Ratios and Proportion
Part of Mathematics
Using ratios to scale recipes, mix materials, divide resources fairly, and calculate unknown quantities from known relationships.
Why This Matters
A ratio is a comparison of two quantities. Proportions are ratios that stay constant as amounts change. These two concepts appear in almost every practical calculation a rebuilding community faces. Mortar is mixed 1 part lime to 3 parts sand — that is a ratio. A medicine is dosed at 5 mg per 10 kg of body weight — that is a proportion. A map shows 1 cm representing 500 m — that is a scale ratio. Dividing 47 kg of grain among 3 families in shares of 2:3:4 — that requires ratio arithmetic.
Ratios and proportions are the mathematical language of recipes, formulas, maps, and fair distribution. Unlike addition or multiplication, which deal with absolute quantities, ratios deal with relationships — and it is relationships that determine material properties, medicine doses, architectural proportions, and economic fairness. A mortar mixture that is off by even 10% in its ratio can fail; a medicine that is wrong by 50% in dose can kill or have no effect.
Mastering ratio and proportion calculation is not just practical — it is a prerequisite for chemistry (see Alcohol and Distillation), medicine dosing, and eventually engineering calculations.
What a Ratio Is
A ratio compares two quantities of the same kind. The ratio 3:4 (read “three to four”) means “for every 3 of the first thing, there are 4 of the second.”
Equivalent ratios: 3:4 = 6:8 = 15:20 = 30:40. All express the same relationship; they are the same ratio at different scales.
Simplifying ratios: Divide both parts by their GCD (greatest common divisor).
- 12:16 → GCD is 4 → 3:4
- 15:25 → GCD is 5 → 3:5
- 100:250 → GCD is 50 → 2:5
Ratios with three or more parts: 2:3:5 means the whole is divided into 2+3+5=10 parts; the first gets 2/10, the second 3/10, the third 5/10.
Proportion: Scaling a Ratio
A proportion states that two ratios are equal:
a/b = c/d (or equivalently, a:b = c:d)
The most useful form of this relationship is the cross-multiplication rule:
If a/b = c/d, then a × d = b × c
This lets you find any unknown if you know the other three values.
Standard setup:
- Known ratio: K₁ : K₂
- Scaled situation: x : S (where x is unknown, S is known)
- Solve: x = (K₁ × S) ÷ K₂
Material Mixing Calculations
Mortar and Concrete
Lime mortar: 1 part lime putty : 3 parts sand by volume
Question: You have 45 liters of lime putty. How much sand do you need?
- Ratio: 1:3
- Sand = 3 × 45 = 135 liters
Question: You want to make 200 liters total of mortar (1:3 ratio). How much of each?
- Total parts = 1 + 3 = 4
- Lime = (1/4) × 200 = 50 liters
- Sand = (3/4) × 200 = 150 liters
- Check: 50 + 150 = 200 ✓
Concrete (rougher mix): 1 part cement : 2 parts sand : 3 parts gravel
For 600 liters of concrete:
- Total parts = 6
- Cement = (1/6) × 600 = 100 liters
- Sand = (2/6) × 600 = 200 liters
- Gravel = (3/6) × 600 = 300 liters
Gunpowder: 75:15:10 (Saltpeter:Charcoal:Sulfur)
For 1 kg total:
- Saltpeter: 750 g
- Charcoal: 150 g
- Sulfur: 100 g
Scaling to 3 kg total:
- Saltpeter: 3 × 750 = 2,250 g
- Charcoal: 3 × 150 = 450 g
- Sulfur: 3 × 100 = 300 g
Dividing Quantities in a Given Ratio
Method:
- Sum the ratio parts
- Divide the total by the sum
- Multiply each part value by the ratio number
Example: Divide 84 kg of grain among three workers in the ratio 2:3:5.
- Sum of parts: 2 + 3 + 5 = 10
- Unit value: 84 ÷ 10 = 8.4 kg per part
- Worker A: 2 × 8.4 = 16.8 kg
- Worker B: 3 × 8.4 = 25.2 kg
- Worker C: 5 × 8.4 = 42 kg
- Check: 16.8 + 25.2 + 42 = 84 ✓
Example: Three families contributed labor in the ratio 4:4:6 (days worked). Divide 140 kg of communal harvest accordingly.
- Sum: 4 + 4 + 6 = 14
- Unit: 140 ÷ 14 = 10 kg
- Family A: 4 × 10 = 40 kg
- Family B: 4 × 10 = 40 kg
- Family C: 6 × 10 = 60 kg
Medicine Dosing by Proportion
Dosage is almost always expressed as a ratio: amount of medicine per unit of body weight.
Example: Willow bark extract dose is 2 g per 25 kg body weight. Patient weighs 70 kg. What dose?
- Known ratio: 2 g : 25 kg
- Unknown: x g : 70 kg
- x = (2 × 70) ÷ 25 = 140 ÷ 25 = 5.6 g
Example: A tincture is diluted 1:10 (1 part tincture to 10 parts water). You have 30 mL of tincture. How much water do you add?
- 10 × 30 = 300 mL water
- Total solution: 330 mL at 1:10 dilution (note: 1:10 means 1 per every 10 of water, total 11 parts; adjust if “1 in 10” total)
- Clarify before preparing: “1:10 dilution” can mean either 1+10=11 total (ratio) or 1 in every 10 (proportion to total). Establish a standard in your community’s medical practice.
Map and Scale Calculations
A scale of 1:500 means 1 unit on the map represents 500 units in reality.
Example: A map at 1:500 scale shows a distance of 4.5 cm between two wells. What is the actual distance?
- Actual = 4.5 cm × 500 = 2,250 cm = 22.5 m
Example: You want to draw a field that is 180 m × 120 m on paper at 1:1000 scale. What size drawing?
- Width: 180 ÷ 1000 = 0.18 m = 18 cm
- Height: 120 ÷ 1000 = 0.12 m = 12 cm
Creating a scale: To fit a 450 m distance onto a 30 cm piece of paper:
- Scale = 450 m ÷ 0.30 m = 1500
- Use 1:1500 scale
The Rule of Three (Cross-Multiplication)
In pre-calculator times, the “Rule of Three” was the standard method for proportion problems in trade and commerce. It solves: “If A gives B, what does C give?”
Procedure:
- Write the three known values in order: “If [A] then [B]; if [C] then [?]”
- Answer = (B × C) ÷ A
Example: If 7 workers can build a wall in 12 days, how long will 4 workers take?
- This is an inverse proportion (fewer workers = more days)
- Answer = (12 × 7) ÷ 4 = 84 ÷ 4 = 21 days
Example: If 12 kg of salt costs 3 tokens, what does 20 kg cost?
- This is a direct proportion (more salt = more cost)
- Answer = (3 × 20) ÷ 12 = 60 ÷ 12 = 5 tokens
Direct vs. inverse proportion:
- Direct: As one quantity increases, the other increases proportionally (price, area, volume)
- Inverse: As one quantity increases, the other decreases proportionally (time vs. workers, speed vs. time)
For inverse proportions, multiply the top values before dividing, exactly as above.
Common Ratio Shortcuts
- Doubling: Ratio 1:2 — double one quantity to get the other
- Halving: Ratio 2:1 — half of one gives the other
- Percentage as ratio: 15% = 15:100 = 3:20
- Unit rate: Express as X per 1 unit for easy scaling. “3 liters per day” is easier to scale than “21 liters per week”
Ratio and proportion form the arithmetic backbone of material science, medicine, trade, and resource distribution. A community that can perform these calculations confidently will make better mixtures, fairer distributions, and more accurate estimates than one relying on memory and guesswork.