Non-routine problem solving (Grade 5)
What Grade 5 tests
Grade 5 problems often hide a fraction or ratio inside a story. The reliable method is to draw a unit bar or tape diagram, label the known part, and find what one unit represents. Percentage problems are ratio problems with a denominator of 100.
Core rules
- Fraction change: if a tank goes from 1/4 full to 2/3 full after 30 L are added, the 30 L equals the difference 2/3 − 1/4 = 5/12 of the capacity.
- Average: average = total ÷ number of items. If you know the average and the count, multiply to recover the total.
- Percentage: "of" means multiply; "what percent" means part ÷ whole × 100.
- Composite area: add non-overlapping pieces, or subtract a cut-out from a surrounding rectangle.
Worked example
A tank is 1/4 full. After 30 L are added, it is 2/3 full. Find the capacity. The added water is the difference: 2/3 − 1/4 = 8/12 − 3/12 = 5/12 of the tank. So 5/12 of the capacity is 30 L. One twelfth is 30 ÷ 5 = 6 L, and the whole tank is 12 × 6 = 72 L. Check: 1/4 of 72 = 18 L; 18 + 30 = 48 L; 2/3 of 72 = 48 L. Answer: 72 L.
Traps
- Subtracting the fractions the wrong way round and getting a negative change.
- Adding percentages of different wholes, such as a discount then a tax on the reduced price.
- Using the average of two speeds instead of total distance ÷ total time.
- Forgetting that 1/4 of the tank is already full, so the 30 L is not 2/3 of the tank.