Proportional Reasoning: Thinking Puzzles

Proportional reasoning shares in a ratio, mixes in the right amounts, scales maps and recipes, and tells direct from inverse proportion.

Key ideas

To share in a ratio, divide the total by the sum of the parts, then multiply by each part.
In a mixture, adding only water leaves the milk unchanged; work from the part that stays the same.
Direct proportion: twice as much of one gives twice as much of the other, like a recipe or a map scale.
Inverse proportion: more workers take fewer days; workers × days stays the same.
Stacking puzzles are solved by trying every order and keeping those that follow all the rules.

Remember

Share ₹600 in 1 : 2 : 3	600 ÷ 6 = 100 → ₹100, ₹200, ₹300
Mixture	20 L milk : water 3 : 1 → 15 L and 5 L; for 3 : 2 add 5 L of water
Inverse proportion	6 workers × 10 days = 60 → 4 workers take 15 days
Map scale 1 cm : 5 km	7 cm on the map = 35 km
Recipe	300 g for 4 people → 450 g for 6
Orders of 4 different blocks	4 × 3 × 2 × 1 = 24

Watch out

Dividing the total by one part instead of by the sum of the parts.
Treating more workers as more days: that is inverse, not direct, proportion.
Changing the milk when only water is added.

Worked example

Four blocks with a circle, a triangle, a square and a star are stacked one on top of another.
(1) The triangle is immediately above the star.
(2) The triangle is somewhere above the square.
(a) How many different stacks follow all the rules? (b) How many follow all of them except rule (2)?

Working: all rules, from the top: circle/triangle/star/square; triangle/star/circle/square; triangle/star/square/circle

Answer: (a) 3; (b) 6