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# Page 1 Mathematics: Applied Study Guide for the TABE

## How to prepare for the TABE Math: Applied Test

### General Information

The Applied section of the TABE allows you to use a calculator, so it is not testing your computing skill. Rather, you will be assessed on how well you can take given numbers and design a means for finding a solution to a problem, including which operation(s) to use and in what order to use them.

On the first two levels (E and M) of the TABE Math: Computation Test, you will be working mostly with whole numbers, along with a few decimals and fractions. In levels D and A, however, you need to be able to perform operations with other types of numbers. Here is a list of the number types and levels in which they appear.

• whole numbers (E,M)
• decimals (E, M, D)
• fractions (M, D, A)
• integers (D, A)
• percents (D, A)
• algebraic (A)

Here are some procedures with which you will need to be fluent. We have just given the basic information about them and you should pursue more information and practice, especially for areas in which you are unsure.

### Skills to Practice

Many students cringe at the thought of “word problems” and the applied math section of the TABE is made up of these. Not only do you have to do a given operation with numbers, but you first have to figure out which operation, or operations, to do. But there are clues in every word problem to help you along and understanding these will be valuable.

#### Choosing the Right Operation(s)

Be familiar with buzzwords and clues because these will make it easier for you to translate words and phrases into numbers and mathematical operations. A mathematical sentence can express an equality or inequality. The words “is” and “equal to” suggest an equality of two statements, and are represented by the equal sign (=). “Greater than” and “less than” suggest an inequality, and are represented by the greater than sign (>) and less than sign (<), respectively. Then there are the phrases “greater than or equal to” and “less than or equal to,” which are written mathematically as $\ge$ and $\le$, respectively.

Watch out for these words and the operations they indicate:

• Plus, combine, total, sum, together, more, and increase are used to indicate Addition
• Minus, less, difference, left, taken away, decreased, and fewer are used to indicate Subtraction
• Times, product, twice, thrice, and of are used to indicate Multiplication
Per, quotient, and ratio are used to indicate Division

#### Identifying Unimportant Information

Irrelevant information may be deliberately inserted in math questions to make simple questions seem complicated or to mislead test-takers. Familiarity and constant practice in solving word problems will help you find what’s useful information in a math question. It helps to know formulas. A given number could be irrelevant if it does not fit in the formula. Sketch and label as you read the question. With a visual guide, an irrelevant piece of information will be easier to spot.

#### Estimation

In many cases you will be required to calculate the exact value of an unknown.. There are situations, though, where your skill in estimating becomes very useful. You can validate answers by estimating, such as what happens when you have painstakingly computed and arrived at a number, but by estimating you know that the answer you computed doesn’t make sense. This tells you that you have to compute again.

You may also estimate answers when you don’t have the time to do precise calculations. Estimating comes in handy when you need an answer quickly, such as this simple addition:

153 + 2508 + 48 + 3091 + 203 + 1999 = ?

150 + 2500 + 50 + 3100 + 200 + 2000 = 8000

The exact sum is 8002, but 8000 is close enough.

When rounding off, it’s important to round up some numbers and round down some, too. This way, you keep the margin of error to a minimum.

#### Measurement

Measuring is a mathematical method of assigning a number or value to the quantity or size of a particular object based on standards, calculations, formulas and units. The distance between two points is a measure of length. The size of a flat surface or plane is measured by determining its area. The capacity of or space occupied by a three-dimensional object is measured by calculating its volume. Measurement can be done on almost anything that can be quantified, including time, weight, mass, force, speed, angles, density, temperature, and many more.

The common measurement units encountered in TABE math are:

• Feet (ft), inches (in), centimeters (cm), meters (m), kilometers (km) – for distances and lengths
• Square meters (sq.m.), square inches (sq.in.), square feet (sq.ft.) – for areas
• Cubic feet (cu.ft.), cubic meters (cu.m.), liters, pints, gallons – for volume and capacity

It is important to be familiar with the process of converting one unit to another. There are two main systems of measurement – the Metric System (meter, grams, liter) and the US Standard Units (feet, pounds, gallons).

A unit can be converted to another unit within its system (from feet to inches), or to another unit in the other system (from meters to inches). The important thing to remember when solving measurement problems or math questions involving measurement units is to convert the different units into a common unit.

For instance, if you need to get the total length of poles with lengths of 3 m, 6 ft, 40 inches, and 5’10”, make sure to convert the lengths to one unit first before proceeding to addition. If the answer choices are given in feet, common sense dictates that you should convert all values to feet.

Here’s a very useful and safe way to convert measurements. To convert 45 m/s (meters per second) to mph (miles per hour), write the value to be converted as a fraction:

Multiply by the conversion factors, also written as fractions:

Tip: Take note that conversion factors are written in such a way that the same units can be cancelled later (e.g., m in the numerator will later cancel out with the m in the denominator of the conversion factor).

Cancel out the same units occurring in the numerator and denominator. Multiply all numerators, and multiply all denominators. Divide the resulting numerator by the resulting denominator. Affix the units that have not been cancelled.

