Mathematics: Levels E, M, D, and A Study Guide for the TABE Test

Ratios and Proportional Relationships

Percentage of Test Level Specifically Assessing Ratios and Proportional Relationships (— = Assumed)

L	E	M	D	A
not tested	not tested	3%	10%	—

Ratios and proportions are used to show how different quantities compare to each other. A proportion is nothing more than writing one ratio equals another. See the example below:

\[\frac{24}{30} = \frac{8}{10}\]

Unit Rate

A unit rate could just as well be called a unit ratio. It is any ratio where the denominator is a single unit.

As an example:

Suppose you drove your motorcycle \(300\) miles and used \(6\) gallons of gas.

You could write this: \(\frac{300 \text{ miles}}{6 \text { gallons}}\) as a ratio, or you could divide both numbers by \(6\) and get \(\frac{50 \text{ miles}}{1 \text { gallon}}\). The second ratio is a unit rate, and what makes it a unit rate is the \(1\) on the bottom.

Common examples of unit rates are speed units (\(\frac{miles}{hour}\)), density units (\(\frac{g}{mL}\)), typing speed (\(\frac{words}{minute}\)), food value (\(\frac{calories}{serving}\)), and many others. Although these don’t commonly have a \(1\) written in the denominator, it is understood to be there.

Associated with 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. It’s not a guarantee, 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 top and bottom by \(24\) and get \(\frac{11 \text{ chocolate chips}}{1 \text { cookie}}\). In words, you would say the rate is \(11\) chocolate chips per cookie.

Associated with Fractions

You may run across a ratio that is made of fractions 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 \(\frac{1}{6}\) of a 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 remember that a complex fraction can be thought of as a division statement. Our complex fraction can be rewritten as a multiplication problem using the reciprocal of the denominator:

\[\frac{\frac{2}{3}}{\frac{1}{6}} = \frac{2}{3} \times \frac{6}{1}\] \[\frac{2}{3} \times \frac{6}{1} = \frac{12}{3}\] \[\frac{12}{3} = \frac{4}{1}\]

Thus, the unit rate is:

\[\frac{4 \text{ cups of sugar}}{1 \text{ cup of butter}}\]

Or you could say \(4\) cups of sugar per cup of butter.

Proportional Relationships

A proportion is an equation between two ratios, like this: \(\frac{3}{4} = \frac{6}{8}\). Here is an example of how it may be used.

What value of \(x\) will make this proportion true?

\[\frac{3}{18} = \frac{x}{24}\]

A good way to solve this is to multiply both sides of the equation by \(24\).

\[\frac{3}{18} \times 24 = \frac{x}{24} \times 24\] \[\frac{72}{18} = x\] \[4 = x\]

Be able to recognize a proportional relationship by remembering that it is any relationship that has two equal rates. Here’s another example.

Let’s say that you drove your car for \(200\) miles and it took you \(4\) hours. Your next trip is going to take \(7\) hours. If you drive the same speed as before, how far will you go?

Is this a proportional relationship? Yes, because it says “the same speed”, so we will have two equal rates.

When you write the proportion, be sure to keep the units in the same positions on both sides.

\[\frac{200 \text{ mi}}{4 \text{ hr}} = \frac{x}{7\text{ hr}}\]

Do the same trick as before and multiply both sides by \(7\).

\[\frac{200}{4} \times 7= \frac{x}{7} \times 7\] \[\frac{1\text{,}400}{4} = x\] \[350 \text{ miles}= x\]

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. Then repeat with the other pair:

\[\frac{200}{4} = \frac{x}{7}\] \[200 \times 7 = 4 \times x\] \[1\text{,}400 = 4x\] \[x = \frac{1\text{,}400}{4} = 350\]

Ratio and Rate Reasoning

To help with reasoning in rate and ratio problems, a good thing to use is called a tape or bar diagram (it’s called that because it looks kind of like pieces of tape). Let’s look at an example to see how to use it.

In a certain class, the ratio of boys to girls is \(3:4\). If there are \(35\) students altogether, how many boys are in the class?

