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Page 2 - Mathematics Study Guide for the PRAXIS Test

More Number and Quantity Concepts

Percent

Percents pop up in life as interest rates, tax rates, discount rates, student grades, and concentrations, to list a few, and as increases or decreases in almost any common quantity.

About Percents

A percent is a sort of ratio, basically meaning per hundred.

\(\frac{75}{100}\) can be thought of as \(75\) per \(100\), and can be written as \(0\text{.}75\)

or as a percent: \(75\%\). That, in a nutshell, is what you need to know about percents. They can be easily interchanged with decimals, ratios, or fractions.

Solving Problems with Percents

Change a Percent To a Decimal—All that is needed is to move the decimal two places to the left. \(\ \ 12\text{ %} = 0\text{.}12\) \(\ \ \ \ 4\text{.}5\text{ %} = 0\text{.}045\)
The opposite is true for changing a decimal to a percent.

Calculate a Percent Of a Quantity—Remember that, in this context, “of” means multiply. What is \(3\text{.}8 \%\) of \(15\text{,}200\)? Change the percent to a decimal and multiply.

\[0\text{.}038 \cdot 15\text{,}200=577\text{.}6\]

Calculate Percent of Change—The population of Smallville went from \(7\text{,}200\) down to \(6\text{,}500\). What was its percent of change? Subtract to find the difference and divide that by the original amount.

\[7,200-6,500=700\] \[\frac{700}{7200}=0.097\bar{2}\] \[0.0972=9.72\text{%}\]

Change a Ratio To a Percent—Suppose a student gets \(37\) out of \(44\) possible points on a math test. What is her percentage correct? Divide \(37\) by \(44\) and change to a percent.

\[\frac{37}{44}=0\text{.}841=84\text{.}1\text{%}\]

Rates

Rates are commonly used to compare some quantity to time, though they are not limited to that. Only constant rates will show up on this test.

What is a Rate?

Rates are a special use of ratios. A few examples of rates are speed (miles per hour), salary (dollars per week), flow rate (liters per minute), and nutrition (calories per gram). Most often, rates are written as unit rates, meaning the number that comes after the “per” is \(1\).

For example, if you drive \(90\) miles in two hours, your unit speed rate is

\[\dfrac{90\text{ miles}}{2\text{ hours}}\]

which is

\[\dfrac{45\text{ miles}}{1\text{ hour}}\]

or

\[45\dfrac{\text{ miles}}{\text{hour}}\]

Solving Rate Problems

Rates With Time—To get a unit rate, divide the given quantity by the given time.

\(r=\frac{Q}{t}\).

This was shown in the previous section. \(Q\) can stand for any quantity, such as distance, dollars, liters, calories, or whatever is being measured.

Depending on what you are looking for, you can easily rearrange the given formula in two different ways:

\(Q=rt\) or \(t=\frac{Q}{r}\)

As an example, suppose a painter can paint a wall at a rate of \(5\) square ft per minute. How long will it take him to paint a wall that is \(9\) ft by \(15\) ft?

The \(9\) ft by \(15\) ft wall is \(135\text{ ft^{2}}\). That will be our quantity, \(Q\).

Now use \(t=\frac{Q}{r}\).

\[t=\frac{135}{5}\] \[t=27\text { minutes}\]

Other Rates—Some ratios that can be treated as rates don’t involve time. Density is one. The weight of one cubic foot of water is \(62\text{.}4\) lb. A slightly different way to say this is to say that water weighs \(62\text{.}4 lb for every\)1$$ cubic foot, or that its density is

\[\dfrac{62.4 \text { lb} }{1\text { ft}^{3} }\text{ or }62.4\dfrac {\text { lb} }{\text { ft}^{3} }\]

You can still use the relation \(r=\frac{Q}{t}\) and its related formulas if you change the \(t\) to some generic variable, maybe \(u\) for unit: \(r=\frac{Q}{u}\). Here, the \(u\), unit, means the unit in the denominator of the rate.

For example, how much would \(24\) cubic feet of water weigh?

Use \(Q=ru\).

\[\require{cancel}\] \[Q=\dfrac{62\text{.}4 \text { lb} }{1\cancel{\text { ft}^{3}} } {\ \ \cdot \ \ 24 \cancel{\text { ft}^3}}=1\text{,}497\text{.}6 \text{ lbs}\]

Notice how the units \(\text{ft}^{3}\) will cancel out in this calculation.

Place Value and Number Order

Place values of digits to the right of the decimal point were mentioned in a previous section, but here is a short review of those digits to the left of the decimal point.

A Review of Place Value

Here is a table of place values for \(531\text{,}769\):

5 3 1 7 6 9
x 100,000 x 10,000 x 1,000 x 100 x 10 x 1
hundred thousands ten thousands thousands hundreds tens ones

Ordering Numbers

Putting different types of numbers in order is discussed in the next three sections, but what if you are given a mixture of all three types to order? For example:

\[\dfrac{2}{5}, \;0.331, \;\dfrac{3}{4}, \;\text{ and } \;2.9\text{ % }\]

Handle this group by changing each one to a decimal (you will have a calculator) and following the method below.

