding house receive?
(Type only the number, e.g., 3)
Review the concepts above.