Worked Solutions
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.
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.