It’s important, of course, to be familiar with basic conversion factors. The following is a list of some basic conversions.It is in no way exhaustive, but it will give you an idea of what to explore from other resources.

1 foot = 12 inches
1 yard = 3 feet
1 inch = 2.54 centimeters
1 mile = 5280 feet

1 ton = 2000 pounds
1 kilogram = 2.2 pounds

1 liter = 1000 milliliters
1 milliliter = 1 cubic centimeter

1 hour = 60 minutes
1 hour = 3600 seconds

#### Geometry Procedures

Geometric concepts are often woven into questions on fractions, percentage, percent change, ratios, and other math and number concepts. TABE word problems often involve plane and solid geometric shapes, and your familiarity with points in a coordinate plane, lines and slopes, angles, polygons, area and perimeter of polygons, Pythagoras’ theorem, circles, area and circumference of circles, and volume of solids.

Points have no dimension, only location or position in the coordinate plane. They are represented by a pair of numbers enclosed in parentheses, i.e. point (5, 10) is a point in Quadrant I of the x-y plane, located 5 spaces to the right of the origin, (0, 0), and 10 spaces above the origin.

Practice adequately on solving areas, perimeters and circumference of flat geometric shapes.

The perimeter of a polygon is simply the sum of the lengths of all its sides.

Circumference is the distance around a circle, and can be computed from this formula:

$C = 2\pi r$, where r is the radius of the circle

To compute the area, here are the commonly-used formulas for areas of various shapes:

Rectangle: $A = l \cdot w$ where l is length and w is width
Square: $A = s^2$ where s is the length of the side
Parallelogram: $A = b \cdot h$ where b is base and h is height
Triangle: $A = \frac{1}{2}b \cdot h$
Trapezoid: $A = \frac{1}{2}(a+b) \cdot h$ where a is the length of the short side and b is the length of the long side
Circle: $A = \pi r^2$ where r is the radius of the circle

When these flat shapes are raised to a certain height, or depth, or thickness, they become solids. These solids occupy space, and that space is called the volume of that solid. Having solved the area, it will be a simple matter of multiplying that area by the thickness to get the volume.

A cube is simply a flat square raised to a thickness equal to the length of the square’s side (s), therefore the volume (V) of a cube is:

Cube: $V = A \cdot s = s^2 \cdot s = s^3$

Likewise, the volume of the other shapes raised to a thickness or height (h) becomes:

Rectangular prism or box with rectangular base: $V = l \cdot w \cdot h$
Cylinder or circular base stretched to a height, h: $V = \pi r^2 h$

Learn to manipulate formulas. Math questions may provide the areas or volumes, and ask for the other elements, instead. In this case, plug in the known values, represent the unknown values with letters, isolate the unknown value to one side of the equation, and solve for the unknown. For instance, a TABE math question may ask for the height of a tank if it has a capacity of 30 cu.m. of water and has a radius of 1.5 m.

Write the appropriate formula and plug in the known values:

Isolate the unknown value and perform the required operations:

#### Using Data

Reading charts may seem daunting initially, but the ability to do so can be developed by familiarizing with the basics. Most of the questions in this category do not even require computation because the answers are already presented in the graph, table, or chart; if not, they can be easily inferred.

Charts or tables are normally presented with a title - this is your first clue. A title that says “Car sales of the different branches from 2001 - 2005” will definitely show these pieces of information: the different branches and the sales in these branches for the years stated. Inspect the heading, columns, rows, and legends. Acquaint yourself with how data is presented.

For example, the heading above the columns indicates the sales performance, each column represents the year, and each row represents the performance for each branch. Tackle the questions next. You will probably be asked about the sales of a particular branch in a particular year, or the branch with the highest sales in 2004, or the branch that showed the biggest difference in sales from 2001 to 2005.

There are several types of graphs - pie, bar, line, dot, pictograph, scatter plot and histogram. Graphs contain information displayed visually, including the answer to your test question. All you need to do is understand these graphs, and know how to make use of the data contained in them.

As in the charts, read the title first. It summarizes what the graph represents. Read the vertical and horizontal labels and the legend. These graphs compare values of one thing/category versus other things/categories over another dimension (a period of time, for instance). A line graph may compare the population of students in classrooms A, B and C, for the past 3 years. The vertical label may indicate the numbers, while the horizontal label may indicate the years. The legend may indicate the different colors designated for the classrooms.

A pie graph is quite unique because it does not have the vertical and horizontal labels that the other graphs have. Instead, it compares values relative to other values using percent, fraction or ratio.

#### Statistics and Probability

The TABE Applied Math test includes statistical concepts, such as finding the central value of a set of data. Computing for the central value is usually done by solving for the mean, median or mode.

