A hard question on number puzzles for P4–P5, with a step-by-step solution, diagrams and the answer. Number puzzles and made-up operations appear in the Hua Xia Cup from the P1 and P2 first rounds to the P6 finals.
The question
A new operation ◎ is defined by a ◎ b = m × a + n × b, where m and n are two fixed whole numbers that we aren't told.
We do know these two results:
3 ◎ 6 = 331 ◎ 6 = 27
What is 5 ◎ 9?
Show the hints (3)
💡Hint 1: Write each result out using the rule: you get two lines about m and n.
💡Hint 2: Multiply the lines so both have the same number of m, then subtract: m disappears.
💡Hint 3: With m gone you get 12 × n = 48.
Worked solution
Have a go first, then read the solution step by step.
1.By the rule: 3 × m + 6 × n = 33 (first line); 1 × m + 6 × n = 27 (second line).
2.First line as it is: 3 × m + 6 × n = 33; second line times 3: 3 × m + 18 × n = 81.
3.Subtract: the 3 m go, leaving 12 × n = 81 − 33 = 48, so n = 4.
4.Back in the first line: 3 × m = 33 − 6 × 4 = 9, so m = 3.
5.So a ◎ b = 3 × a + 4 × b, and 5 ◎ 9 = 3 × 5 + 4 × 9 = 51.
⚠️Common mistake: guessing m and n from one result. One line alone allows many pairs; it takes both lines to pin them down.
Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download