Ordering Decimals

Here’s a quick trick to put a group of decimals in order. Find the number that has the most decimal places. Let’s say it has four decimal places. Move the decimal point on all the numbers four places to the right, adding zeros if you need to. Presto, no more decimal places, only whole numbers. Now it’s easy. For example:

Order these numbers from highest to lowest: \(0.107, \;1.29, \;0.0566, \;0.020\)
After moving the decimal point four places: \(1070, \;12,900, \;566, \;200\)
Final order: \(12,900, \;1070, \;566, \;200 \rightarrow 1.29, \;0.107, \;0.0566, \;0.020\)

Ordering Fractions

There will be an on-screen calculator on this test, so the easiest way to order fractions is to change each fraction to a decimal, and order them as above. The only difference is that you will be in charge of rounding each decimal rather than having them given. The method above will still work, as long as you don’t do something silly like round everything to a single digit. When in doubt, keep a few extra decimal places when you round.

Ordering Percents

Percents can easily be changed to decimals by moving the decimal point two places to the left. When they are decimals, put them in order as shown above.

Properties of Whole Numbers

Solving even the simplest problems calls for knowing the properties of whole numbers such as the ones listed below. You should be very familiar with them. Some of them can also help you evaluate whether an answer you get makes sense and is actually correct.

Factors and Divisibility

Consider \(2\times 5\times 7= 70\)
We could say that \(2\), \(5\), and \(7\) are factors of \(70\).
We could just as well say it another way: \(70\) is divisible by \(2\), \(5\), and \(7\) (also by \(10\), \(14\) and \(35\)).

What about factors of \(42\)? They are \(2\), \(3\), \(6\), \(7\), \(14\), and \(21\).
Do \(70\) and \(42\) have any factors in common? Yes, \(2\), \(7\), and \(14\).
What is the greatest of these common factors? \(14\)

For kind of obvious reasons, we call \(14\) the greatest common factor (GCF) of \(42\) and \(70\). The GCF is said to be useful for simplifying fractions. Divide the top and bottom by the GCF and the fraction will be simplified in one round of division.

Multiples

Multiples are products of whole numbers. Multiples of \(5\), for example, are \(10\), \(15\), \(20\), and \(25\).

You should know how to find the least common multiple of two (or more) integers. This doesn’t mean the rarest multiple, though it does sound that way. It means the lowest common positive multiple of two or more integers.

Think of 6 and 9.
Some multiples of 6 are 12, 18, 24, 30, 36.
Some multiples of 9 are 18, 27, 36, 45.
What is the lowest of those that is common to both groups? 18, right?
The number 36 is also a common multiple, but it’s not the lowest.
This can come in handy when you are finding a common denominator to add or subtract fractions.

Even and Odd Numbers

Well-known properties of even integers include always being divisible by 2, and always ending in 0, 2, 4, 6, or 8.

Also, if you add, subtract, or multiply even numbers, the result is always even. (A mathematician would say that the set of even numbers is closed under addition, subtraction, and multiplication.)

You also will always get an even integer by adding two odd integers. In fact, the only way you can get an odd result from these three operations is by adding an odd and an even, or multiplying two odds.

Division is more complicated, so it’s not very useful as a quick check on an answer.

These properties are a little bit trivial, but they can help you decide fairly quickly whether an answer makes sense or not.

Prime Numbers

Prime numbers were mentioned earlier in the section about Concepts Concerning Integers. A prime number is a whole number greater than 1 that is not divisible by any whole numbers other than itself and 1. Small numbers that are examples of primes include 2, 3, 5, 7, 11, 13, 17, …

Mathematicians are kind of fascinated with primes and probably have parties where they each try to come up with the biggest prime number of the night. As of about a year ago, the biggest prime number known had 24,862,048 digits, so this is now way past the party game stage. (A number with 24,862,048 digits will not be on the test.)

Counterexamples

It can be quite hard to prove a claim because no matter how many examples you may show where it appears to be true, they don’t prove that there may not be some situation where it isn’t true. On the other hand, to disprove a claim, all you have to do is come up with one case where it is false. This case is known as a counterexample.

As an example, suppose someone claims that the division of two even integers always produces an even result. They cite several examples: \(12\div6=2 \text{ , }8 \div2=4\text{ , } 24\div6=4\). You probably already see how to spoil their fun. Point out a counterexample such as \(12\div4=3\). Bingo. Disproved.

Problem-Solving Techniques

No one has come up with the perfect method for teaching or learning how to solve math problems, which is one reason they can be hard. There are tips that can help and some of them are listed here. Persistence is your friend if you have the time for it.

Identifying Relevant Information

  • Know what you have.—Read the problem. Then read it again. Write down all given numbers with their units. Identify each one with a variable: \(t\) for time, \(c\) for cost, \(l\) for length, etc.

