Percentages and Ratios in Engineering Work

In plain English

A percentage is a share out of one hundred: 15% means 15 parts in every 100. A ratio compares two or more quantities: a 1 : 2 : 4 concrete mix has twice as much sand as cement and four times as much aggregate. Both appear everywhere in engineering — efficiencies, scrap rates, tolerances, price changes, mixtures and drawing scales.

The arithmetic is simple, but a few traps catch even experienced people: percentages of different bases, “percent” versus “percentage points”, and increases that do not cancel decreases.

Technical definition

A percentage is a ratio multiplied by 100: p% = (part ÷ whole) × 100. Percentage change compares a new value with an original: (new − old) ÷ old × 100. A ratio a : b states that the quantities are in proportion a to b; it can be simplified by dividing all terms by their greatest common divisor, and a proportion a : b = c : x is solved by cross-multiplication, x = b × c ÷ a. The difference between two percentages is measured in percentage points.

How it works and where it is used

Always ask “percent of what?”. A 20% discount followed by a 20% price rise does not bring a price back where it started, because the second percentage is taken of a smaller amount. In the same way, efficiencies of machines in a chain multiply rather than add: a motor, gearbox and pump of 95%, 92% and 98% efficiency together give 0.95 × 0.92 × 0.98 = 85.7%.

Formula

p% of x = x × p ÷ 100     % change = (new − old) ÷ old × 100

a : b = c : x ⇒ x = b × c ÷ a     Part_i = T × term_i ÷ Σ terms

T = total to be divided. Reverse percentage: original = final ÷ (1 + p/100) for an increase, or final ÷ (1 − p/100) for a decrease.

Worked example

• Percentage of a value: 15% of 240 = 240 × 15 ÷ 100 = 36.
• Percentage change: a part’s price rises from 1200 to 1380: (1380 − 1200) ÷ 1200 = +15%.
• Percentage points: scrap falls from 4% to 3%. That is a fall of 1 percentage point, but a 25% reduction in scrap (1 ÷ 4).
• Reverse percentage: an invoice total of 1170 includes 17% sales tax. The price before tax is 1170 ÷ 1.17 = 1000 (not 1170 − 17% = 971.10).
• Up then down: +10% then −10% gives 1.10 × 0.90 = 0.99, a net −1%.
• Combined efficiency: 95% × 92% × 98% = 85.7%.
• Mix ratio: 0.84 m³ of dry materials in a 1 : 2 : 4 mix → one part = 0.84 ÷ 7 = 0.12 m³: cement 0.12, sand 0.24, aggregate 0.48 m³.
• Scale: on a 1 : 50 drawing, a line of 37 mm represents 37 × 50 = 1850 mm = 1.85 m.

In the workshop and in practice

On the shop floor, percentages describe OEE, yield, scrap, utilisation and efficiency; in purchasing, discounts, taxes and price changes; in the laboratory and on site, concentrations and mix proportions. Writing the base of every percentage (“% of planned time”, “% of good parts started”) avoids most misunderstandings in reports and meetings.

Common mistakes and misconceptions

• Subtracting a percentage to remove tax or a markup. Divide by (1 + p/100) instead.
• Adding efficiencies or losses in series. Multiply the efficiencies.
• Confusing percent with percentage points. A change from 4% to 3% is 1 point, or 25%.
• Mixing ratio bases. Mix ratios by volume and by weight are different; check which one the specification uses.

Related calculators

• Percentage Calculator — P% of a number, what % one value is of another, % change, increase/decrease, reverse % and % difference.
• Ratio & Proportion Calculator — Simplify a ratio A:B(:C), solve a proportion A:B = C:x, or share a total in a ratio.
• Fractions Calculator — Add, subtract, multiply and divide fractions and mixed numbers; simplify; decimal to exact fraction.

Related topics

• Mean, Standard Deviation and the Normal Curve in Quality Control — Using the average and the spread of measurements to judge how well a process meets its tolerance.

References

• ISO 80000-1: Quantities and units — General (ratios and percentages)
• BIPM: The International System of Units (SI Brochure) — use of the per cent sign