Fractions and Ratios

Part of Mathematics

Using ratios and proportions in practical applications from mixing concrete to calculating gear trains.

Why This Matters

Ratios are the language of recipes, mixtures, and mechanical advantage. Every concrete mix is specified as a ratio (1:2:3 cement-sand-gravel). Every gear train multiplies force or speed through ratios. Every medicine dosage is a concentration expressed as a fraction. Every map represents territory at some scale ratio. Without understanding ratios, you cannot reliably reproduce any mixture, scale any design, or calculate any mechanical advantage.

Fractions appear everywhere that whole numbers are insufficient. You will rarely cut a board to exactly 6 feet — it will be 6 and 3/8 inches, or 6 feet 4 and 1/2 inches. Land measurements, timber dimensions, recipe quantities, and angular measurements all require working fluently with fractions. In a rebuilding scenario, the inability to work with fractions means wasted materials, failed mixtures, and imprecise construction.

The practical difference between a community that understands ratios and one that does not is the difference between consistent, reproducible results and guesswork. When your mortar mix works perfectly, you need the ratio to make it work the same way tomorrow. When your gear reduction gives the right speed for your grinding mill, you need the ratio to build another one.

Working with Fractions

The Basics

A fraction represents a part of a whole: numerator (top) ÷ denominator (bottom).

• 1/4 means 1 part out of 4 equal parts
• 3/8 means 3 parts out of 8 equal parts
• 7/4 means 7 quarter-parts, or 1 and 3/4

Adding and Subtracting Fractions

Fractions must have the same denominator before you can add or subtract:

1. Find the lowest common denominator (LCD) — the smallest number both denominators divide into evenly
2. Convert each fraction to the LCD
3. Add or subtract the numerators
4. Simplify if possible

Adding 3/8 + 1/4

• LCD of 8 and 4 is 8
• Convert: 1/4 = 2/8
• Add: 3/8 + 2/8 = 5/8

Multiplying Fractions

Multiply numerators together and denominators together:

• 3/4 × 2/3 = 6/12 = 1/2
• 5/8 × 4/5 = 20/40 = 1/2

Cancel Before Multiplying

Look for common factors between any numerator and any denominator before multiplying. In 3/4 × 2/3, the 3s cancel (top-left and bottom-right) giving 1/4 × 2/1 = 2/4 = 1/2. This keeps numbers small and manageable.

Dividing Fractions

Flip the second fraction and multiply:

• 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 1 and 1/2
• This answers: “How many halves fit in three-quarters?” Answer: one and a half.

Common Fraction-Decimal Equivalents

Memorize these for quick mental conversion:

Fraction	Decimal	Fraction	Decimal
1/2	0.500	1/8	0.125
1/3	0.333	3/8	0.375
2/3	0.667	5/8	0.625
1/4	0.250	7/8	0.875
3/4	0.750	1/16	0.0625
1/5	0.200	1/6	0.167
1/10	0.100	5/6	0.833

Ratios and Proportions

Understanding Ratios

A ratio compares two or more quantities. Written as a:b or a:b:c.

• 1:3 means “one part to three parts” — the total has 4 parts
• 1:2:3 means “one part to two parts to three parts” — the total has 6 parts

Common Mistake

A 1:3 ratio does NOT mean “one-third.” It means one part out of four total. The fraction equivalent is 1/4 and 3/4. Getting this wrong doubles your error in mixing.

Solving Proportions

When two ratios are equal, you can cross-multiply to find an unknown:

If a/b = c/d, then a × d = b × c

Scaling a Recipe

A mortar recipe calls for 8 shovels of sand for every 2 shovels of lime. You have 13 shovels of sand. How much lime do you need?

• Set up: 8/2 = 13/x
• Cross multiply: 8x = 2 × 13 = 26
• Solve: x = 26/8 = 3.25 shovels of lime

The Unitary Method

Find the value of one unit first, then scale:

1. If 5 workers dig a trench in 12 days, how long for 8 workers?
2. Total work = 5 × 12 = 60 worker-days
3. With 8 workers: 60 ÷ 8 = 7.5 days

This method works for any proportion problem and is easy to understand intuitively.

