Fractions and measurements

Why it matters

In the US trades, most measurements are still in inches and fractions of an inch. Electricians measure box heights and conduit runs; mechanics measure shaft lengths, shims and bolt sizes; everyone reads a tape measure many times a day. A mistake of 1/8 inch can mean a bracket that does not line up or a part that does not fit. Fraction fluency is also a big part of apprenticeship aptitude tests.

Core idea: reading a tape measure

On a standard imperial tape, each inch is divided into 16 equal parts. The marks get shorter as the fractions get smaller:

• The longest mark in the middle of each inch is 1/2.
• The next longest marks are 1/4 and 3/4.
• Then 1/8, 3/8, 5/8, 7/8.
• The shortest marks are the sixteenthsThe smallest marks on a standard imperial tape, which divides each inch into 16 equal parts.: 1/16, 3/16, 5/16 and so on.

Count the small marks from the last whole inch. If the measurement is 9 inches and 11 small marks, it is 9 11/16 in. If the count is even, simplify: 10/16 = 5/8, so 9 and 10 marks is 9 5/8 in.

Most tapes also mark every 16 inches (a common stud spacing) and show feet: 7' 4 1/2" means 7 feet 4 and a half inches, or 7 x 12 + 4.5 = 88.5 inches.

Adding and subtracting fractions

To add fractions, they need a common denominatorThe same bottom number for two fractions, needed before they can be added or subtracted. (the same bottom number). Tape fractions make this easy because 2, 4, 8 and 16 all divide into 16.

Example 1: 3/8 + 5/16

• Convert 3/8 to sixteenths: multiply top and bottom by 2, giving 6/16.
• Add: 6/16 + 5/16 = 11/16.

Example 2: two pieces, 5 3/8 in and 2 7/16 in, laid end to end

• Add the whole inches: 5 + 2 = 7.
• Add the fractions: 3/8 = 6/16, and 6/16 + 7/16 = 13/16.
• Total: 7 13/16 in.

Example 3: 7/8 + 3/4

• 7/8 stays 7/8; 3/4 = 6/8.
• 7/8 + 6/8 = 13/8, which is more than one whole inch.
• 13/8 = 1 5/8 (because 8/8 = 1, with 5/8 left over).

The general rule (works for any fractions): a/b + c/d = (a x d + c x b) / (b x d), then simplify by dividing top and bottom by their greatest common divisorThe largest number that divides both the top and bottom of a fraction, used to simplify it.. For 1/2 + 1/4: (1 x 4 + 1 x 2) / (2 x 4) = 6/8 = 3/4.

Subtraction works the same way. A 96 in stick minus a 35 5/8 in cut: 96 = 95 8/8, and 95 8/8 - 35 5/8 = 60 3/8 in (ignoring the width of the saw cut, the "kerfThe width of material removed by a saw cut, which you allow for when cutting.", which you should allow for in real work).

Converting decimals and metric

Calculators and digital calipers give decimals. To turn a decimal inch into the nearest sixteenth, multiply by 16 and round:

• 0.30 in x 16 = 4.8, which rounds to 5, so 5/16 in.
• 2.69 in: the fraction part 0.69 x 16 = 11.04, so 2 11/16 in.

Common equivalents worth memorising: 1/8 = 0.125, 1/4 = 0.25, 3/8 = 0.375, 1/2 = 0.5, 5/8 = 0.625, 3/4 = 0.75, 7/8 = 0.875.

Metric: 1 inch = 25.4 mm exactly. Machinery from many manufacturers uses metric bolts, so mechanics switch often: 3/4 in = 0.75 x 25.4 = 19.05 mm, which is why a 19 mm wrench nearly fits a 3/4 in bolt (but use the correct one: "nearly" strips heads).

Worked example: centring a box

An electrician needs to centre a light box on a wall that is 11' 5 1/4" wide.

1. Convert to inches: 11 x 12 + 5 1/4 = 137 1/4 in.
2. Halve it: 137 1/4 / 2. Half of 137 is 68 1/2; half of 1/4 is 1/8. Total: 68 1/2 + 1/8 = 68 4/8 + 1/8 = 68 5/8 in.
3. Mark 68 5/8 in from either end, and check from the other end.

Common mistakes

• Miscounting marks. Count from the last whole inch, and double-check by reading the next larger mark (for example, 10/16 should sit on the 5/8 mark).
• Adding denominators. 1/2 + 1/4 is not 2/6. Convert to a common denominator first.
• Forgetting to simplify. 12/16 should be written 3/4.
• Mixing feet and inches. Convert everything to inches before adding.
• Hooking errors. The tape's end hook slides slightly to account for its own thickness on push and pull measurements. Do not "fix" it; keep the hook clean and undamaged.

Try it

Open the Do tab. A small JavaScript fraction calculator is already written except for one line: the numerator when two fractions are added. Fill it in, then use the calculator to check the examples from this lesson.

Quick recap

• A standard tape divides each inch into 16ths; simplify even counts.
• To add fractions, use a common denominator: a/b + c/d = (ad + cb) / bd, then simplify.
• Decimal to nearest sixteenth: multiply by 16 and round.
• 1 in = 25.4 mm exactly.