Mathematics Study Guide for the TABE Test

Numbers and Operations: Base Ten—Operations 3

This section will review some additional ideas that help us thoroughly understand and best use operations with whole numbers as well as decimals.

Inverse Operations

Addition and subtraction are opposite or “inverse” operations. One of them undoes what the other one does. For example, in the first calculation below, $6$ is subtracted from $18$ to get a result of $12$. The second calculation adds $6$ to the $12$, getting us back to $18$ and reversing what was done to the $18$:

\[18 - 6 = 12\] \[12 + 6 = 18\]

Similarly, these are inverse operations:

\[35 \div 5 = 7\] \[7 \times 5 = 35\]

This idea comes in handy when you want to check your answer. It will also be important when you’re doing algebra later.

Addition and Subtraction

Knowing that addition and subtraction are inverse operations allows you to think about a problem from the opposite direction. For example, the result of the problem $14 - 8 = ?$ can also be thought of as $8 + ? = 14$. Written this way, you can see that the question is really, “What number makes $14$ when added to $8$?”

Multiplication and Division

Memorizing the products of any two numbers up to $12 \times 12$ will be very helpful. Not only will this make the multiplication process quicker, but it will save you time when you’re doing division as well. For example, if you know that $6\times 7 = 42$, then the inverse of that is $42 \div 7 = 6$ or $42 \div 6 = 7$.

Operations with Decimals

The basic operations work the same way for decimals as for whole numbers. The main difference is that you need to pay attention to the location of the decimal point. You should be able to add, subtract, multiply, and divide decimals out to the hundredths place.

Adding and Subtracting Decimals

As with whole numbers, when adding or subtracting decimal numbers, write them vertically, with digits aligned according to their place value. All decimal points must be aligned, as well.

Decimal addition and subtraction follow in much the same manner as the addition or subtraction of whole numbers shown earlier, with the only difference being the decimal point that separates the whole and decimal numbers. Regrouping will be necessary if a column’s sum is over $9$, just like in regular addition. In subtraction, regrouping will again be necessary when a digit is larger than the digit it must be subtracted from.

Operations with Money

One of the most common situations in which you’ll encounter decimals in the real world is when dealing with money. Because money is written in dollars and cents, it naturally involves decimals, with the cents representing the hundredths place. Being able to add, subtract, multiply, and divide decimals will help you handle everyday tasks like shopping, budgeting, and making financial decisions.

Let’s look at a practical example.

Maria buys three items at a store that cost $\$4.75$, $\$2.60$, and $\$3.45$. How much does she spend in total?

Solution

To find the total cost, we add the decimal numbers, being sure to line up the decimal points:

\[\begin{align} \ \ 4.75 \\ \ \ 2.60 \\ +\underline {3.45} \\ 10.80\\ \end{align}\]

Therefore, Maria spent a total of $\$10.80$.

Multiplying Decimals

Multiplying decimal numbers starts in the same way we multiply whole numbers, differing only in the last few steps. In fact, at first, you will simply multiply the numbers while ignoring the decimals.

For instance, if you multiply $7.65$ by $9.8$, you will work as if you were multiplying $765$ by $98$. The answer to that is $74\text{,}970$.

Now, these steps are where decimal multiplication differs:

Count the total number of decimal places in both the multiplier and the multiplicand (in this case, $1 + 2=3$ decimal places).
Move the decimal that number of places from the right toward the left. In this case, we move the decimal three places to get $74.970$.
Drop any trailing zeros after the decimal, as the number will retain its value without them. So, our final answer is $74.97$.

Dividing Decimals

Division of decimals is similar to division of whole numbers, except for the presence of the decimal point. When doing long division, add these three steps:

Move the decimal point in the divisor to the right until the divisor becomes a whole number.
Move the decimal point in the dividend to the right the same number of places that you moved the decimal in the divisor.
Put a decimal point in the quotient area, directly above its spot in the dividend. That is where it will be in the answer.

For instance, say we were dividing $85.28$ by $4.1$. We’d set up the basic long division:

\[\require{enclose} \begin{array}{r} 4.1 \enclose{longdiv}{85.28} \\[-3pt] \end{array}\]

Once we move the decimals, we have:

\[\require{enclose} \begin{array}{r} 41 \enclose{longdiv}{852.8} \\[-3pt] \end{array}\]

We do the division as we did before to get:

\[\require{enclose} \begin{array}{r} 20.8\phantom{.} \\[-3pt] 41 \enclose{longdiv}{852.8} \\[-3pt] \underline{82} \phantom{2.8.} \\[-3pt] 32 \phantom{.8.} \\[-3pt] \underline{0} \phantom{.8.} \\[-3pt] 328 \; \phantom{.} \\[-3pt] \underline{328} \; \phantom{.} \\[-3pt] 0 \; \phantom{.} \\[-3pt] \end{array}\]

Note: You may add zero in the dividend if necessary and proceed with the division up to the required number of decimal places.