  • Know what you want.—Write down what the problem is asking for. Label it with an appropriate variable. Try to figure out the units the answer should have. Example: \(P\) = ____ meters.

  • Look at known formulas or theorems.—Many problems have standard formulas that could apply: \(d=rt,\ \ A=lw, \ \ C=2\pi r, \ \ y=mx+b\). Look at the variables that are given and wanted and try to match them to a formula.

  • Make it easier.—Often the numbers in a problem are unfriendly. Try putting simpler numbers in and solving the problem with those. Your brain will work better with those and may see a way to do the problem with the original numbers.

  • Make a drawing.—If the problem is geometric, a labeled drawing is almost a necessity.

Be aware that the problem may have extra information that isn’t actually needed, though most test questions seem to give only what is needed.

Selecting Operations

There are common clue words or phrases that show up in math problems that can help you decide what operation(s) need(s) to be done.

Total, sum, perimeter, in all can indicate addition.
Difference, exceeds, how much more, how much less, can indicate subtraction.
Product, times, area, total, indicate multiplication, though total can mean addition too.
Quotient, average, ratio, can indicate division.

Many problems need more than one step to solve. If you can’t figure out the final answer, figure out anything you can as a first step and you may be able to use that quantity to figure out the final answer.

Rounding

Some problems will state how the answer should be rounded. If it’s a money problem, it’s understood that it should be rounded to \(2\) decimal places. Otherwise, just be reasonable.

Units of Measurement in Problems

Word problems often have measurement units attached to the numbers. You should remember common units for two reasons. First, it will help you decide if your answer makes sense. Second, it can help you decide what operations you will need to solve the problem. For example, if you’re looking for area, you are probably going to be multiplying two length units. If it’s volume, you’ll be multiplying three length units.

Measurement Systems

Because the U.S. has never jumped whole-hog into the metric (SI) system, we have to deal with U.S. customary units as well as metric units. It’s a pain, but what are you going to do?

Length

You will find that these U.S. customary units of length are found in many math problems: inches, feet, yards, and miles. Know how to convert between these units.

In the metric system, all lengths are based on the meter (m), and the following units are often seen: millimeters (mm), centimeters (cm), maybe decimeters (dm), and kilometers (km). Know what the prefixes mean. There are lots more, but you’re not as likely to run into them.

Volume

Take each length unit above, cube it, and you have volume units. This is done in both U.S. customary and metric systems. The units may be abbreviated with a cu. in front of the unit (cu. ft. for cubic feet) or an exponent of 3 for the length unit (\(ft^{3}\text{ or } m^{3}\)). In addition to cubic units, the metric system has liters (L) and milliliters (mL), and the U.S. system has pints, quarts, and gallons.

Weight

Again, you have to be aware of both measurement systems. Weight in U.S. customary units is usually done in ounces (oz.), pounds (lb.), and tons (T.).

Common weights in the metric system are grams (g) and kilograms (kg), though these units are more correctly known as units of mass.

Temperature

The Fahrenheit temperature is well known to those of us in the U.S., in which \(32^{\circ}F\) is the freezing point of water and \(212^{\circ}F\) is boiling.

The Celsius scale has \(0^{\circ}C\) as freezing and \(100^{\circ}C\) is boiling. There are formulas for converting from one system to another that you don’t have to know but should know how to use, given that they are pretty basic algebraic equations.

Time

The metric (SI) base unit of time is the second (s), and prefixes define other time units, such as millisecond (ms), kilosecond (ks), and so forth. Minutes, hours, and days are used in both systems.

Converting Units

Converting metric units is done by simply moving the decimal point. The trick is to know which way and how far. For example, to go from \(5.3\) kg to g units, remember that a kilogram is \(1000\) g. If you’re converting to grams, you are going to a smaller unit, so there will be more of them. To get more of them, move the decimal point to the right, in this case, three places: \(5.3\) kg = \(5.300\) g.

Another method is to use a conversion factor. Use \(\dfrac{1000 \text{ g}}{1 \text{ kg}}\) to multiply times the given number: \(\require{cancel} 5.3 \cancel{\text{ kg}} \cdot \dfrac{1000 \text{ g}}{1 \cancel{\text{ kg}}}=5.300 \text{ g}\).

Notice how the kg units cancel out. That’s how you know you have the conversion factor right side up. If you were going from g to kg, you would use the factor \(\dfrac{1 \text{ kg}}{1000 \text{ g}}\) .

Converting U.S. units to other U.S. units, or from one system to another, can also be done with a conversion factor.

For example to go from \(1320\) ft to miles, remember that one mile is \(5280\) ft and do this:

\(1320 \cancel{\text{ ft}} \cdot \dfrac{1 \text{ mile}}{5280 \cancel{\text{ ft}}} = 0.25 \text{ mi}\).

Again, notice how the ft units cancel out.

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