Year 8 maths test: free practice questions with answers

Year 8 sample test

10 questions, one from each of 10 topics. Work them out on paper, then open each answer to mark it.

1. Question 1Linear relations in real-life problems

   A mobile plan charges a flat monthly fee of $15 and $0.08 per text message sent. In a month where Maya sends $n$ messages, her bill is given by $C = 0.08n + 15$. How many text messages did Maya send if her bill was $36.20? Give your answer as a whole number.

   Show the answer and working

   Answer:

   $265$

   Key idea: Substitute the known value of $C$ into the linear equation and solve for the unknown $n$.

   1. Write the equation with $C = 36.20$:
      $0.08n + 15 = 36.20$
   2. Subtract 15 from both sides:
      $0.08n = 21.20$
   3. Divide both sides by $0.08$:
      $n = \frac{21.20}{0.08} = 265$
   4. Check: $0.08 \times 265 + 15 = 21.20 + 15 = 36.20$ ✓

   Answer: Maya sent $265$ text messages.

   More linear relations in real-life problems practice →

2. Question 2Volume

   A trapezoidal prism has a cross-section with parallel sides of $6$ cm and $10$ cm and a perpendicular height of $4$ cm. The prism is $9$ cm long. Find the volume of the prism in cm³.

   Show the answer and working

   Answer:

   $288 \text{ cm}^3$

   Key idea: Volume of a right prism = area of cross-section × length.

   1. Identify the cross-section: a trapezium with parallel sides $a = 6$ cm and $b = 10$ cm, and perpendicular height $h = 4$ cm.
   2. Find the area of the trapezium: $A = \frac{1}{2}(a + b)h = \frac{1}{2}(6 + 10) \times 4 = \frac{1}{2} \times 16 \times 4 = 32 \text{ cm}^2$
   3. Multiply by the length of the prism: $V = A \times l = 32 \times 9 = 288 \text{ cm}^3$
   4. Check units: all measurements were in cm, so volume is in cm³.

   Answer: $288 \text{ cm}^3$

   More volume practice →

3. Question 3Percentages

   Priya sells cars and earns a 4% commission on every sale, plus a weekly base pay of $320. In one week she sells a car for $18 500 and a car for $7 250. How much does Priya earn that week in total?

   Show the answer and working

   Answer:

   $1 350

   Key idea: Commission is a percentage of the total sales value. Add base pay separately.

   1. Find total sales: $18\,500 + 7\,250 = 25\,750$
   2. Calculate 4% commission on total sales: $25\,750 \times 0.04 = 1\,030$
   3. Add base pay: $1\,030 + 320 = 1\,350$

   Answer: Priya earns $1 350 that week.

   More percentages practice →

4. Question 4Fractions and decimals

   Convert $\dfrac{9}{16}$ to a decimal.

   Show the answer and working

   Answer:

   $0.5625$

   Key idea: A fraction whose denominator has only factors of 2 and 5 produces a terminating decimal.

   1. Factorise the denominator: $16 = 2^4$, so the only prime factor is 2. The decimal will terminate.
   2. Perform the division: $9 \div 16$.
   3. $9 \div 16 = 0.5625$ (multiply numerator and denominator by $5^4 = 625$ to see it clearly: $\dfrac{9 \times 625}{16 \times 625} = \dfrac{5625}{10000}$).
   4. Verify: $0.5625 \times 16 = 9$. ✓

   Answer: $0.5625$

   More fractions and decimals practice →

5. Question 5Time and time zones

   Convert these times: (a) 7:45 am to 24-hour time, (b) 20:30 to 12-hour time.

   Show the answer and working

   Answer:

   (a) 07:45 and (b) 8:30 pm

   Key idea: In 24-hour time, hours run from 00 to 23. For pm times after 12 noon, add 12 to the hour. For 24-hour times from 13:00 onwards, subtract 12 and write pm.

   1. (a) 7:45 am is before noon, so the hour stays the same. Write the hour with two digits: $07{:}45$.
   2. (b) $20 - 12 = 8$, so 20:30 is 8:30 pm.

   Answer: (a) $07{:}45$ and (b) 8:30 pm

   More time and time zones practice →

6. Question 6Pythagoras' theorem

   A ladder leans against a wall. The base of the ladder is $1.8$ m from the wall and the ladder reaches $2.4$ m up the wall. How long is the ladder, correct to 2 decimal places? Give your answer in metres.

   Show the answer and working

   Answer:

   $3.00$ m

   Key idea: The ladder is the hypotenuse of a right-angled triangle with legs $1.8$ m and $2.4$ m.

