Convert Customary, Metric, Square and Cubic Units

A unit equation says that two measurements name the same amount. By the definition of a foot, you are given that $1\text{ ft} = 12\text{ in}$. You can turn this equation into a fraction equal to $1$.

The Division Property of Equality says that dividing both sides of an equation by the same nonzero quantity keeps the sides equal. The divisor must be nonzero because division by zero is undefined. Here, $12\text{ in}$ and $1\text{ ft}$ are positive measures, so you may divide by either one. The Multiplicative Inverse Property says that a nonzero quantity divided by itself equals $1$. Together, these two properties give you two fractions:

Customary units are the units in everyday use in the United States, such as feet, pounds and gallons. Weight tells how heavy an object is, and capacity tells how much a container holds. Customary sizes follow no single pattern, so you take each relationship from a table of unit equations.

Metric units change size by powers of 10, which are numbers such as $1{,}000$ and $\frac{1}{100}$ made by repeatedly multiplying or dividing $1$ by $10$. In this lesson, the base units are the meter (m) for length, the gram (g) for mass, which is the amount of matter in an object, and the liter (L) for capacity. A prefix is a word part placed in front of a base unit to set its size. Each prefix means the same thing with every base unit, so one pattern covers all three columns of the table.

To cross between systems, you need a unit equation that links them. In the table below, the inch equation is exact, because an inch is defined as exactly $2.54\text{ cm}$. The other three equations use rounded values. A fraction made from a rounded value is close to $1$ but not exactly $1$, so write $\approx$ and treat the result as an approximation: $10\text{ mi} \times \frac{1.609\text{ km}}{1\text{ mi}} \approx 16.09\text{ km}$. The divisor $1\text{ mi}$ is positive, so it is nonzero. To go the other way, flip the factor so that the metric unit is in the denominator: $\frac{1\text{ mi}}{1.609\text{ km}}$.

Pounds measure weight and kilograms measure mass. On Earth, an object that weighs $1\text{ lb}$ has a mass of about $0.454\text{ kg}$.

Area is the amount of surface a shape covers, and volume is the amount of space a solid fills. A square unit is the area of a square with sides $1$ unit long. A cubic unit is the volume of a cube with edges $1$ unit long. So $1\text{ ft}^2$ means $(1\text{ ft})^2$, and $1\text{ ft}^3$ means $(1\text{ ft})^3$.

Every foot inside these expressions must be converted. The Substitution Property of Equality lets you replace a quantity with an equal one, so replace $1\text{ ft}$ with $12\text{ in}$. The Power of a Product Property, $(ab)^n = a^n b^n$, then gives $(12\text{ in})^2 = 144\text{ in}^2$ and $(12\text{ in})^3 = 1{,}728\text{ in}^3$. This matches counting: a square with $1$-ft sides holds $12$ rows of $12$ one-inch squares, and a cube with $1$-ft edges holds $12$ layers of $144$ one-inch cubes.

So square the length factor for square units and cube it for cubic units. An exact factor equals $1$, and $1^2 = 1^3 = 1$, so the amount stays the same.

Here is the idea to keep: to convert, multiply by a conversion factor, with the unit you want to remove in the nonzero denominator. The amount stays the same because the factor equals $1$; if the factor is rounded, it is only close to $1$, so write $\approx$. For square units, square the length factor, and for cubic units, cube it. Then check that your result is positive and that the count changed in the direction the unit size predicts.