Sample Challenging Problems (Grades 4–6)

Number & Algebra

1. Find the smallest positive integer n such that 2n, 3n, and 5n each end with the digit 2.
2. Mary thinks of a number. She triples it, subtracts 7, and then divides by 4 to get 11. What number did she start with?
3. There are two consecutive primes that differ by 2 (a twin prime pair). Give an example and explain why larger twin primes are rarer.

Fractions, Decimals & Ratios

1. A recipe needs 2 1/2 cups of flour for every 3/4 cup of sugar. How much flour is needed for 5 cups of sugar?
2. Three friends split $48 in the ratio 2:3:5. After spending $6, $4, and $10 respectively, who has the most money and how much remains?

Geometry & Visual Reasoning

1. Square ABCD has side 10. A point P inside the square has distances PA=6, PB=8, PC=6. Find PD.
2. A composite figure consists of a rectangle 12×8 and a semicircle of diameter 8 attached to one short side. Find its area (use π≈3.14).

Rates, Work & Mixtures

1. Pipe A fills a tank in 6 hours, Pipe B in 8 hours. Both open for 2 hours, then A is closed. How long more does B take to fill the tank?
2. A 10% salt solution is mixed with a 3% solution to get 5 liters of a 6% solution. How much of each was used? (Explain why this is or isn’t possible.)

Logic & Multi-step

1. In a 5×5 grid, how many different rectangles (of any size) can you draw whose sides lie on the grid lines?
2. At a fair, each ride costs 3 tokens. Sophie buys tokens for $12; each token costs the same. If she gets 3 extra tokens as a bonus, how many rides can she go on and how many tokens remain?

These samples illustrate CML-style reasoning: multi-step thinking, integrating concepts, and visual/number sense. Full worksheets contain dozens of similar problems with detailed solutions.