Mathematics Study Guide for the TABE Test
Page 6
Numbers and Operations: Ratios and Proportional Relationships
Ratios and proportions are used to show how different quantities compare to each other. We use them in everything from cooking to financial analysis. While they may seem like complex concepts at first, just know that they build off of what you’ve already learned about fractions.
Ratios
A ratio is a mathematical statement that establishes a relationship between two quantities. For example, if there are two girls for every boy in a class, the ratio of girls to boys is “two to one.” This can be written as \(2\text{:}1\) or \(\frac{2}{1}\).
It’s important that you present the ratio in the correct order. Consider a bag that is holding four times as many green marbles as white marbles. The ratio of green marbles to white marbles is \(4\text{:}1\). However, if you want to state the ratio of white marbles to green marbles, it would be \(1\text{:}4\).
Unit Rate
A unit rate is any ratio where the denominator is a single unit. This is also sometimes called a unit ratio or the constant of proportionality. In the real world, you’ll most often come across this concept when measuring speed or distance traveled.
Suppose you were riding a motorcycle and traveled \(300\) miles. At the end of the trip, you see that you used six gallons of gas. As a ratio, that would be:
\[\frac{300 \text{ mi}}{6 \text { gal}}\]If you divide both numbers by \(6\), you will get the unit rate:
\[\frac{300 \div 6}{6 \div 6}= \frac{50 \text{ mi}}{1 \text { gal}}\]You can tell this is a unit rate because the denominator is a single unit, one gallon.
Common examples of unit rates are speed units, such as miles per hour (mph), density units, such as grams per milliliter (g/mL), typing speed, such as words per minute, food value, such as calories per serving, and more. Although these don’t commonly have a \(1\) written in the denominator, it is understood to be there.
Unit Rates and Ratios
You should be able to take any ratio and express it as a unit rate. A big clue that you are looking at a unit rate is the use of the word per. That won’t always be the case, but it’s usually true.
Suppose you use \(264\) chocolate chips to make a batch of \(24\) chocolate chip cookies. The ratio of chocolate chips to cookies is \(\frac{264}{24}\). You could write this as a unit rate by dividing the top and bottom by \(24\) to get \(\frac{11}{1}\). You would say the rate is \(11\) chocolate chips per cookie.
Ratios and Fractions
You may run across a ratio that is expressed as a fraction, and you should be able to express that as a unit rate just like you did with whole numbers. For example, if a recipe calls for \(\frac{2}{3}\) cup of sugar and \(\frac{1}{6}\) cup of butter, what is the unit rate of sugar to butter? Since we have fractions to deal with, our ratio is going to be a complex fraction:
\[\frac{\frac{2}{3}}{\frac{1}{6}}\]Changing this to a unit fraction means getting a \(1\) on the bottom. One way to do this is to know that a complex fraction can be thought of as a division statement. Remember dividing by fractions? Our complex fraction can be rewritten as a multiplication problem using the reciprocal of the denominator:
\[\frac{\frac{2}{3}}{\frac{1}{6}} = \frac{2}{3} \times \frac{6}{1}\] \[\frac{2}{3} \times \frac{6}{1} = \frac{12}{3}\] \[\frac{12}{3} = \frac{4}{1}\]That is our unit rate. Because we were dividing the sugar by the butter, we would say there are four cups of sugar per cup of butter.
Ratio and Rate Reasoning
One effective tool for reasoning out rate and ratio problems is a tape or bar diagram (called such because it looks kind of like pieces of tape). This approach helps you visualize the problem and its solution. Let’s look at an example to see how to use it.
Let’s say the ratio of boys to girls in a classroom is \(3\text{:}4\). If there are \(35\) students altogether, how many boys are in the class?
To create a tape diagram, first draw two bars, one with three equal blocks and the other with four equal blocks. This makes the block ratio the same as the student ratio:
You can see that there are seven blocks altogether. Remember that these seven blocks represent \(35\) students. How many students must be in each block? Divide \(35\) by seven and you will see there are five kids in each block. The boys have three blocks, and at five boys per block, that makes \(15\) boys in the class.