Practical Mixing Ratios

Construction Mixes

Material	Ratio	Parts	Per Batch (buckets)
Standard concrete	1:2:3 (cement:sand:gravel)	6 total	1 + 2 + 3
Strong concrete	1:1.5:2.5 (cement:sand:gravel)	5 total	2 + 3 + 5
Mortar (laying)	1:1:6 (cement:lime:sand)	8 total	1 + 1 + 6
Mortar (pointing)	1:1:4 (cement:lime:sand)	6 total	1 + 1 + 4
Lime plaster	1:3 (lime putty:sand)	4 total	1 + 3
Cob (earthen wall)	1:1:1 (clay:sand:straw)	3 total	1 + 1 + 1

Measuring by Volume, Not Weight

In field conditions, measure dry ingredients by volume (shovelfuls, bucketfuls). Be consistent — use the same container for every measurement. Level off the top, do not heap. Heaped measurements can vary by 30% or more.

Medicine and Solution Ratios

Concentration ratios are critical for safety:

• 1:10 dilution = 1 part concentrate + 9 parts water (NOT 1 part in 10 parts water)
• 10% solution = 10 grams solute per 100 mL total solution
• Parts per thousand = grams per liter

Life-Critical

A 1:100 dilution and a 1% solution are the same thing. But mixing “1 part per 100” when you mean “1 part plus 100 parts” gives a 1:101 ratio — close enough for concrete, dangerously wrong for medicine. Always clarify whether ratios include or exclude the first ingredient.

Gear Ratios and Mechanical Advantage

Simple Gear Ratios

When two gears mesh, the ratio of their teeth determines speed and torque:

• Gear ratio = driven teeth ÷ driving teeth
• A 40-tooth gear driven by a 10-tooth gear: ratio = 40/10 = 4:1
• The driven gear turns 4 times slower but with 4 times the torque

Compound Gear Trains

Multiply the individual ratios:

1. First stage: 10-tooth drives 30-tooth = 3:1
2. Second stage: 10-tooth (on same shaft as 30-tooth) drives 40-tooth = 4:1
3. Total ratio: 3 × 4 = 12:1

Grinding Mill Design

A water wheel turns at 8 RPM. You need the grinding stone to turn at 96 RPM.

• Required ratio: 96/8 = 12:1 (speed increase, so invert for gears)
• Use two stages: 3:1 × 4:1 = 12:1
• First pair: 30-tooth on waterwheel shaft drives 10-tooth
• Second pair: 40-tooth on intermediate shaft drives 10-tooth
• Result: 12x speed multiplication at 1/12 the torque

Pulley Ratios

Every additional pulley in a block-and-tackle doubles the mechanical advantage but halves the speed of lift:

Pulleys	Mechanical Advantage	Pull 1 lb to lift…	Rope pulled per 1 ft lifted
1	1:1	1 lb	1 ft
2	2:1	2 lbs	2 ft
3	3:1	3 lbs	3 ft
4	4:1	4 lbs	4 ft
6	6:1	6 lbs	6 ft

Scale and Proportion in Design

Map Scales

A map scale of 1:1000 means 1 unit on the map equals 1000 units in reality:

• 1 cm on map = 10 meters on ground
• 1 inch on map = 1000 inches (83.3 feet) on ground

To create a useful site map:

1. Choose a scale that fits your paper (1:100 for a building, 1:1000 for a village, 1:10000 for a district)
2. Divide every real measurement by the scale factor
3. Draw at the reduced measurements
4. Label the scale on the map for anyone else who uses it

Scaling Structural Designs

Scaling Limits

Structures do NOT scale linearly. A beam twice as long needs to be more than twice as strong because the bending moment increases with the square of the length. Doubling all dimensions of a building multiplies its weight by 8 (cubed) but its cross-sectional strength only by 4 (squared). This is why insects can carry many times their weight but elephants cannot — it is called the square-cube law. Always consult structural principles when scaling up designs significantly.

Proportion in Practice

The golden ratio (approximately 1:1.618) and simple whole-number ratios like 2:3, 3:5, and 5:8 produce aesthetically pleasing proportions for doors, windows, and facades. These are not arbitrary — cultures worldwide have converged on similar proportions because they reflect natural structural efficiencies and visual balance.

For practical building:

• Door proportions: roughly 2:5 (width to height) — a 3-foot door is about 7 feet tall
• Window proportions: roughly 1:2 or 2:3 (width to height)
• Room proportions: 2:3 or 3:4 (width to length) tend to feel balanced

These proportions guide initial design. Structural requirements and available materials will always take priority.