First, draw two bars, one with \(3\) equal blocks and the other with \(4\) equal blocks. This makes the block ratio the same as the student ratio:

You can see that there are \(7\) blocks altogether. Remember that they represent \(35\) students. How many students must be in each block? Divide \(35\) by \(7\) and get \(5\) kids in each block. The boys have \(3\) blocks, and at \(5\) boys per block, that makes \(15\) boys in the class.

This approach can help you to visualize the problem and its solution.

Another approach is purely algebraic. Take the ratio \(3:4\) and from that, write two expressions in \(x\):

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 \(3\times 5=15\).

Multi-Step Ratio and Percent Problems

Here are some examples of problems you should be able to do that use ratios and proportional relationships.

Calculating Commissions

A salesman gets paid a base salary of \(\$2\text{,}000\) each month plus a commission of \(8\%\) of his sales. If he sells \(\$7\text{,}500\) worth of product in June, what will his salary be for that month?

First, calculate his commission:

\[8\% \text{ of }\$7\text{,}500\] \[.08 \times 7\text{,}500 = 600\]

Then, add the commission to his base salary:

\[2\text{,}000 + 600 =2\text{,}600\]

Using a Proportion:

The new Steinway Tower in New York City is a very thin building. It has a height of about \(1\text{,}440 \text{ ft}\) and a width of about \(60\text{ ft}\). The Empire State Building has a north-south width of about \(200\text{ ft}\). If the Empire State Building had the same height to width ratio as the Steinway Tower, how tall would it be?

First, calculate the height-width ratio of the Steinway Tower.

\[\frac{1\text{,}440}{60}\]

Divide \(60\) into itself and \(1\text{,}440\) and get \(24\), so we get a ratio of \(\frac{24}{1}\).

Second, write a height-to-width proportion between building dimensions. We don’t know the height of the Empire State Building so we write \(x\) in its place.

\[\frac{24}{1} = \frac {x}{200}\]

Multiply both sides by \(200\).

\[24 \times 200 = x\] \[x = 4\text{,}800 \text{ ft}\]

Note: In reality, the Empire State Building is nowhere near this tall, because it doesn’t have the same ratio as the Steinway Tower.

The Number System

Percentage of Test Level Specifically Assessing the Number System (— = Assumed)

L	E	M	D	A
not tested	not tested	5%	21%	—

The skills below take us into the system of rational numbers. To review, a rational number is any number that can be written as a fraction with whole numbers (integers). These numbers are all rational numbers:

\(\frac{3}{8}\) because \(3\) and \(8\) are whole numbers
\(15\) because it can be written as \(\frac{15}{1}\)
\(7\frac{3}{5}\) because it can be written as \(\frac{38}{5}\)
\(8.6\) because it can be written as \(\frac{86}{10}\)

What is not a rational number? Many roots, like \(\sqrt{3}\) or \(\sqrt[3]{21}\). Also \(\pi\).

Using Fractions to Solve Problems

You should be able to calculate quotients of fractions and recognize when a word problem needs this kind of a calculation. Take a look at the situation below.

Some small engines, like chainsaw engines, require you to mix oil with gasoline. If a gallon of gasoline needs one thirty-second of a gallon of oil, how many gallons can you treat with one half gallon of oil?

The basic question here is how many \(\frac{1}{32}\) gallons of oil are in \(\frac{1}{2}\) gallon of oil. Because you are dividing something into smaller parts, this sounds like a job for division.

Divide \(\frac{1}{2}\) by \(\frac{1}{32}\):

\[\frac{1}{2} \div \frac{1}{32}\] \[\frac{1}{2} \times\frac{32}{1}\] \[\frac{32}{2}\] \[16\text{ gallons }\]

Extending Multiplication and Division Skills

We’re going to take a look at a few other kinds of problems that take us a little further in multiplication and division of rational numbers.