Another approach for solving ratio and rate problems is purely algebraic. We will take the ratio \(3\text{:}4\), and from that we will write two expressions in terms of a letter such as \(x\) (this is known as a variable, which you’ll learn more about later). Let the boys’ expression be \(3x\) and the girls’ be \(4x\). We know that there are \(35\) students altogether, so we can write and solve this equation:
\[3x+4x=35\] \[7x = 35\] \[x= 5\]Since the boys are \(3x\) in number, they must be \(3\times 5=15\).
Note: We’ll cover algebraic concepts in greater detail later in this study guide. This is just a simple introduction to the ideas.
Proportions
A proportion is an equation between two ratios, such as \(\frac{3}{4} = \frac{6}{8}\). Proportions are useful in scaling up or down real-world ratios, like recipes. In proportion problems, you will often be given one complete ratio and one ratio with a missing value that is again represented by a letter such as \(x\).
For instance, what value of \(x\) will make this proportion true?
\[\frac{3}{18} = \frac{x}{24}\]One way to solve this is to multiply both sides of the equation by \(24\).
\[\frac{3}{18} \times 24 = \frac{x}{24} \times 24\] \[\frac{72}{18} = x\] \[4 = x\]You can recognize a proportional relationship by remembering that it is any relationship that has two equal rates. Let’s try an example problem.
You’re taking a two-part trip to see family and friends. First, you will drive your car \(200\) miles to visit your grandparents, and it will take you four hours. After that visit, you’re going to meet a friend one state over, and you know that trip is going to take seven hours. If you drive the same speed the entire trip, how far is the second part of your trip?
Solution
First, determine if this is a proportional relationship. We know it is because it says “the same speed,” so we have two equal rates (speed is distance divided by time).
When you write the proportion, be sure to keep the units in the same positions on both sides:
\[\frac{200 \text{ mi}}{4 \text{ hr}} = \frac{x}{7\text{ hr}}\]Use the method from above and multiply both sides by \(7\):
\[\frac{200}{4} \times 7= \frac{x}{7} \times 7\] \[\frac{1\text{,}400}{4} = x\] \[350 \text{ mi}= x\]So, the second part of your trip is \(350\) miles.
Cross-Multiplication
There is another way to handle a proportion that is often faster than multiplying both sides by the same number. It’s called cross-multiplication, and it gets rid of both denominators at once. To do it, just multiply the top of one ratio by the bottom of the other ratio, and vice versa:
\[\frac{200}{4} = \frac{x}{7}\] \[200 \times 7 = 4 \times x\] \[1\text{,}400 = 4x\] \[x = \frac{1\text{,}400}{4} = 350\]Both methods will get you to the right answer, so use whichever you feel will be the fastest.
Exponents and Radicals
Exponents are a way of showing repeated multiplication of the same number. Instead of writing a number multiplied by itself many times, we use a smaller raised number, the exponent, to show how many times the base is used. Exponents make it easier to write and work with repeated multiplication, especially when dealing with larger numbers.
For example, in \(2^3\), this means the number \(2\) is being multiplied by itself three times:
\[2^3 = 2 \cdot 2 \cdot 2 = 8\]Radicals are expressions that represent the opposite, or inverse, of exponents. Instead of multiplying a number by itself, radicals help us find a number that was multiplied by itself to get the original value. In other words, just as exponents build numbers through repeated multiplication, radicals “undo” that process by finding the original base value.
For example, \(\sqrt{16} = 4\) because \(4 \cdot 4 = 16\).
Radicals can also represent higher roots. For example, \(\sqrt[3]{27} = 3\) because \(3 \cdot 3 \cdot 3 = 27\).
