Basic Math for the Trades

Trades math is not calculus. It is adding, subtracting, multiplying and dividing with real things you can see and touch. It tells you how much material to order, whether a cut will fit, and whether a corner is square. A wrong answer wastes material, delays the job and can sometimes create a hazard, so it is worth getting comfortable with these skills. Use a calculator whenever you like. The real skill is knowing which numbers go in and whether the answer makes sense.

What You Will Be Able to Do

• Add, subtract, multiply and divide fractions and mixed numbers.
• Convert between fractions and decimals.
• Use percentages for waste factors and ratios for mixes and scales.
• Convert units before you calculate.
• Find area and volume, and order material with a sensible allowance.
• Use the Pythagorean theorem to find a diagonal and to check for square.

Fractions

Adding and Subtracting

Fractions need the same bottom number, called the denominator, before you can add or subtract them.

Worked example: A 3/8-inch washer plus a 5/16-inch shim.

1. Change 3/8 to sixteenths: 3/8 = 6/16.
2. Add the top numbers: 6/16 + 5/16 = 11/16.
3. Total thickness: 11/16 inch.

Worked example: 7/8 minus 3/16.

1. 7/8 = 14/16.
2. 14/16 - 3/16 = 11/16.

Reduce the answer when you can. Divide the top and bottom by the same number: 12/16 = 6/8 = 3/4.

Mixed Numbers

A mixed number has a whole number and a fraction, such as 3-1/2. The simplest way to add them is to add the whole numbers and the fractions separately.

Worked example: 3-1/2 + 2-3/4.

1. Whole numbers: 3 + 2 = 5.
2. Fractions: 2/4 + 3/4 = 5/4 = 1-1/4.
3. Total: 5 + 1-1/4 = 6-1/4.

Multiplying and Dividing

To multiply, multiply the tops, then the bottoms: 3/4 x 2/3 = 6/12 = 1/2.

To divide, flip the second fraction and multiply: 3/4 / 1/2 = 3/4 x 2/1 = 6/4 = 1-1/2. If a plate is 3/4 inch thick and you remove 1/2 inch per pass, it takes 1-1/2 passes, which means 2 passes in real life.

Fractions and Decimals

Construction often uses fractions. Machining and many specs use decimals. To change a fraction to a decimal, divide the top by the bottom: 3/8 = 3 / 8 = 0.375.

To go the other way, multiply the decimal by the denominator you want, then round.

Worked example: Change 0.347 to the nearest 1/16.

1. 0.347 x 16 = 5.55.
2. Round to 6.
3. 6/16 = 3/8.

Percentages

A percentage is a number out of 100. To work with one, move the decimal two places: 15% = 0.15.

• A percentage of a number: 15% of 200 = 200 x 0.15 = 30.
• What percent one number is of another: 45 out of 200 = 45 / 200 = 0.225 = 22.5%.
• Adding a percentage: multiply by 1 plus the percent. Add 10%: multiply by 1.10.

Worked example: A floor measures 14 feet by 15 feet, and you plan on 10% extra for cuts and breakage.

1. Area: 14 x 15 = 210 square feet.
2. With waste: 210 x 1.10 = 231 square feet.

Waste allowances differ by material and layout. Around 10% is common for plain layouts, with more for diagonal patterns.

Ratios and Proportions

A ratio compares amounts. A 1:2:3 mix means 1 part of the first material, 2 of the second and 3 of the third, by volume.

Worked example: You are filling a mixer with 12 buckets of dry material at 1:2:3.

1. Total parts: 1 + 2 + 3 = 6.
2. One part: 12 / 6 = 2 buckets.
3. So 2 buckets of the first, 4 of the second, and 6 of the third.

A proportion keeps a ratio the same at a different size. If 3 gallons cover 900 square feet, then 1 gallon covers 300 square feet, and 1,500 square feet needs 1,500 / 300 = 5 gallons.

Scale is also a proportion. On a drawing at 1/4" = 1'-0", every 1/4 inch stands for 1 foot. A line 3-1/2 inches long is 3.5 / 0.25 = 14 feet.

Units and Conversions

Put everything in the same units before you calculate. This one habit prevents most big errors.

Worked example: Change 4 feet 9 inches to decimal feet. 9 / 12 = 0.75, so it is 4.75 feet.

Temperature: F = (C x 9/5) + 32, and C = (F - 32) x 5/9. So 20°C is (20 x 9/5) + 32 = 68°F.