Rounding Numbers

In some cases, we just want a quick approximate answer to a problem. For example, suppose you pick up three items in the grocery store for $\$4.98, \$3.10,$ and $\$2.95$. You could do some quick rounding and adding to get an estimate of your bill to make sure you have enough money. Your thinking would go something like this:

Round the numbers to the nearest dollar ($5, 3, 3$) and add them. You have $5 + 3 + 3 = \$11$. That’s not the actual total, but it’s close enough.

Rounding Whole Numbers

Whole numbers can be rounded to any place value represented within that number. The basic idea is this:

Look at the digit in the place you need to round to.
If the number to its right is $5$ or greater, add $1$ to the digit and change all the other digits to its right to $0$.
If the number to the right of that digit is less than $5$, the digit remains the same and all the numbers to its right change to $0$.

Let’s try a couple examples.

Round $136$ to the nearest ten.

Solution

The digit $3$ is in the tens place. Look to its right and you see the $6$. That’s greater than $5$, so add $1$ to the three and write $0$ in place of the $6$, giving you $140$. This is called rounding up ($136$ was rounded up to $140$).

Round $5\text{,}470$ to the nearest thousand.

Solution

The digit $5$ is in the thousands place. Look to its right and see the $4$. That’s less than $5$, so you don’t add $1$ to the $5$, and you write $0$ in place of the $4$ and the $7$. This is rounding down ($5\text{,}470$ was rounded down to $5\text{,}000$).

Rounding Decimals

Rounding decimals is exactly like rounding whole numbers, except after rounding, you don’t add any zeros to the right. Instead, you simply drop those digits.

Here are a couple examples.

Round $9.268$ to the nearest hundredth.

Solution

The digit $6$ is in the hundredth place. So, we look to its right and see $8$. That’s greater than $5$, so we add $1$ to the $6$, drop the $8$, and end up with $9.27$.

Round $0.3642$ to the nearest tenth.

Solution

The digit $3$ is in the tenths place. Look to its right and see the $6$. That’s greater than $5$, so add $1$ to the $3$ and drop the $642$, giving you $0.4$.

Note: Don’t worry about the zero to the left of the decimal point. That’s just there to help keep us from missing the decimal point. It doesn’t affect the rounding process.

Mental Math and Estimation

There are certain basic operations that you should be prepared to do quickly in your head. For instance, you need to be able to mentally add or subtract $10$ from any two-digit number. Luckily, it’s not hard. Just add or subtract $1$ from the tens place.

For example, what is $10$ more than $58$? You don’t need a calculator, just add $1$ to the $5$ and write $68$. What is $10$ less than $32$? Just subtract $1$ from the $3$ and write $22$.

It works because by adding or subtracting $1$ in the tens place, you are actually adding or subtracting $10$ to or from the original number.

Multiplying by $10$ is also easy to do in one’s head, as you just add a $0$ to the multiplicand: $5 \times 10 = 50$.

Similarly, dividing by $10$ is straightforward. Simply drop the final $0$ or, if the final number is a number other than $0$, move the decimal point one space to the left: $35 \div 10 = 3.5$.

There will also be situations where your skill in estimating becomes very useful. You can validate answers by estimating. For instance, if you’ve done calculations to arrive at an answer, but it isn’t close to a basic estimation, chances are you need to do your calculations again. You may also estimate when you don’t have the time to do precise calculations or when you have a range of answer choices and you need to narrow down the possibilities.

Consider this problem:

\[153 + 2\text{,}508 + 48 + 3\text{,}091 + 203 + 1\text{,}999 =\]

By rounding off the addends, you can do much easier mental addition:

\[150 + 2\text{,}500 + 50 + 3\text{,}100 + 200 + 2\text{,}000 = 8\text{,}000\]

The exact sum is $8\text{,}002$, but $8\text{,}000$ is close enough. On a multiple-choice question, if the answer choices varied widely, doing this estimation would allow you to confidently guess the correct answer.

When rounding off, choose the lowest place value that still allows you to do the math easily in your head. This will keep the margin of error to a minimum. For instance, rounding $1\text{,}204$ to $1\text{,}200$ will provide a more accurate answer than $1\text{,}000$.