   1. Write Pythagoras' theorem: $c^2 = a^2 + b^2$
   2. Substitute the known values: $c^2 = 1.8^2 + 2.4^2$
   3. Evaluate the squares: $c^2 = 3.24 + 5.76 = 9.00$
   4. Take the square root: $c = \sqrt{9.00} = 3$
   5. Round to 2 decimal places: $c = 3.00$ m

   Answer: $3.00$ m

   More pythagoras' theorem practice →

7. Question 7Rates

   Orange juice is sold in two sizes. A 600 mL bottle costs $3.00 and a 1.5 L bottle costs $6.60. Which size is the better buy?

   Show the answer and working

   Answer:

   The 1.5 L bottle (0.44 cents/mL vs 0.50 cents/mL)

   Key idea: Convert both options to the same unit, then find the cost per millilitre (unit price) for each. The lower unit price is the better buy.

   1. Convert 1.5 L to millilitres: $1.5 \text{ L} = 1500 \text{ mL}$
   2. Unit price of the 600 mL bottle: $\frac{300}{600} = 0.50 \text{ cents/mL}$
   3. Unit price of the 1500 mL bottle: $\frac{660}{1500} = 0.44 \text{ cents/mL}$
   4. Compare: $0.44 < 0.50$, so the 1.5 L bottle has the lower unit price.

   Answer: The 1.5 L bottle is the better buy at 0.44 cents/mL.

   More rates practice →

8. Question 8Integer operations

   Evaluate $-14 + 9 - (-5)$.

   Show the answer and working

   Answer:

   $0$

   Key idea: Subtracting a negative is the same as adding its positive. Then work left to right.

   1. Rewrite the subtraction of a negative: $-14 + 9 - (-5) = -14 + 9 + 5$
   2. Add the positive integers first as a mental strategy: $9 + 5 = 14$, so $-14 + 14$
   3. Evaluate: $-14 + 14 = 0$

   Answer: $0$

   More integer operations practice →

9. Question 9Probability

   A bag contains 3 cards numbered 1, 2 and 3. One card is drawn and not replaced, then a second card is drawn. Construct a two-way table of all possible outcomes and find the probability that the sum of the two cards is greater than 4.

   Show the answer and working

   Answer:

   $\dfrac{1}{3}$

   Key idea: In a two-way table for sampling without replacement, a card cannot appear twice in the same draw, so outcomes where both draws give the same number are impossible and are crossed out.

   1. Set up a table with the first draw as rows (1, 2, 3) and the second draw as columns (1, 2, 3).

   First \ Second	1	2	3
   1	,	(1,2)	(1,3)
   2	(2,1)	,	(2,3)
   3	(3,1)	(3,2)	,

   2. There are 6 equally likely outcomes.
   3. Find the sums: $(1,2)\to 3$, $(1,3)\to 4$, $(2,1)\to 3$, $(2,3)\to 5$, $(3,1)\to 4$, $(3,2)\to 5$.
   4. Outcomes with sum $> 4$: $(2,3)$ and $(3,2)$, that is 2 outcomes.
   5. $P(\text{sum} > 4) = \frac{2}{6} = \frac{1}{3}$

   Answer: $P(\text{sum} > 4) = \dfrac{1}{3}$

   More probability practice →

10. Question 10Statistics

    A news article states: "9 out of 10 dentists recommend ShineBright toothpaste." The survey contacted 10 dentists by email and received responses from 10 of them. All 10 were dentists employed by the company that makes ShineBright. Identify two problems with this statistical claim.

    Show the answer and working

    Answer:

    (1) The sample size of 10 is too small to be reliable. (2) All dentists surveyed work for the company making the product, creating a conflict of interest and bias.

    Key idea: When evaluating a statistical claim, consider whether the sample is representative of the population, whether the sample size is large enough, and whether there are any conflicts of interest that could introduce bias.

    1. The sample size is only 10 dentists. This is far too small to represent the views of all dentists, so the claim cannot be reliably generalised.
    2. All 10 dentists are employed by the company that makes ShineBright. This is a conflict of interest: they may feel pressure to recommend their employer's product, introducing strong bias into the results.
    3. Together, these problems mean the claim "9 out of 10 dentists recommend ShineBright" is not a reliable or fair representation of dentists' opinions in general.

    Answer: (1) The sample of 10 dentists is far too small to represent all dentists. (2) All dentists surveyed work for the company that makes ShineBright, creating a conflict of interest and strong bias in favour of the product.

    More statistics practice →