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.
Level 1 · Warm-up
1 Work out 125 × 48.
Show worked solution
48 = 8 × 6 and 125 × 8 = 1000, so 125 × 48 = 1000 × 6 = 6000. 6250 is 125 × 50, rounding 48 up and never correcting; 5000 is 125 × 40, which drops the 125 × 8; 600 loses a zero.
2 Use a difference of two squares to work out 86 × 94. First write it as 902 − b2.
Show worked solution
86 = 90 − 4 and 94 = 90 + 4, so 86 × 94 = 902 − 42 = 8100 − 16 = 8084.
3 What is 62.5% of 96?
Show worked solution
12.5% = 1/8, so 62.5% = 5/8. 96 ÷ 8 = 12 and 12 × 5 = 60.
Level 2 · Core
4 Which is closest to \(\sqrt{0.0098 \times 4010}\)?
Show worked solution
0.0098 ≈ 0.01 and 4010 ≈ 4000, so the product is about 40. 62 = 36 and 72 = 49, so √40 is between 6 and 7: 6.3. 63 is √4000, which forgets the 0.01; 0.63 is √0.4 and 630 is √400 000, each with the decimal point moved.
5 Write (63 × 50) ÷ (45 × 28) as a fraction in lowest terms.
Show worked solution
63 and 28 share 7: 63 ÷ 7 = 9 and 28 ÷ 7 = 4, leaving (9 × 50) ÷ (45 × 4). 45 ÷ 9 = 5, leaving 50 ÷ (5 × 4) = 50 ÷ 20 = 5/2. Numerator 5, denominator 2.
6 A whole number n has n3 = 103 823. Find n.
Show worked solution
403 = 64 000 and 503 = 125 000, so n is in the 40s. Cubes of 1, 2, …, 9 end in 1, 8, 7, 4, 5, 6, 3, 2, 9, and only 73 = 343 ends in 3. So n = 47. Check: 472 = 2209 and 2209 × 47 = 103 823.
Level 3 · Challenge
7 Three statements:
1 999 × 1001 = 999 999
2 172 − 132 = 120
3 12.5% of 0.48 is 0.6
Which statements are correct?
1 999 × 1001 = 999 999
2 172 − 132 = 120
3 12.5% of 0.48 is 0.6
Which statements are correct?
Show worked solution
999 × 1001 = 10002 − 12 = 999 999 (1 ✓). 172 − 132 = (17 − 13)(17 + 13) = 4 × 30 = 120 (2 ✓). 12.5% = 1/8 and 0.48 ÷ 8 = 0.06, not 0.6 (3 ✗). 1 and 2.
8 899 = a × b, where a and b are whole numbers greater than 1 and a < b. Find a and b.
Show worked solution
899 = 900 − 1 = 302 − 12 = (30 − 1)(30 + 1) = 29 × 31. So a = 29 and b = 31. Both are prime, so no other pair works.
9 Work out 3 × 992 + 6 × 99 + 3.
Show worked solution
3 × 992 + 6 × 99 + 3 = 3(992 + 2 × 99 + 1) = 3(99 + 1)2 = 3 × 1002 = 30 000.