Fraction verification and exact numerical probes
Recognise a valid transformation and its original domain, then use the particular result displayed by the task
- Record restrictions first Every original denominator must be nonzero. When dividing, the divisor must also be nonzero
- Cancel whole factors Factorise where needed. Divide a complete numerator and denominator by the same nonzero factor; do not cancel addends
- Multiply or use a reciprocal Multiply numerators and denominators. For division, reverse only the divisor before multiplying
- Use a common denominator For addition and subtraction, convert both fractions to the same denominator. Subtract the entire second numerator
- Keep the original domain A cancelled factor still excludes its original zero. An equivalent simplified rule is used only at the original allowed inputs
- Original domain first Exclude zeros of every original denominator. For division, also exclude inputs making the divisor zero. A cancelled factor does not restore an excluded input
- Read the chosen displayed form Use the numerator and denominator of the supplied simplified result for the numerical probe. An equivalent rescaled fraction can have different numerator and denominator values
- One check, not a proof Matching at one allowed input is a useful check, not an identity proof. The algebraic transformation and original domain must hold throughout the allowed domain
Worked example
Simplify, retain the original domain, and check the displayed result at x = 2.
- \(\displaystyle x\ne-3,\quad x\ne1,\quad x\ne0\)The first two restrictions come from original denominators; the divisor is zero at 0.
- \(\displaystyle \frac{(x-3)(x+3)}{x+3}\times\frac{x-1}{2x}\)Factor the difference of squares and reverse only the divisor
- \(\displaystyle \frac{(x-3)(x-1)}{2x},\quad x\ne-3,0,1\)Cancel the whole factor x + 3, retaining every original restriction
- \(\displaystyle N(2)=-1,\quad D(2)=4,\quad \frac{N(2)}{D(2)}=-\frac14\)Evaluate this displayed form at the allowed input. One numerical agreement is not a general identity proof
Common mistakes
Difficulty levels
Questions about this topic
Why does cancelling a factor not restore an input?+
The original denominator was zero at that input. The simplified rule agrees with the original only on its original domain
What extra restriction appears in division?+
The divisor must be defined and nonzero. Record zeroes of its original denominator and its numerator before reversing it
What does a zero coefficient mean?+
The corresponding power is absent. For example, x² − 4 has x coefficient zero, and x has constant zero
Are the three probe entries independent algebra skills?+
No. They are components of one numerical check using a supplied expression, not separate successful fraction manipulations
Can equivalent numerator and denominator pairs be used in the probe?+
The quotient can be represented in many ways, but this probe asks for the particular displayed numerator and denominator values. Follow the named form rather than changing its scale
Does the verification bank assess freely written fractions?+
No. Its setup and result responses are choices. The preceding topic constructs a canonical coefficient table; unrestricted expression writing and a particular written derivation remain outside both response contracts
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 2: Algebraic foundations