Unit conversions

For the ACT test, there is a list of units that you should be familiar with:

In this chapter, you’ll learn how to convert from one unit to another within the same category (for example, grams to kilograms, which are both in the “Mass” category).

Most students find metric conversions trickier, so that will be the main focus. Standard units (yards, feet, and inches) are often more familiar, but we’ll review them briefly before moving on to metric units.

Standard units

From smallest to largest, there are 12 inches in a foot, and 3 feet in a yard. In the number line below, you can see the units from smallest to largest, and how many of the smaller unit make 1 of the next larger unit (shown by the arrows above the number line).

How many inches are in a yard?

This is the main set of standard conversions you should be comfortable with:

• inches → feet → yards
• 12 inches = 1 foot
• 3 feet = 1 yard

Metric unit conversions by number line

A helpful way to visualize metric conversions is to place the units from one category on a number line. Below is the number line we’ll reference as we discuss metric conversions.

You’ll notice that most steps on the metric number line involve 1,000. This means it takes 1,000 of a smaller unit to make 1 of the next larger unit. For example, it takes 1,000 millimeters to make 1 meter.

The main exception is centimeters, which sit between millimeters and meters:

• 100 cm = 1 m

One way to remember this is that the prefixes relate to familiar number ideas:

• milli- sounds like “millennium” (1,000 years)
• centi- sounds like “century” (100 years)

You’ll also see similar roots in other languages (Latin: milia, centum; Spanish: mil, cien; French: mil, cent).

Once you know how many of one unit make the next, you can convert using the number line. For example, suppose you have 30,000 milliliters and want liters.

• 1,000 mL = 1 L
• So you divide by 1,000 to move from mL to L:

$$30,000÷1,000=30$$

That means 30,000 mL is equivalent to 30 L.

Try converting 250,000 milligrams to kilograms. How many kg do we have when we have 250,000 mg?

Unit conversions by T-chart (dimensional analysis)

This section shows a different method for unit conversions. Use whichever method feels clearer to you.

The idea of T-chart (dimensional analysis) conversions is to lay out the units so you don’t lose track of whether you should multiply or divide. You’ll multiply all numbers on the top row, multiply all numbers on the bottom row, and then divide the top total by the bottom total.

Start with your first unit in the top-left box. When you move to the next column, place matching units diagonally so they cancel (because you’re dividing a unit by itself).

• If you’re converting to the second unit, you can stop after the second column (for example, mg → g).
• If you need a third unit (or more), keep adding columns (for example, mg → g → kg), each time placing the repeated unit diagonally.

Take a look at this example converting from 3,000,000 mm to km:

Let’s go through the steps:

1. Write what you have in the top left.

2. In the box diagonal to the bottom-left, write how many of the 1st unit make 1 of the next unit.

3. Write 1 of the 2nd unit in the top of the 2nd column.

   This second column shows that there are 1,000 mm in 1 m. The mm is on the bottom so it’s diagonal to the mm in the top-left. That diagonal match shows the mm units will cancel when you divide.

4. Repeat these steps with another column if you need to keep converting to a larger unit.

5. When you have everything written out,

   1. Multiply everything on the top together.

   2. Multiply everything on the bottom together.

   3. Divide the total on top by the total on bottom.

   4. The number you get is your final answer.

      Important to note: the unit of your final answer will be the top-right unit in the table (it does not cancel because there is no matching unit diagonal to it).

Remember, you don’t have to use this technique. Many students find it less intuitive at first. However, it’s a reliable method and helps you avoid confusion about whether to multiply or divide.

Converting with units of area

In area problems, units are written as ${\text{length}}^2$ (for example, ${\text{m}}^2$, ${\text{ft}}^2$, square yards).

This is because area is found by multiplying two lengths. For example, if both sides are measured in inches, then:

• $\text{inches}\ast \text{inches}={\text{inches}}^2$

When you convert area units, your conversion factors must be squared. Follow this example:

The conversion relationships are the same as before, but each factor is squared:

• 12 inches = 1 foot becomes $12^2$ inches${}^2$ = 1 foot${}^2$
• 3 feet = 1 yard becomes $3^2$ feet${}^2$ = 1 yard${}^2$

This is the key adjustment to watch for when converting area units.