Mathematics Study Guide for the ACT

Measurement

We use measurements every day. They are part of our daily lives. We use measurements to quantify various physical quantities such as weight, mass, length, time, and money. Two common systems of measurement used throughout the world are the United States (US) system and the metric system. Let us take a look at each of the measurement units and how to convert them within the same system.

Money

The currency system of the United States consists of coins and bills. The coins are the penny, nickel, dime, quarter, and half dollar. The penny is worth one cent, or $\$0.01$, the nickel is worth five cents, or $\$0.05$, the dime is worth $10$ cents, or $\$0.10$, the quarter is worth $25$ cents, or $\$0.25$, and the half dollar is worth $50$ cents, or $\$0.50$. There are one, five, $\boldsymbol{10}$, $\boldsymbol{20}$, $\boldsymbol{50}$, and $\boldsymbol{100}$ dollar bills.

When converting between these units, it is important to understand their relative worth. For example, five pennies are worth the same as one nickel (five cents). Similarly, five dimes are worth the same as a half dollar ($50$ cents).

Length

Length is the measurement of distance between two points. In the US system, the most commonly used units of measurement of length are inches, feet, yards, and miles. One foot is equivalent to $12$ inches, one yard is equivalent to three feet or $36$ inches, and one mile is equivalent to $5\text{,}280$ feet or $1\text{,}760$ yards.

In the metric system, the most commonly used units to measure length are meters, centimeters, kilometers, and decimeters. One centimeter is equivalent to $0.01$ meters, one decimeter is equivalent to $0.1$ meters, and one kilometer is equivalent to $1\text{,}000$ meters.

Many regard the metric system as simpler to use because all the conversion factors between different units of measurement are based on powers of ten.

Weight

Weight is the measurement of the force exerted on an object due to gravity. In the US system, weight is commonly measured in ounces, pounds, and tons. One ounce is equal to $\frac{1}{16}$ of a pound and one ton is equal to $2\text{,}000$ pounds.

In the metric system, weight is measured in grams, kilograms, and metric tons (or tonnes). There are $1\text{,}000$ grams in a kilogram and one metric ton is equivalent to $1\text{,}000$ kilograms.

To convert between different units of weight within the same system, we use conversion factors. For example, if we want to know how many ounces there are in five pounds, we will multiply $5$ by $16$ since there are $16$ ounces in a pound. The unit for the answer will be ounces.

Mass

In our everyday lives, mass and weight are almost interchangeable. But in physics they are different concepts. Mass is the amount of matter in an object, while weight is a measure of the force exerted on an object due to gravity. The US system of measurement uses grains, ounces, and pounds to measure mass. One pound is equal to $16$ ounces, one ton is equal to $2\text{,}000$ pounds, and a grain is equal to $\frac{1}{7\text{,}000}$ of a pound.

The metric system uses milligrams, grams, and kilograms. One gram is equal to $1\text{,}000$ milligrams and $1\text{,}000$ grams is equivalent to $1$ kilogram. This system also uses the tonne (metric ton), which is equal to $1\text{,}000$ kilograms.

To convert between different units of mass within the same system, we use conversion factors. For example, if we want to know how many pounds there are in two tons, we will multiply $2$ by $2\text{,}000$ since there are $2\text{,}000$ pounds in a ton.

Time

Time measurement allows us to organize events and activities. We use both the standard form of time and military time in our everyday lives.

Most of the world uses the standard timekeeping system. It is also known as the $12$-hour time system. It is based on the $24$-hour day, which is divided into two $\boldsymbol{12}$-hour periods: a.m. (from midnight to noon) and p.m. (from noon to midnight). An hour is equal to $60$ minutes, and a minute is equal to $60$ seconds.

For example, the times $12\text{:}35$ p.m. and $8\text{:}31$ a.m. are written in standard form. Here, the first number (before the colon) represents hours and the second number (after the colon) represents minutes. Hours in standard time are always expressed using a number from $1$ to $12$.

Military time is most commonly used by the military (as the name suggests), emergency services, and other organizations where the $\boldsymbol{24}$-hour clock is necessary. It is also called the $24$-hour time system. In military time a day is divided into $24$ periods, but these hours are numbered from $0$ to $23$, with $0$ representing midnight and $23$ representing $11\text{:}00$ p.m.

We use the following formatting to express time in military time:

$1545$—This is the time $3\text{:}45$ p.m. in the standard system.

$2110$—This is the time $9\text{:}10$ p.m. in the standard system.

The first two digits represent the hour and the last two digits represent the minutes.