To solve for the mean of a list of numbers, add all the numbers then divide the sum by how many numbers there are in the list. The mean of 40, 56, 38, 67, 50, 40, 55 and 70 is 52 because:

To find the median, arrange the numbers in ascending or descending order. Find the middle number (or numbers) of the sorted list. That middle number is the median. If there are two numbers in the middle, as in this case where there is an even number of elements in the list, add the middle pair and divide by 2. The result is the median. The median of this list is:

Sorted list in ascending order: 38, 40, 40, 50, 55, 56, 67, 70
Middle pair: 50 and 55
Median: $(50 + 55) \div 2 = 52.5$

The mode of a set of numbers is the number that occurs most frequently. In this set, 40 occurred twice while the rest occurred only once each. Therefore, the mode is 40.

Probability is a mathematical concept of predicting or calculating the chance that an event will happen or occur. It is given by the formula:

A clown has prepared 10 tricks for the party, including pulling a bunny out of a hat. If the time allows him to perform only 1 trick, what is the likelihood that he will perform that bunny-from-the-hat trick?

Probability = $\frac{1}{10}$

This number and counting concept is expanded to include combinations and permutations.

Combination is a concept of counting the possible number of ways a thing can be done or the pairing of things where order of elements does not matter. Combination is represented by this formula:

where n is the number of things to choose from r is the number of choices

Example: A salad bar has 9 kinds of fruits. Customers may mix 4 kinds of fruits per serving. How many possible fruit combinations can customers come up with?

Permutation is a similar concept except that the order of the things matters. Permutation is computed using these formulas:

(Repetition allowed) $P = n^r$

(Repetition not allowed) $P = \frac{n!}{n-r)!}$

#### Patterns

A math question may show a series of numbers that has a pattern and ask you to identify the nth element. First, you need to know whether the pattern shows an arithmetic sequence or a geometric sequence.

You know that a pattern is an arithmetic sequence if the same value is added to an element to get the next element. The series “5, 8, 11, 14, 17, 20, …” is an arithmetic sequence because 3 is added to get the next element. A question may go about asking you to give the next, the 100th, or the missing element in a pattern. Here is a formula to help you predict a specific element without going through repetitive addition. The formula for computing the nth element, $x_n$ is:

where $x_n$ is the nth element $x_1$ is the first element $d$ is the common difference between elements

In the given arithmetic sequence, we can solve the 100th element using the formula:

The elements may also be added using this formula for the Sum of Arithmetic Sequences:

To take the sum of the first 6 elements of the above sequence without adding them one by one:

This series of numbers, “3, 6, 12, 24, 48, 96, …” is obviously not an arithmetic sequence because the difference between elements is not constant. This is a geometric sequence. You get the next element by multiplying the previous element by a constant; in this sequence, the constant multiplier is 2.

The formula for finding the nth element is given by:

where $x_n$ is the nth element $x_1$ is the first element $r$ is the common ratio, or the common divisor

For the given series, the 12th element can be computed using the formula this way:

The formula for the sum of a geometric sequence is:

To find the sum of the first six elements using the formula:

There are more complicated patterns that you may also explore - square number sequences, Fibonacci sequences, triangular sequences, etc.. The patterns presented here, however, are the basics.

#### Functions

A function has three elements: the input, the output and their relationship. Every input relates to an output. They are also called ordered pairs and can be represented by (x, f(x)). The usual way of representing a function is to write it as f(x), read as “f of x”, but it can be written in many other ways, such as

$g(r) = r + 5$
$h(\theta) = \theta^2$

In this function, “$f(n) = - 3n + 10$”, what is f(n) if $n = -7$?

Substitute the value of n in the equation to get f(n):
$f(-7) = -3\cdot(-7) + 10 = 31$
This means that an input of -7 for this function produces an output of 31.

#### Algebra

Algebra is a very broad subject that involves exponents, radicals, logarithms, polynomials, simplifying and factoring, linear and quadratic equations, functions, word problems, and math concepts learned previously. Many real-life situations can be solved using algebra.

For example, a person travels by car and wants to know how fast he must drive to get from one city to another to catch a meeting in 1.5 hours. The distance between the two cities is 60 miles. The question can be rephrased as “What speed (?) times the travel time of 1.5 hours will result in 60 miles? In algebra, we assign variables to unknowns, instead of blanks or question marks. There are also rules followed and standard symbols used. That question is better written as:

Note:

In this equation, x represents the unknown value (for speed), but other letters in the alphabet may be used.

The word times is understood as “multiplication”, and will result in is written simply as “=”.

Isolate the unknown value because we want to get the end result of “x = something”. To do that, divide the right side by 1.5 and the left side by 1.5:

Rule: Keep the equality of both sides by doing on one side what you do on the other side. When we isolated x, we had to introduce division by 1.5 on the left side to get rid of the existing 1.5. To keep the balance of the equality, the expression on the right side must also be divided by 1.5.

We have arrived at the step that says $x = 40$, so we have solved the unknown; hence, we have solved the algebraic problem.

That’s a very simple illustration, and there are a lot more complicated situations in algebra. The basic thing is to be able to figure out an equation that describes the question, designate symbols and variables, make use of known values and formulas, and follow algebraic rules.