Comprehensive Math and Science Conversion & Proportion Formulas

Cross-Multiplication

A method used to solve proportions by multiplying diagonally.

Proportion

Two equal ratios, such as a/b = c/d.

How to solve a proportion

Cross-multiply, then divide to solve for the unknown.

Percentage

A number out of 100.

To find a percent of a number

Change the percent to a decimal and multiply by the whole.

To find what percent one number is of another

Part ÷ whole × 100.

Percent increase/decrease

Change ÷ original × 100.

Dimensional Analysis

Converting from one unit to another using known conversion factors.

Railroad Tracks

Setting up conversion factors so unwanted units cancel and the desired unit remains.

Conversion Factor

A ratio equal to 1 that is used to convert units.

Metric Prefix Order

kilo → hecto → deka → base → deci → centi → milli.

Kilo (k)

1000 times the base unit.

Hecto (h)

100 times the base unit.

Deka (da)

10 times the base unit.

Base Unit

The standard unit, equal to 1.

Deci (d)

0.1 of the base unit.

Centi (c)

0.01 of the base unit.

Milli (m)

0.001 of the base unit.

Micro (μ)

0.000001 of the base unit.

Nano (n)

0.000000001 of the base unit.

Important metric rule

When converting between multiple prefixes, return to the base unit.

1 inch

2.54 centimeters.

1 mile

5280 feet.

1 year

52 weeks.

1 hour

3600 seconds.

1 kilogram

2.2 pounds.

1 gallon

16 cups.

1 mL

1 cm³.

1 pound

454 grams.

Density

How compact the particles in a substance are.

Density Formula

D = M ÷ V.

Mass Formula

M = D × V.

Volume Formula

V = M ÷ D.

Density Unit

Usually g/mL or g/cm³.

Mass

The amount of matter in an object; measured using an electronic balance.

Volume

The amount of space an object takes up.

Volume of a rectangular object

Length × width × height.

Displacement Method

Find volume by subtracting initial water volume from final water volume.

Displacement Formula

Final volume − Initial volume.

Density of water

1 g/mL.

Object with density greater than 1 g/mL

Sinks in water.

Object with density less than 1 g/mL

Floats in water.

Object with density equal to 1 g/mL

Has the same density as water.

Scientific notation with a positive exponent

Move the decimal to the right.

Scientific notation with a negative exponent

Move the decimal to the left.

Numerator

The top number of a fraction or ratio.

Denominator

The bottom number of a fraction or ratio.

Unit cancellation

Units that appear on both the top and bottom cancel out.

Why use dimensional analysis?

To correctly convert measurements from one unit to another.

Percent Yield

Actual yield ÷ theoretical yield × 100.

Theoretical Yield

How much product should be produced.

Actual Yield

How much product was actually produced.

Scaling a recipe

Multiplying each ingredient by the number of batches needed.

Supply and Demand

Used to help businesses decide how much to charge for a product.

Density and dimensional analysis together

You may need to use a density formula and then convert the answer into another unit.