Dividing Fluently

Be able to use standard long division to divide numbers with several digits the way you see below:

\[\require{enclose} \begin {array}{r} 13.5\\ 24 \enclose{longdiv}{324.0}\\ \underline{24}\phantom{4.0}\\ 84\phantom{.0}\\ \underline{72}\phantom{.0}\\ 120\\ \underline{120}\\ \end{array}\]

Another important skill is knowing how to multiply and divide mixed numbers. Before you multiply or divide, each mixed number has to be rewritten as an improper fraction (the bigger number on top). Check out the example below.

\[4\frac{1}{2} \div 1\frac{1}{8}\]

Changing into improper fractions gives us:

\[\frac{2 \cdot 4+1}{2} \div \frac{8 \cdot 1+1}{8}\] \[\frac{9}{2} \div \frac{9}{8}\]

Invert the second fraction and multiply:

\[\frac{9}{2} \times \frac{8}{9}\] \[\frac{72}{18}\] \[4\]

Using Greatest Common Factors and Least Common Multiples

Be able to find the greatest common factor (GCF) of any two numbers up to \(100\). Also, find the least common multiple (LCM) of any two numbers up to \(12\). Let’s look at the greatest common factor. To phrase this a little differently, what we are looking for is a factor that both numbers have in common*, and we are looking for **the biggest one of those. Here’s an example of finding the greatest common factor of \(30\) and \(48\).

List all factors of both numbers.

Factors of \(30:\ \ \ \ 1, \,2, \,3, \,5,\, 6, \,10,\, 15,\, 30\)
Factors of \(48:\ \ \ \ 1, \,2,\, 3, \,4,\, 6, \,8, \,12, \,16,\, 24, \,48\)

Pick out the factors that are common (the same) to both numbers: \(1, \,2,\, 3,\, 6\).
Which is the greatest common factor? \(6\).
That’s it. \(6\) is the GCF of \(30\) and \(48\).

When you’re thinking of the least common multiple, the first thing to do is remember that least common means lowest common multiple. Let’s say we want the lowest common multiple of \(4\) and \(5\). We do this a lot like we did above with the GCF, but here we list multiples of the two numbers, not factors.

Multiples of \(4:\ \ \ \ 4, \,8, \,12, \,16, \,20, \,24, \,28,\, 32, \,36, \,40, \,44\, …\)
Multiples of \(5:\ \ \ \ 5, \,10,\, 15, \,20, \,25,\, 30,\, 35,\, 40, \,45\, …\)

Pick out the multiples that are common (the same) to both numbers: \(20, 40\).
Which is the lowest one? \(20\).
That’s it. \(20\) is the LCM of \(4\) and \(5\).

Using Properties

Suppose you have the sum of two numbers that have a common factor, say \(21+49\) because they both have \(7\) for a factor. You could factor out the common factor (\(7\)) and write the result like this: \(7(3+7)\). You can show that these two expressions will be equivalent: \(21+49=7(3+7)\).

Using the distributive property on the right side and adding the left, we get \(70=70\).

Working with Positive and Negative Numbers

If we want to count the money in our pocket, but find nothing there, we have exactly zero cents. Can we have anything less than zero cents? Well, in a way, yes. What if our pockets are empty but we also owe our mom the \(\$20\) she loaned us for lunch last week. We’re worse than broke. Someone would have to give us \(\$20\) just to get us up to broke. This kind of situation is why negative numbers were invented. To show our wealth in this case we would write \(-\$20\) to show that our wealth is \(\$20\) less than zero.

Other situations also use negative numbers to mean an amount less than zero. Temperature is one such situation. Any temperature below zero is a negative temperature. This zero is different than zero dollars because it doesn’t mean there is no temperature. It’s just a temperature that someone (Daniel Fahrenheit or Anders Celsius, for example) decided to call zero. Still, this kind of negative temperature is useful so we use it.

In physics, velocity is labeled positive or negative. One direction is picked to be positive and the opposite direction is thought of as negative. Zero velocity means no velocity, just like zero cents means no money.

The point of this all is that zero often means none. No money or no velocity. Sometimes, though, zero is just a measurement that is labeled zero, like zero degrees. You should be able to explain the meanings of zeros in different situations.

Using the Number Line

Just as a number line can show integers (whole numbers), so can it show non-integer rational numbers. We’ve seen this before.

Absolute Value

Absolute value basically means to ignore any negative sign that may be attached to the quantity inside the absolute value symbols. It’s easier to see than to explain.

\[\vert 12 \vert = 12\ \ \ \ \ vert -12 \vert =12\ \ \ \\]

If there is any operation inside the symbols, do that first, then make the result positive, as below:

\[\vert 5-8 \vert = \vert -3 \vert = 3\]

Absolute values are always positive.

