Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
26 questions · official format, shuffled order
1 Expand and simplify (x + 5)(x − 3) − (x − 1)2.
Show worked solution
(x2 + 2x − 15) − (x2 − 2x + 1) = x2 + 2x − 15 − x2 + 2x − 1 = 4x − 16. (4x − 14 writes (x − 1)2 as x2 − 2x − 1; 2x − 16 writes it as x2 + 1, losing the middle term.)
2 What is the next number in this series?
Show worked solution
The gaps are the odd numbers 3, 5, 7, 9, so the next gap is 11: 26 + 11 = 37. The terms are also 12 + 1, 22 + 1, 32 + 1, …, and 62 + 1 = 37 agrees. (35 repeats the gap of 9.)
3 The price of a book rises from $80 to $92. What is the percentage increase?
Show worked solution
The rise is $12, and 12/80 = 15/100 = 15%. (12% treats the $12 as a percentage; 13% divides by the new price $92; 115% is the new price as a percentage of the old.)
4 In a code, each letter is swapped with the letter the same distance from the other end of the alphabet (A ↔ Z, B ↔ Y, C ↔ X, …). GOLD is written as TLOW. How is RING written?
Show worked solution
R (18) ↔ I (9), I (9) ↔ R (18), N (14) ↔ M (13), G (7) ↔ T (20): IRMT. (SJOH just moves each letter one place on; IRNT leaves the N unchanged.)
5 Work out (6 × 105) × (4 × 10−2) in scientific notation.
Show worked solution
6 × 4 = 24 and 105 × 10−2 = 103, so 24 × 103 = 2.4 × 104. (2.4 × 103 moves the decimal point but forgets to raise the index; 2.4 × 107 adds 5 and 2, ignoring the negative; 24 × 104 is not in scientific notation and is ten times too big.)
6 What letter comes next in this series?
Show worked solution
The jumps are 2, 3, 4, 5 places, so the next is 6: O is the 15th letter and the 21st is U. (T repeats the jump of 5.)
7 What is the simple interest on $2 400 invested at 3.5% per year for 4 years?
Show worked solution
3.5% of $2 400 = 3.5 × $24 = $84 a year, and 4 × $84 = $336. ($84 is one year only; $2 736 is the total amount, not the interest; $288 uses 3%.)
8 In this pyramid, each brick is the sum of the two bricks directly below it. Which number replaces the question mark?
Show worked solution
Row 2: n + 3, n + 5, 7. Row 3: 2n + 8, n + 12. Top: 3n + 20 = 44, so 3n = 24 and n = 8. Check: 11, 13, 7 → 24, 20 → 44 ✓.
9 Which line passes through (4, 1) and is perpendicular to y = 2x − 1?
Show worked solution
The perpendicular gradient is −½, and 1 = −½ × 4 + c gives c = 3: y = −½x + 3. (y = −2x + 9 changes only the sign; y = ½x − 1 only flips the fraction; y = 2x − 7 is parallel.)
10 The same rule links the three numbers in each row. Which number replaces the question mark?
Show worked solution
The third number is the first number squared, minus the second: 36 − 10 = 26, 9 − 2 = 7, 25 − 4 = 21. So 64 − ? = 55 and ? = 9. (63 adds 8 and 55.)
11 What is the total surface area of this rectangular prism?
Show worked solution
2 × (20 + 15 + 12) = 2 × 47 = 94 cm2. (47 cm2 counts each face once; 60 is the volume, in cm3.)
12 Solve x2 = 5x + 14.
Show worked solution
x2 − 5x − 14 = (x − 7)(x + 2) = 0, so x = −2 or x = 7. Check −2: 4 = −10 + 14 ✓. (x = −7 or x = 2 has the signs the wrong way round; "7 only" loses the negative root.)
13 Mia cycles 60 km to a town at 30 km/h and returns along the same road at 60 km/h. What is her average speed for the whole trip?
