Build and interpret grouped models
Brackets preserve the event order; fractions can represent a share of a quantity or division into equal groups
- Define the unknown State what the variable measures and its unit
- Write the whole relationship Include every part of a total, every group and any fixed or individual charge
- Solve and check Clear numerical denominators, solve exactly and substitute into the original description
- Interpret and revise Whole counts must be feasible. Measurements can be fractional. Revise the equation if the group size, fee or grant changes
Worked example
Pieces measuring one third and one quarter of the original ribbon are used, leaving 5 metres with no material lost. Find the original length
\[\frac{L}{3}+\frac{L}{4}+5=L\]
- \(\displaystyle 4L+3L+60=12L\)Multiply every term by 12
- \(\displaystyle 60=5L\)Collect the length terms
- \(\displaystyle L=12\)Divide by 5 and report 12 metres
- \(\displaystyle 4+3+5=12\)The original used and remaining lengths sum correctly
Common mistakes
Subtracting a grant from each share
✕\(\displaystyle \frac{B}{6}-24=19\)✓\(\displaystyle \frac{B-24}{6}=19\)
The $24 is deducted once from the full bill, before equal sharingLeaving out the remainder
✕\(\displaystyle \frac{L}{3}+\frac{L}{4}=L\)✓\(\displaystyle \frac{L}{3}+\frac{L}{4}+5=L\)
The used portions and remaining piece together make the whole ribbonDifficulty levels
introRecognise equations for a described calculation, a rectangle or two related shares
standardSolve number descriptions, grouped calculations and equal expressions; find related dimensions or shares; find maximum affordable counts with whole-number budget boundaries
challengeFind three related shares with a multiplier and offset, or a maximum whole count from a non-integer budget boundary
examChoose the requested numerical answer for the same six model types
Questions about this topic
Can the answer be a fraction?+
A fractional measurement can be valid. A required whole count must satisfy its original conditions; do not round an exact allocation to make it fit
Why is a per-person charge outside the division?+
It is added to each individual share. A single shared charge would instead be included in the total before dividing
What should I report?+
State the requested quantity, its unit, the original check and assumptions such as equal sharing and no extra fees
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 8: Linear equations and inequalities
In this chapter
8A Linear equations and expressions: Solve a linear equationRecognise linear equations8A Linear equations and expressions: Set up and solve a practical linear equation8B Equations with brackets and fractions: Solve a linear equation8B Equations with brackets and fractions: Set up and solve a practical linear equationSolve equations that contain algebraic fractionsSubstitute values into a formulaSubstitute into and rearrange a formulaRead, draw and match inequality number lines8D Linear inequalities: Solve and interpret a linear inequationRead the point of intersection of two linesSolve a pair of simultaneous linear equationsSolve simultaneous equations by substitution8F Simultaneous equations by substitution: Solve a practical problem with simultaneous equations8F Simultaneous equations by substitution: Set up a pair of simultaneous equations from a description8F Simultaneous equations by substitution: Build and use a formula in two unknownsSolve simultaneous equations by elimination8G Simultaneous equations by elimination: Solve a practical problem with simultaneous equations8G Simultaneous equations by elimination: Set up a pair of simultaneous equations from a description8G Simultaneous equations by elimination: Build and use a formula in two unknowns8H Linear inequalities in the coordinate plane: Solve and interpret a linear inequation