Math101 · Independent Ontario learning library
Math101learn.math101.caMultiplying Fractions
Multiplying fractions finds a fraction **of** another quantity. The product $\frac ab\cdot\frac cd=\frac{ac}{bd}$ follows from taking $a/b$ of $c/d$ of one whole.
Fraction multiplication models portions, probabilities, scale factors, area, and rate calculations. Cancellation builds number sense and prepares learners to simplify algebraic products.
Intuition and core definition
Multiplying fractions finds a fraction of another quantity. The product $\frac ab\cdot\frac cd=\frac{ac}{bd}$ follows from taking $a/b$ of $c/d$ of one whole. Denominators $b$ and $d$ must be nonzero. Unlike addition, common denominators are not required.
Notation, language, and conditions
A whole number $n$ may be written $n/1$, and a mixed number must be converted before multiplication. Cancellation divides a numerator and a denominator by the same nonzero common factor; it is simplification of the overall product, not deletion of nearby digits.
Why this idea matters
Multiplying fractions scales a quantity by a part, and factor cancellation reveals the size of that scaling without producing unnecessarily large intermediate numbers.
A dependable method
- Convert mixed numbers to improper fractions and attach the product’s sign.
- Factor numerators and denominators enough to see common factors.
- Cancel common factors across any numerator–denominator pair.
- Multiply remaining numerators and remaining denominators.
- Reduce, convert form if requested, and estimate the size.
Worked example
Representations and interpretation
An area model shades $3/4$ of a rectangle in one direction and $2/3$ in the other; the overlap covers $6/12=1/2$ of the whole. This shows why both numerators and denominators multiply.
Reasoning about variations
Multiplying a positive quantity by a proper fraction makes it smaller; multiplying by an improper fraction greater than one makes it larger. This size reasoning assumes positive factors—negative signs also determine direction on the number line.
Common mistakes
How to check your work
- Estimate with benchmarks such as $0,1/2,$ and $1$.
- Convert terminating fractions to decimals and compare products.
- Undo with division by a nonzero factor and recover the other factor.
Practice
- Compute $\frac79\cdot\frac{27}{14}$.
- Find $\frac35$ of $40$.
- Compute $-1\frac14\cdot\frac8{15}$.
Answers and brief solutions
Show answers
- $\frac32$ Cancel $7$ with $14$ and $27$ with $9$ to obtain $3/2$.
- $24$ $\frac35\cdot40=3\cdot8=24$.
- $-\frac23$ $-\frac54\cdot\frac8{15}$ cancels to $-2/3$.
Synthesis and transfer
Finding two-thirds of three-quarters of a garden uses nested scaling; an area model and the product of numerators and denominators should describe the same occupied region.
The occupied portion is $\frac23\cdot\frac34=\frac12$: first shade three quarters of a rectangle, then retain two thirds of that shaded region. Reversing the order produces the same half, illustrating commutativity through the overlapping area. Units also clarify the operation: multiplying a fraction of the garden by a fractional allocation leaves a fraction of the whole garden. Cancelling before multiplying is a numerical efficiency, not a change to the model, because it divides matching numerator and denominator factors by the same nonzero amount. An estimate guards against an inverted factor: multiplying two positive proper fractions must produce a value smaller than either original factor.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Compute $\frac79\cdot\frac{27}{14}$.
- Cancel $7$ with $14$ and $27$ with $9$ to obtain $3/2$.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
