Applied Mathematics Study Guide for the WorkKeys

Skills Needed for Level 4 Questions

Calculations in Level $4$ are likely to involve two steps, and may include an extra piece of information that will not be needed to solve the problem. The information may not be presented in the order it will be used and you may see simple graphs, charts, or diagrams.

Using Two Operations

There are a lot of problems that can’t be done in one step. You will need to calculate one thing first and then use that to calculate a second thing (the answer to the problem). Here’s an example.

So far today I’ve eaten $20$ m & m™s. If I eat $12$ an hour for the next $7$ hours, how many will I have eaten in total?

Step $1$: Calculate how many I will eat in the next $7$ hours.
Step $2$: Add that number to the number I’ve already eaten.

Step $1$ : $12$ per hour $\times \,7$ hours = $84$
Step $2$ : $20 + 84 = 104$ total m & m™s

You can decide to string the two calculations together into one expression like this:

\[12\times7+20\]

If you do that, make sure to do the multiplying first, and then the addition.
Even if you write it like this $20+12\times7$, do $12\times7$ first, then add the $20$.

There are rules for doing operations in the right order. Make sure you know them.

Remember to follow the order of operations (PEMDAS):

Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division (MD), in order, left to right.
Then do all addition and subtraction (AS), in order, from left to right.
Good way to remember: Please Excuse My Dear Aunt Sally

Finding the Average (Mean)

The average of a set of numbers is also called the mean (arithmetic mean, if you want to get technical.) It is a way of taking a bunch of numbers, maybe your grades, and condensing them down to one number that represents the whole set. Finding the average is a two step problem.

Example: What is the average of these summer temperatures?

\[68, \,77, \,72, \,81, \,91, \,84, \,89\]

Step $1$ : Add all the numbers.

\[68+77+72+81+91+84+89= 562\]

Step $2$ : Divide the total by how many numbers there are.

$562\div7=80.3$ (rounded to the nearest tenth)

It’s common to end up with a decimal in the average. Also, the numbers you add up may very well have decimals in them. This shouldn’t be a worry—your calculator won’t care.

Ratios, Proportions, and Rates

You will need to know how to recognize a ratio, proportion, or rate, and perhaps write simple examples from a written description.

Finding a Ratio

A ratio is a way to compare two quantities, best understood by looking at some examples. You very likely have $2$ feet and $10$ toes.. To compare them, we can say that the ratio of feet to toes is $2$ to $10$, written $2 \text{:}10$. We could just as well say the ratio of toes to feet is $10 \text{:} 2$.

Like fractions, ratios can be reduced. $10 \text{:}2$ is equivalent to $5 \text{:}1$, and $2 \text{:}10$ is the same as $1 \text{:}5$.

In practice, ratios are very often written as fractions, so $2 \text{:}10=\dfrac{2}{10}=\dfrac{1}{5}$.

Finding a Proportion

A proportion is just an equation with a ratio on each side.
It can be written like this:

\[\dfrac {2}{3}=\dfrac {12}{18}\]

Or like this:

\[2 \text{:}3=12 \text{:}18\]

A proportion is usually used to solve for an unknown quantity and will look like this:

\[\dfrac {3}{8}=\dfrac {x}{32}\]

This may not come up in Level $4$, but it could show up in higher levels.

Finding the Rate

Rates are special cases of ratios, very often having to do with time. Most often rates are written as unit rates, meaning the bottom number in the ratio is $1$.

Speed is a rate of distance traveled per time unit: $\dfrac{65\text{ miles}}{1\text{ hour}}$

The number $1$ is generally understood to be there and is not written:

\[\dfrac{65\text{ miles}}{\text{ hour}}\]

If the word per is written in the problem, you should take that to mean “divided by” when you use it in the problem. For example, if an oil well is producing $45$ gal. per min., you would write that rate as:

\[\dfrac{45 \text{ gal.}}{\text{min.}}\]

Other abbreviations are used to represent rates, for example m.p.h. or mph for miles per hour or mpg for miles per gallon (an example of a rate that doesn’t involve time.)

Adding Fractional Numbers

At this level, only very common simple fractions will show up.

Fractions

To add fractions with the same bottom numbers (denominators), just add the top numbers (numerators) and write their total over the common denominator.

