Ratios, proportions and scale drawings

A ratio compares two quantities. Mortar is mixed one part cement to three parts sand. A small engine runs on fuel mixed fifty to one with oil. A drawing shows the building at one-fiftieth or one-forty-eighth of its real size. A crew of four lays a certain length of pipe in a day, so a crew of six should lay more. All of these are ratio problems, and they are solved with one idea: when two ratios are equal, you can find any missing number by cross-multiplying.

On Kingsway Tower, Maya meets ratios every day without calling them that: reading the scale on a drawing, mixing glycol for the heating loop test, and working out how long the floor rough-in will take if Luis gets another apprentice. This lesson makes the method explicit so that it works every time, including when the numbers are awkward.

Ratios and parts

A ratio such as 1 : 3 means that for every 1 part of the first thing there are 3 parts of the second. The parts can be any unit, as long as both are measured the same way: shovels, buckets, litres, gallons or kilograms. The total number of parts is the sum of the numbers in the ratio, and each ingredient's share of the total is its number divided by that sum.

The last row is a different kind of ratio. The water-to-cement ratio of concrete is a single number, the mass of water divided by the mass of cement. It controls the strength of the concrete: more water makes concrete easier to place but weaker. That is why adding water to a ready-mix truck on site without the supplier's or engineer's approval is not allowed on most jobs.

Solving proportions by cross-multiplying

A proportion says that two ratios are equal. If three of the four numbers are known, the fourth is found by cross-multiplying: multiply diagonally across the equals sign and divide by the remaining number.

The key is to set up the ratios with the same units in the same positions: litres over litres, or metres over hours, on both sides. If you write the first ratio as glycol over water, the second must also be glycol over water.

Scale drawings

Every drawing on Kingsway Tower is drawn to scale: each line on the paper is a fixed fraction of the real length. The scale is written in the title block and often under each view. Knowing how to read it lets you check a dimension that is missing, estimate a length quickly, or sketch your own layout to scale.

Canadian drawings use metric ratio scales. A scale of 1 : 50 means 1 mm on the drawing represents 50 mm on site. Floor plans are often drawn at 1 : 100 or 1 : 50, details at 1 : 20, 1 : 10 or 1 : 5, and site plans at 1 : 200 or 1 : 500. Some drawings for imperial-sized products, and older drawings, use architectural scales such as 1/4 in = 1 ft 0 in, so it is worth knowing both.

There is an important caution: never rely on scaling a drawing when a dimension is given or can be calculated from given dimensions. Paper stretches, prints are often reduced, and PDF copies are frequently printed at fit-to-page instead of full size. Many drawings carry a note such as "do not scale" for exactly this reason. Scaling is for estimating and checking, and for finding approximate locations so you can ask the right question. When a critical dimension is missing, raise it with the foreman so that it can go to the designer as a request for information.

A quick way to check whether a print is at true size is to measure the graphic scale bar printed on most drawings. If the bar labelled 5 m measures 100 mm on a 1 : 50 drawing, the print is full size; if it measures 71 mm, the sheet was reduced (in that case to an A-size smaller, or from 24 in by 36 in down to 11 in by 17 in), and every scaled length must be adjusted.

Direct and inverse proportion: crews and rates

Two quantities are directly proportional if doubling one doubles the other. Material and length are directly proportional: twice the length of pipe needs twice the hangers. Many job-planning problems use a rate, an amount per unit of time or per unit of something else, and direct proportion.

Two quantities are inversely proportional if doubling one halves the other. In ideal conditions, crew size and time are inversely proportional: twice the workers finish in half the time. Real jobs are not quite ideal, because people get in each other's way and some tasks cannot be split, but the inverse rule is a useful first estimate.

Ratios and proportions in code

This program splits a total into parts by a ratio and converts lengths with a drawing scale, the two calculations in this lesson that come up most often.

def split(total, *parts):
    """Divide `total` in the ratio given by `parts`, e.g. split(85, 3, 7)."""
    s = sum(parts)
    return [total * p / s for p in parts]

def real_from_drawing(measured, ratio):
    """Real length from a length measured on a drawing at 1:ratio (same units)."""
    return measured * ratio

print("mortar 1:3   ", [round(v, 3) for v in split(0.12, 1, 3)])
print("concrete 1:2:3", [round(v, 3) for v in split(1.5, 1, 2, 3)])
print("glycol 3:7   ", [round(v, 2) for v in split(85, 3, 7)])
for m in [47, 112]:
    print(f"{m} mm on a 1:50 drawing is {real_from_drawing(m, 50):g} mm on site")

mortar 1:3    [0.03, 0.09]
concrete 1:2:3 [0.25, 0.5, 0.75]
glycol 3:7    [25.5, 59.5]
47 mm on a 1:50 drawing is 2350 mm on site
112 mm on a 1:50 drawing is 5600 mm on site

The scale function works for any ratio scale. For a 1 : 50 drawing, 112 mm on paper is 5600 mm on site.

Comparing by unit price

A ratio also answers the everyday question of which package is the better buy. Divide each price by its quantity to get a unit price, then compare like with like. A box of 100 fasteners for 18.50 costs 0.185 each; a box of 250 for 41.25 costs 0.165 each, so the bigger box is cheaper per piece, provided the job will actually use them. The same method compares a bulk bag of sand with bags, or a long coil of tubing with straight lengths. Write the unit next to every unit price, such as per metre, per foot, per piece or per kilogram, so that two prices with different units are never compared by mistake.