Everyday maths: proportions, units and weighted averages

The rule of three is probably the first piece of mathematics you use outside a classroom and the last one you forget. It scales a recipe, converts currency, works out a discount or splits a bill. Its only tricky point is telling direct from inverse proportion, and that is where it pays to slow down.

Direct proportion: more of one, more of the other

Two quantities are directly proportional when multiplying one by a factor k also multiplies the other by k. Examples: price and quantity (if you buy twice as much, you pay twice as much); time and distance at constant speed. Formula: if A/B = C/X, then X = B × C / A.

Inverse proportion: more of one, less of the other

Two quantities are inversely proportional when multiplying one by k divides the other by k. Examples: number of workers and time to complete a task (more workers → less time); speed and time to cover a fixed distance. Formula: A × B = C × X, so X = A × B / C.

How to identify the type

Ask yourself: if I increase the first quantity, does the second increase (direct) or decrease (inverse)? If 3 workers take 10 days, do 6 workers take more or less? → Less. It's inverse. If 3 kg cost €9, do 6 kg cost more or less? → More. It's direct.

Solved example: mixtures

A recipe for 4 people needs 300 g of flour. How much for 7 people? Direct: 4 people → 300 g; 7 people → X. X = 7 × 300 / 4 = 525 g. Another example: if one tap fills a tank in 6 hours, how long do 4 taps take? Inverse: 1 × 6 = 4 × X; X = 6/4 = 1.5 hours.

The compound rule of three

When three or more quantities are involved, the proportions are chained one at a time. If 6 workers build 3 walls in 10 days, how long will 9 workers take to build 5? Analyse each relationship separately: more workers means fewer days (inverse, factor 6/9), and more walls means more days (direct, factor 5/3). The result is 10 × (6/9) × (5/3) = 11.1 days.

When proportionality breaks down

Remember that the rule of three describes a model, and reality does not always cooperate. If one painter takes 6 hours to paint a room, two painters do not always take 3: they can get in each other's way, share a single ladder or need to coordinate. Likewise, buying twice as much does not always cost twice as much, because volume discounts appear.

The most deceptive case is average speeds. If you travel somewhere at 100 km/h and return at 50 km/h, your average speed is not 75 km/h but about 66.7 km/h, because you spend twice as long travelling slowly. Whenever a problem mixes proportions with averages, check the result against a concrete numerical case.

The Pythagorean theorem and its real uses

Few formulas are learned with such reluctance and used so much afterwards. Pythagoras' theorem is not a classroom exercise: it is the tool used to check that a wall is square, to work out the distance between two points on a map, and to measure the television you have just bought.

The theorem and its formula

In any right-angled triangle: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle) and a, b are the legs. It has more than 370 known proofs — more than any other mathematical theorem. The most visual: build squares on each side; the area of the large one equals the sum of the two small ones.

The condition is worth dwelling on, because it is where nearly everyone slips: the triangle has to be right-angled. If none of its angles measures 90 degrees the equality does not hold, and you need the law of cosines, which is its generalisation. Checking where the right angle sits is always the first step.

Applications in architecture and construction

Builders use the theorem to verify that an angle is right without a protractor: if the sides measure 3, 4 and 5 metres (or any multiple), the angle opposite the 5-metre side is exactly 90°. This Pythagorean triple (3-4-5) has been used since Ancient Egypt to mark out land and build pyramids.

GPS and 3D distances

The distance between two points in three-dimensional space is calculated using the theorem's extension: d = √(Δx² + Δy² + Δz²). GPS systems calculate your phone's position by triangulating distances to four satellites using precisely this formula, extended to include relativistic correction.

Screens and resolutions

Screen diagonals (TVs, laptops, phones) are measured in inches using the theorem: if a screen measures 16 × 9 inches, the diagonal is √(16² + 9²) = √(256 + 81) = √337 ≈ 18.4 inches. Manufacturers use the diagonal because it is the largest and most impressive number.

That gives rise to a marketing trap worth knowing: two televisions with the same diagonal can have very different surface areas if their aspect ratios differ. A 21:9 panel at 34 inches has considerably less area than a 16:9 at the same 34 inches, because it is more elongated. The diagonal impresses, but it is the area that determines what you see.

Useful Pythagorean triples

• •3-4-5: the best known, and its multiples 6-8-10, 9-12-15, 30-40-50
• •5-12-13: handy when 3-4-5 is too small
• •8-15-17 and 7-24-25: useful for larger layouts
• •20-21-29: the most balanced of the small ones

These combinations are valuable because they give whole-number sides, so an exact right angle can be marked out with nothing but a tape measure and no decimals. It is precisely the method Egyptian surveyors used with a twelve-knot rope, and the one builders still use today to set out a site.