Operations

Addition and Subtraction

To add integers with like signs, add in the usual manner, then affix the common sign to the sum:

\[-23 + (-44) = -67\] \[4 + 98 = 102\]

To add integers with unlike signs, get the difference of the integers, then affix the sign of the larger integer:

\[-56 + 42 = -14\] \[-15 + 36 = 21\]

To subtract integers, first change the sign of the subtrahend, then proceed to the addition of integers:

\[100 - (-4) = 100 + (4) = 104\] \[698 - (29) = 698 + (-29) = 698 - 29 = 669\]

Multiplication and Division

Multiply and divide integers with like signs as usual, then affix a positive sign (or no sign):

\[-25 \cdot -20 = +500 = 500\] \[14 \cdot 29 = 406\] \[-75 \div -5 = +15 = 15\]

Multiply and divide integers with unlike signs as usual, then affix a negative sign:

\[400\text{,}000 \cdot (-1) = -400\text{,}000\] \[-95 \cdot 4 = -380\] \[5 \div -2 = -2.5\]

Graphing

Be able to plot points in any of the four quadrants in the \(x-y\) plane. Check yourself by looking at each lettered point on the graph below and writing down its coordinates. Then look below the graph to see if you have the right values.

A = \((2, 3)\)
B = \((1,-4)\)
C = \((-3, 6)\)
D = \((-6, -3)\)

Another thing you should know how to do is find the distance between two points, such as \((-3, 4)\) and \((6, 4)\). Since they are both on the \(y=4\) line, the question comes down to “What is the horizontal distance between the \(x=-3\) line and the \(x=6\) line?” Very likely, you can look at the graph and see what that distance will be. If not, this always works: find the absolute value of the difference between the two numbers. It doesn’t matter in which order you write them. See the two examples below.

\[\vert 6-(-3)\vert = \vert 6+3\vert = \vert 9 \vert = 9\] \[\vert -3 -6\vert = \vert -9\vert = 9\]

Here’s an example of a problem where graphing negative values comes up.

The graph below shows the morning temperatures starting at \(6\) a.m. and warming up. Assuming the graph followed the same straight path down into negative temperatures, what temperature was it at \(5:30\) a.m.?

The line is splitting each graph square exactly in half. If you judge carefully, you can see that extending it downward to \((0,-10)\) will show that at \(5:30\) the graph line is exactly halfway between \(0\) and \(-10\). That puts the temperature at \(-5\).

Extending Addition and Subtraction Skills

Adding or subtracting rational numbers can be shown on a number line just like whole numbers can. This shows adding \(2\frac{1}{2} + 3\frac{1}{4}\):

This gives us a nice visual, but it isn’t very practical for actually adding mixed numbers. Here’s a little reminder of how to add two mixed numbers. We’ll use the same numbers as above.

\[2\frac{1}{2} + 3\frac{1}{4}\]

Line up the whole numbers and the fractions:

\[\begin{array}{r} 2\frac{1}{2}\\ +3\frac{1}{4}\\ \hline \end{array}\]

Make the fractions have the same denominator and add them, and add the whole numbers:

\[\begin{array}{r} 2\frac{2}{4}\\ +3\frac{1}{4}\\ \hline 5\frac{3}{4}\\ \end{array}\]

There are complications that can come up and you should brush up on these skills so you can handle them.

Approximating Irrational Numbers

Irrational numbers are a bit weird when put in decimal form for two reasons: the decimals never have a repeating pattern and they never end. That means we can never write the complete exact value of an irrational number. Whatever we write down will be an approximation, though we can get a really good approximation by figuring out more decimal places.

Keeping that in mind, what could we do without a calculator to decide which is bigger, \(4\) or \(\sqrt{18}\)? For starters, we can compare \(\sqrt{18}\) to some known square roots. We know these common square roots:

\[\sqrt{4}=2 \ \ \ \ \sqrt{9}=3 \ \ \ \ \sqrt{16}=4 \ \ \ \ \sqrt{25} = 5\]

\(18\) is between \(16\) and \(25\) so \(\sqrt{18}\) must be between \(4\) and \(5\). That means that \(\sqrt{18}\) is more than \(4\) but less than \(5\), and we have our answer.

The above tells us that \(\sqrt{18} = 4\) point something.