Examples:

\[\dfrac{2}{5} + \dfrac{1}{5} = \dfrac {3}{5}\] \[\dfrac{3}{8} + \dfrac{5}{8} = \dfrac {8}{8}=1\]

Decimals

The most important thing to do when adding decimals is to carefully line up the decimal points and the columns. Once you do that, the rest is just addition.

Example: $0.308 + 2.94$

\[\begin {array}{cccccc} &0&.& 3 & 0 & 8 \\ +&2&.& 9 & 4 && \\ \hline &3&.& 2 & 4 & 8 \\ \end{array}\]

Percentages

Since percentages are a form of decimals, you add them the same way you add decimals.

Example: $75.3\% + 9.5\%$

\[\begin {array}{cccc} 7&5& .& 3 & \% \\ +&9& . & 5 &\% \\ \hline 8&4& .& 8 & \% \\ \end{array}\]

Adding and Subtracting Fractions

We’re sticking with fractions with common denominators at this level, so, as we saw earlier, adding means adding only the top numbers. By the same token, subtracting means subtracting only the top numbers.

When you’re adding, one thing to watch for is getting a sum larger than one.

\[\dfrac{3}{5} + \dfrac{4}{5} = \dfrac {7}{5}\]

You generally shouldn’t leave a fraction in improper form (top number is larger than the bottom), so, to rewrite it as a mixed number, think of:

$\dfrac {7}{5}$ as $\left( \dfrac {5}{5} + \dfrac {2}{5}\right)$

Rewrite that as:

\[\left(1 + \dfrac {2}{5}\right)\]

or better yet as:

\[1 \dfrac {2}{5}\]

Subtracting is not tricky. Subtract the second top number from the first one. Reduce the answer if you can.

\[\dfrac{7}{8} - \dfrac{3}{8} = \dfrac {4}{8} = \dfrac{1}{2}\]

If you need to subtract from a mixed number, you may want to change it to an improper fraction. For example:

\[1\dfrac{3}{8} - \dfrac{7}{8}\] \[\left(1 + \dfrac{3}{8}\right) - \dfrac{7}{8}\] \[\left(\dfrac{8}{8} + \dfrac{3}{8}\right) - \dfrac{7}{8}\] \[\dfrac{11}{8} - \dfrac{7}{8}\] \[\dfrac {4}{8}\] \[\dfrac{1}{2}\]

See the next section for a shorter way to do this.

Multiplying Mixed Numbers

When you’re working with mixed numbers, first change each one into an improper fraction. The method above shows steps for doing this, but there is a shorter way to do it.

Consider this mixed mixed number: $2\dfrac{4}{5}$

To change it to an improper fraction, multiply the $5$ by the $2$ and add that to the $4$. Write that result on top and keep the $5$ on the bottom.

$5\times 2=10 \text{ and } 10+4=14$, so the improper fraction is $\dfrac{14}{5}$.

If you need to multiply this result by a whole number, say $3$, set it up this way:

\[3\times \dfrac{14}{5}\] \[\dfrac{3}{1} \times \dfrac{14}{5}\] \[\dfrac{3\times 14}{1\times 5}\] \[\dfrac{42}{5}\]

To change back to an improper fraction, divide $42$ by $5$ and you will get $8$ with a remainder of $2$, which can be written as:

\[8\dfrac{2}{5}\]

If you are doing a problem where you need to multiply a mixed number by a decimal it’s usually best to change the mixed number to a decimal.

\[0.35\times 2 \dfrac{4}{5}\]

Divide $4$ by $5$ to get the decimal part of $2 \dfrac{4}{5}$:

\[\dfrac{4}{5}=0.8\] \[0.35 \times 2.8 =0.98\]

Sorting Information in a Question

Often the information you’re given in a problem is not in the same order it will be used in your calculation. Suppose the sales tax on a new car is $\$1 \text{,}800$. If the total purchase price is $\$34 \text{,}470$ which also includes a dealer prep fee of $\$550$, how much was the price of just the car?

A table of values can be your friend here because it can help you organize your thoughts.

\[\begin {array}{l|r} \text{Car} & \text{?} \\ \hline \text{Prep Fee}&+ \$550\\ \hline \text{Tax}& +\$1800\\ \hline \text{Total}&\$34,470 \\ \end{array}\]

Or you may decide that this is a good way to lay it out:

\[\begin {array}{l|r} \text{Total}&\$34,470 \\ \hline \text{Prep Fee}&- \$550\\ \hline \text{Tax}& -\$1800\\ \hline \text{Car} & \$32,120 \\ \end{array}\]