The Properties of Exponents
An exponential expression, or power, consist of two main parts: the base and the exponent. Consider this exponential expression:
\[3^4\]In this example, \(3\) is the base and \(4\) is the exponent. This translates as three raised to the power of four, which means multiplying three by itself four times:
\[3 \cdot 3 \cdot 3 \cdot 3 = 81\]There are two specific types of exponential expressions with special names: squares and cubes. A square has an exponent of \(2\), such as \(3^2\) and \(4^2\). A cube has an exponent of \(3\), such as \(3^3\) and \(4^3\).
Rules of Exponents
In addition to those basic parts of exponential expressions, there are certain rules to follow when dealing with exponents. Here are the most crucial ones:
- Two numbers with exponents can only be added or subtracted if they have the same base and same exponent:
- Any exponential expression with an exponent of \(1\) is equal to the base number:
- When multiplying two quantities with the same base, you add their exponents:
- When dividing two quantities with the same base, you subtract their exponents (the top one minus the bottom one):
- When you have a power of a power, like \((x^3)^4\), you multiply the exponents:
Common Squares and Cubes
A square number, also known as a perfect square, is created when multiplying a whole number by itself. The number \(4\) is a perfect square because it is the result of \(2 \times 2\). You should know the perfect squares of the first \(12\) whole numbers:
\[\begin{array}{ccc} 1^2=1&2^2=4&3^2=9\\ 4^2=16 &5^2=25&6^2=36\\ 7^2 =49&8^2=64&9^2=81\\ 10^2 =100&11^2=121&12^2=144\\ \\ \end{array}\]Likewise, it’s good if you know the first few cubes (or perfect cubes):
\[\begin{array}{cc} 1^3=1&2^3=8\\ 3^3=27&4^3=64\\ 5^3=125&6^3=216\\ \end{array}\]The Properties of Radicals
Radical expressions consist of three parts: the radical sign (\(\sqrt{\ }\)), the radicand, and the index. Consider this radical expression:
\[\sqrt[3]{27}\]In this example, \(27\) is the radicand and \(3\) is the index.
As with exponents, we have two names for specific types of radicals. If the index is a \(2\), then it is a square root. If the index is a \(3\), then it is a cube root. The example above was a cube root. If the index is not specifically stated, the expression is a square root by default.
Fractional Exponents
Radicals can be written using fractional exponents. For example, \(\sqrt{2}\) can be written as \(2^{\frac{1}{2}}\). Likewise, \(\sqrt{11}\) can be written as \(11^{\frac{1}{2}}\) and \(\sqrt[3]{22}\) is \(22^{\frac{1}{3}}\).
A fractional exponent may seem odd, but there is a simple mathematical explanation for why it’s true.
Suppose there is some exponent, \(x\), such that \(2^x \times 2^x = 2\). We know \(2=2^1\). We also know that when numbers with the same base are multiplied, the exponents are added. So, if \(2^x \times 2^x = 2^1\), then \(x+x\) must equal \(1\) and \(x=\frac{1}{2}\). Putting this together, we can add exponents and write:
\[2^{\frac{1}{2}} \times 2^{\frac{1}{2}} = 2\]We also know that \(\sqrt{2} \times \sqrt{2} = 2\). Since both of these things are true, then \(2^{\frac{1}{2}}\) is the same as \(\sqrt{2}\).
What if you have an exponent inside a radical like \(\sqrt{x^3}\)? All you need to do is rewrite the root as an exponent:
\[\sqrt{x^3} = (x^3)^{\frac{1}{2}}\]Then you simplify by multiplying the exponents:
\[(x^3)^{\frac{1}{2}} = x^{\frac{3}{2}}\]Common Square Roots
If you realize square roots are the inverse of squares, you should already know the first \(12\) common square roots:
\[\begin{array}{ccc} \sqrt{1}=1&\sqrt{4}=2&\sqrt{9}=3\\ \sqrt{16}=4&\sqrt{25}=5&\sqrt{36}=6\\ \sqrt{49}=7&\sqrt{64}=8&\sqrt{81}=9\\ \sqrt{100}=10&\sqrt{121}=11&\sqrt{144}=12\\ \\ \end{array}\]All Study Guides for the TABE Test are now available as downloadable PDFs