Weighted averages: proportions with weights

Your grade average decides scholarships, master's places and points in a public examination, and yet almost nobody knows exactly how theirs is calculated. There is a reason for the confusion: Spain runs two different scales alongside several ways of weighting, so the same transcript can yield different figures depending on who does the sums.

Arithmetic mean vs. weighted average

The arithmetic mean gives equal weight to all subjects: add up the grades and divide by the number of subjects. The weighted average gives more weight to subjects with more credits: (sum of grade × credits) / (total credits). For master's and public exam applications, the weighted average is almost always used.

The difference is not academic. A 9 in a 3-credit elective and a 6 in a 12-credit core subject give an arithmetic mean of 7.5 but a weighted average of 6.6. That gap of nearly a full point can be exactly what puts you inside or outside a selection process, and it explains why effort is best concentrated on the subjects carrying the most credits.

The official university scale (Royal Decree 1125/2003)

In Spain, the university grade average is calculated under RD 1125/2003 using numerical values: Fail = 0, Pass = 1, Merit = 2, Distinction = 3, Highest Distinction = 4. This gives a scale of 0 to 4 (not 0 to 10), which can be confusing when universities convert it to a 10-point scale.

Conversion to 0–10 scale

For comparisons with other countries or international applications, many universities offer conversion to a 0–10 scale. The conversion is not straightforward: a grade average of 2.8 out of 4 (high Merit) does not equal 7 out of 10. Each university may use its own equivalence table.

Why the grade average can vary depending on who calculates it

Your university's registry calculates the official grade average using the actual credit weighting of your study plan. Some master's or public exam application platforms use their own formulas that may differ. Always request an official academic certificate with the grade average calculated by your university.

• •Recognised or transferred subjects: some universities include them with their original grade, others exclude them from the calculation
• •Failed subjects: normally only the passed sitting counts, but not always
• •Placement credits and the final project: they weigh heavily because of their high credit load
• •Highest Distinction: worth 4 points on the official scale, not 10

How to check your own calculation

If your estimate does not match the registry's, the cause is usually one of three things: you used the wrong credit values, you included recognised subjects that do not count, or you mixed the 0–4 and 0–10 scales. Check those three before lodging a complaint, and always ask for the subject-by-subject breakdown to locate the discrepancy.

Converting units: proportions with fixed factors

Converting units looks like a mechanical formality, and that is precisely why it goes wrong so often: it gets done without thinking. Conversion errors have brought down aircraft, sunk budgets and destroyed spacecraft, almost always because someone assumed everyone was using the same system.

The Mars Climate Orbiter accident (1999)

NASA lost a $327 million probe because one team used imperial units (pound-force seconds) and another used SI (newton-seconds) without converting. The spacecraft entered the Martian atmosphere at the wrong angle and was destroyed. It is the most expensive unit conversion error in history.

It is not an isolated case. In 1983 an Air Canada Boeing 767 ran out of fuel mid-flight because refuelling was calculated in pounds when the aircraft measured in kilograms: they loaded less than half of what was needed. The plane glided to a disused airfield and landed without casualties, and the incident has been known ever since as the «Gimli Glider».

Temperature: the conversion that isn't multiplication

Unlike almost all other units, temperature is not converted by multiplying by a factor. °C to °F: F = C × 9/5 + 32. °F to °C: C = (F − 32) × 5/9. The zero points of Fahrenheit and Celsius do not coincide. Hence 0°C = 32°F and 100°C = 212°F. Kelvin is multiplicative from absolute zero: K = C + 273.15.

Common length confusions

• •1 statute mile = 1,609.34 m ≠ 1 nautical mile (1,852 m)
• •1 foot = 30.48 cm; 1 inch = 2.54 cm
• •1 yard = 0.9144 m (3 feet)
• •1 light year = 9.461 × 10¹² km (not a unit of time)

Mass: the problem of the 'pound'

In the imperial system there are several 'pounds': the avoirdupois pound (the common one, 453.59 g), the troy pound (used in precious metals, 373.24 g) and the pound-force (not a unit of mass but of force). Gold is quoted in troy ounces (31.10 g), not avoirdupois ounces (28.35 g). The difference is ~10%.

Mass and weight are not the same

In everyday speech they are used interchangeably, but physically they are different quantities. Mass measures the amount of matter and is expressed in kilograms; weight is the force gravity exerts on that mass and is measured in newtons. Your mass is identical on Earth and on the Moon; your weight on the Moon is a sixth. That is why a bathroom scale, which actually measures force, is calibrated for Earth's gravity.

The method that prevents almost every error

The safe way to convert is to write the factor as a fraction whose numerator and denominator are equal in value, positioned so the unwanted unit cancels. To turn 90 km/h into m/s: 90 km/h × (1,000 m / 1 km) × (1 h / 3,600 s) = 25 m/s. The «km» and «h» cancel out and only m/s remains. If the units at the end are not what you expected, the factor was the wrong way up.