Number & Algebra — Worked Solutions (Year 8 Maths)
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Worked examples for Year 8 Maths Number & Algebra. Each shows where the marks are awarded, the key idea, and the full solution explained by your choice of tutor — Stella, Ella or Cassie.
In short: full step-by-step worked solutions for Year 8 Maths — Number & Algebra. Every question is worked through with the method and reasoning shown, so you can check how to get the answer, not just the final result.
How to use these
Try each question first, then check your working. Use the tutor tabs to read the full solution in the style that suits you: Stella is direct and challenging, Ella is warm and explains the why, and Cassie is concise and analytical.
Example 1 — Ratios, rates and percentages
Question
A recipe for fruit punch mixes orange juice and soda water in the ratio $3 : 5$. A café makes a batch using $1.2$ litres of orange juice.
(a) How much soda water is needed?
(b) After making the batch, the café increases the total volume by $20\%$ for a party. What is the new total volume of punch?
Solution
The ratio is $3 : 5$ (juice : soda), and $3$ parts equals $1.2$ L, so one part is $1.2 \div 3 = 0.4$ L.
(a) Soda is $5$ parts: $5 \times 0.4 = 2.0$ L.
(b) Original total is $1.2 + 2.0 = 3.2$ L. Increasing by $20\%$ means multiply by $1.2$: $3.2 \times 1.2 = 3.84$ L.
Find the value of one part first — it makes both ratio questions trivial. And use the multiplier $1.2$ for a $20\%$ increase rather than working out the extra separately.
A ratio tells us how the parts compare, not the actual amounts — so the first job is to find what one "part" is worth. The juice is $3$ parts and that's $1.2$ L, so each part is $1.2 \div 3 = 0.4$ L.
(a) Soda water is $5$ parts, so it's $5 \times 0.4 = 2.0$ L.
(b) The whole batch is the juice plus the soda: $1.2 + 2.0 = 3.2$ L. To increase by $20\%$, think of the punch as $100\%$ becoming $120\%$, which is the same as multiplying by $1.2$: $3.2 \times 1.2 = 3.84$ L.
The reason the multiplier trick works is that the original amount is $100\%$ of itself, so $100\% + 20\% = 120\% = 1.2$ of the original.
Ratio juice : soda $= 3 : 5$.
- One part $= 1.2 \div 3 = 0.4$ L
- (a) Soda $= 5 \times 0.4 = 2.0$ L
- Total $= 1.2 + 2.0 = 3.2$ L
- (b) $+20\%$: $3.2 \times 1.2 = 3.84$ L
Answers: $2.0$ L soda; $3.84$ L total.
Where the marks go
- 1 mark: Finds the value of one part ($0.4$ L)
- 1 mark: Correct soda volume of $2.0$ L
- 1 mark: Correct new total of $3.84$ L using a $20\%$ increase
Key idea
Find the value of one part of a ratio first; a $20\%$ increase means multiplying by $1.2$.
Example 2 — Expanding, factorising and solving
Question
Solve for $x$: $3(2x - 4) = 5x + 3$.
Solution
Expand the bracket first: $3(2x - 4) = 6x - 12$.
So $6x - 12 = 5x + 3$. Get the $x$ terms on one side: subtract $5x$ from both sides to get $x - 12 = 3$.
Then add $12$: $x = 15$.
Always expand before you collect terms, and check by substituting: $3(2 \times 15 - 4) = 3(26) = 78$ and $5 \times 15 + 3 = 78$. Equal — done.
The bracket is in the way, so we expand it first — multiply everything inside by $3$: $3 \times 2x = 6x$ and $3 \times (-4) = -12$, giving $6x - 12$.
Now the equation reads $6x - 12 = 5x + 3$. We want all the $x$'s together, so subtract $5x$ from both sides; whatever we do to one side we must do to the other to keep it balanced. That leaves $x - 12 = 3$.
Finally, add $12$ to both sides: $x = 15$.
To be sure, put $x = 15$ back in: the left side is $3(30 - 4) = 78$ and the right is $75 + 3 = 78$. They match, so $x = 15$ is correct.
Expand, then collect.
- $3(2x - 4) = 6x - 12$
- $6x - 12 = 5x + 3$
- $-5x$ both sides: $x - 12 = 3$
- $+12$: $x = 15$
Check: $3(26) = 78 = 75 + 3$. $x = 15$.
Where the marks go
- 1 mark: Expands the bracket correctly to $6x - 12$
- 1 mark: Collects $x$ terms to one side correctly
- 1 mark: Correct solution $x = 15$
Key idea
Expand brackets before collecting like terms, and keep the equation balanced by doing the same operation to both sides.
Frequently asked questions
Step-by-step solutions to Number & Algebra questions in Year 8 Maths, with the full method shown for each — so you can follow the reasoning, not just the final answer.
Attempt each question yourself first, then compare your method — not just your answer — against the worked solution. The aim is to learn the approach so you can handle unfamiliar questions, rather than memorising answers.
Yes — they follow the NESA syllabus for Year 8 Maths, using the methods and notation expected in exams and assessments.