Show worked solution
Out: 60 ÷ 30 = 2 h. Back: 60 ÷ 60 = 1 h. 120 km in 3 h is 40 km/h. (45 km/h averages the two speeds, but she spends twice as long at the slower one.)
14 What is the distance between P(1, −2) and Q(7, 6)?
Show worked solution
√(62 + 82) = √100 = 10. (14 adds the across and up distances; 100 forgets the square root.)
15 4 identical machines fill 240 bottles in 5 minutes. How many bottles do 6 of these machines fill in 8 minutes?
Show worked solution
One machine fills 240 ÷ (4 × 5) = 12 bottles a minute, so 6 × 8 × 12 = 576. (360 changes only the number of machines; 384 changes only the time.)
16 In this right-angled triangle, what is tan θ?
Show worked solution
Opposite θ is 15, adjacent is 8, so tan θ = 15/8. (8/15 is the wrong way up; 15/17 is sin θ; 8/17 is cos θ.)
17 Factorise 2x2 − 18 fully.
Show worked solution
2x2 − 18 = 2(x2 − 9) = 2(x − 3)(x + 3). (2(x − 9)(x + 9) uses 9 instead of √9; 2(x − 3)2 expands to 2x2 − 12x + 18.)
18 A car uses 7.5 litres of fuel per 100 km. Fuel costs $2 a litre. How much does the fuel cost for a 360 km trip?
Show worked solution
7.5 × 3.6 = 27 litres, and 27 × $2 = $54. ($27 is the litres, not the cost; $270 slips a power of ten.)
19 The box plot summarises a set of test scores. About what percentage of the scores lie between 10 and 30?
Show worked solution
10 is the lower quartile and 30 is the maximum, so three quarters of the scores lie between them: 75%. (50% is the box alone, 10 to 20.)
20 How many times is the digit 7 written when all the whole numbers from 1 to 100 are written down?
Show worked solution
Units: 7, 17, 27, …, 97 is 10. Tens: 70 to 79 is 10 more. Total 20. (19 counts 77 only once; 10 counts only one place.)
21 1 March is a Tuesday. What day of the week is 1 June of the same year?
Show worked solution
31 + 30 + 31 = 92 days, and 92 = 13 × 7 + 1, so 1 June is one day later in the week than Tuesday: Wednesday. (Monday counts every month as 30 days; Thursday adds an extra day.)
22 In a class of 30, 18 students play soccer, 12 play basketball and 5 play both. A student is chosen at random. What is the probability that the student plays neither?
Show worked solution
18 + 12 − 5 = 25 play at least one, so 5 play neither: 5/30 = 1/6. (0 forgets that the 5 were counted twice; 5/6 is the chance of playing at least one.)
23 What is the next number in this series?
Show worked solution
Each number is the sum of the two before it: 23 + 37 = 60. (51 adds the last gap of 14 again; 74 doubles 37.)
24 Two cylinders are similar. The smaller is 4 cm tall and holds 64 cm3. The larger is 6 cm tall. What is the volume of the larger cylinder?
Show worked solution
(3/2)3 = 27/8, and 64 × 27/8 = 216 cm3. (96 uses the length factor 3/2; 144 uses the area factor 9/4.)
25 Pens are sold only in packs of 3 for $2. Notebooks cost $5 each. Tom spends exactly $24 and buys at least one pack of pens and at least one notebook. What is the greatest number of pens he can buy?
Show worked solution
$24 − $5n must be even, so n = 2 or 4. With 2 notebooks, $14 buys 7 packs = 21 pens. (36 buys no notebook; 27 and 33 pens would leave $6 or $2, which cannot buy a whole number of notebooks.)
26 What is the next number in this series?
Show worked solution
The steps alternate ×2 and +3: 1 ×2 → 2, +3 → 5, ×2 → 10, +3 → 13, ×2 → 26, so next is +3: 29. (52 doubles again, out of turn; 39 adds 13.)