Perimeter and Area

Perimeter is the distance around a shape. Area is the space it covers. Use perimeter for baseboard, fencing and trim, and area for flooring, paint, drywall and roofing.

The radius is half the diameter. The distance around a circle is 3.14 x diameter.

Worked example: A room is 12 feet by 15 feet.

1. Floor area: 12 x 15 = 180 square feet.
2. Perimeter: 2 x (12 + 15) = 54 linear feet, before you subtract doorways.

Worked example: A triangular gable is 10 feet wide and 8 feet tall. Area = 1/2 x 10 x 8 = 40 square feet.

Worked example: A pipe has a 6-inch diameter. The radius is 3 inches. Area = 3.14159 x 3 x 3 = 28.27 square inches.

To find the area of an odd shape, split it into rectangles and triangles, then add them up.

Volume

Volume is the space inside something. Use it for concrete, gravel, tanks and excavation.

• Rectangular volume: length x width x height.
• Cylinder volume: 3.14 x radius x radius x height.

Worked example: A concrete slab is 12 feet by 20 feet and 4 inches thick. How much do you order, with about 10% extra?

1. Change inches to feet: 4 / 12 = 0.333 feet.
2. Volume: 12 x 20 x 0.333 = 80 cubic feet.
3. Change to cubic yards: 80 / 27 = 2.96 cubic yards.
4. Add waste: 2.96 x 1.10 = about 3.25 cubic yards.

Worked example: A tank is 3 feet across and 5 feet tall.

1. Radius: 1.5 feet.
2. Volume: 3.14159 x 1.5 x 1.5 x 5 = 35.3 cubic feet.
3. Gallons: 35.3 x 7.48 = about 264 gallons.

Right Triangles

A right triangle has one 90-degree corner. The Pythagorean theorem says that if a and b are the two short sides and c is the long side, then a x a + b x b = c x c.

Worked example: A slab form is 20 feet by 30 feet. How long should each diagonal be?

1. 20 x 20 = 400.
2. 30 x 30 = 900.
3. 400 + 900 = 1,300.
4. Square root of 1,300 = 36.06 feet.

If both diagonals match that number, the form is square.

The 3-4-5 Check

A triangle with sides of 3, 4 and 5 always has a square corner, and so does any multiple of it, such as 6-8-10 or 9-12-15.

1. Measure 3 feet along one line and mark it.
2. Measure 4 feet along the other line and mark it.
3. Measure between the marks.
4. If it reads exactly 5 feet, the corner is square. If not, adjust and measure again.

For big layouts, use larger multiples, since they catch small errors better.

Good Math Habits

1. Estimate first. If a 10 x 12 room seems to need 1,200 square feet of tile, you misplaced a decimal.
2. Convert first. Do not mix feet and inches in one formula.
3. Write it down, on paper, on the material or in your phone.
4. Do it twice when material or money depends on the answer.
5. Round the right way. Round up when ordering material, since you cannot buy half a bag. Keep full precision until the last step.

Common Mistakes

Check Your Understanding

1. You stack a 2-3/8-inch block and a 1-3/4-inch block. How tall is the stack?

2 + 1 = 3, and 3/8 + 6/8 = 9/8 = 1-1/8. The total is 4-1/8 inches.

2. What is 5/8 as a decimal?

5 / 8 = 0.625.

3. A wall is 10 feet long and 8 feet high, with a window of 12 square feet. A gallon of paint covers 350 square feet. Do you need more than one gallon for two coats?

Wall area is 80 - 12 = 68 square feet. Two coats make 136 square feet, so one gallon is plenty.

4. A footing is 2 feet wide, 1 foot deep and 40 feet long. How many cubic yards is that?

2 x 1 x 40 = 80 cubic feet. 80 / 27 = 2.96 cubic yards. Add waste before you order.

5. You measure a 6-foot and an 8-foot side from a corner. What diagonal shows it is square?

10 feet. It is the 3-4-5 triangle doubled.

Key Takeaways

• Fractions need a common denominator to add or subtract.
• Divide the top by the bottom to change a fraction to a decimal.
• Multiply by 1.10 to add 10% for waste.
• Put all numbers in the same units before you calculate.
• Cubic feet divided by 27 gives cubic yards.
• Use 3-4-5 and the Pythagorean theorem to find diagonals and check for square.