Mathematics Study Guide for the TABE Test

Numbers and Operations: Base Ten—Operations 2

In this section, we will cover the other two basic operations: multiplication and division. Multiplication is another way to combine numbers and division is an additional method of separating numbers.

Multiplication

Multiplication is repeated addition, with the numbers being combined to reach a higher result, known as the product. It’s useful when you need to combine many identical numbers. For example, if you have eight pails, each one with nine potatoes, how many potatoes do you have? If you know that eight groups of nine is the same as \(8 \times 9\) and \(8 \times 9 = 72\), you won’t have to add all those nines together.

The key words that tell you to multiply in a word problem are times and product. Also, the word of means multiply in phrases like “one-fifth of” and “thirty-five percent of.”

Note: In addition to the common times symbol (e.g., \(5 \times 6\)), you may also see an operation with a dot (e.g., \(5 \cdot 6\)). This is just another symbol for multiplication.

Modeling Multiplication

The process of multiplication can be represented by a rectangular collection of squares. For instance, consider the rectangle below that has been subdivided into multiple squares:

We can look at the rectangle and see that there are seven rows, each having \(12\) squares. The total number of squares, if you care to count them, is \(84\), which is also the product of \(7\) times \(12\). This helps show why multiplication works the way it does.

Let’s try an example problem.

How many boxes are in the array below?

\[\begin {array}{cccccc} &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ \end{array}\]

Solution

You could count every box, but multiplication is much faster. Each row and column is the same, so we just need to count how many boxes are in each and multiply those numbers:

\[6 \times 7=42\]

Multiplication in Columns

You won’t have to do anything too complex on the TABE, like multiply two four-digit numbers, but you should be able to multiply a four-digit number by a one-digit number or a two-digit number by another two-digit number. Some problems you will be able to do in your head; at other times, you may need to write it down. Being able to do simple multiplication is a valuable and time-saving skill.

To perform long-hand or manual multiplication, start off by writing the factors (or the numbers to be multiplied) vertically. The number written on top is the multiplicand, while the one below is called the multiplier. As you did with addition and subtraction, you’ll align digits according to their place value, with ones on the right and increasing place values toward the left. So, to multiply \(23\) by \(56\), first write it as such:

\[\begin{align} 23& \\ \times \underline{\quad 56}& \\ \end{align}\]

When multiplying, you will often need to regroup (also called carrying), just as you did with addition and subtraction. This happens when the product of two digits is greater than \(9\). In that case, you write down the ones digit of the result and carry the tens digit to the next place value. This carried value is then added to the next multiplication step.

Let’s walk through the regrouping in this problem step by step. We start by multiplying the ones digit (\(6\)) of the multiplier by each digit of the multiplicand:

1. Multiply \(6\) by \(3\) to get \(18\). Write down the \(8\) in the ones place and carry the \(1\) to the tens place.

2. Multiply \(6\) by \(2\) to get \(12\), and add the carried \(1\) to get \(13\).

\[\begin{align} 23& \\ \times \underline{\quad 56}& \\ 138& \ \end{align}\]

We then proceed by multiplying the tens digit (\(5\)) of the multiplier by every digit of the multiplicand. Since this \(5\) is in the tens place (representing \(50\)), we place a zero as a placeholder in the ones column before multiplying:

1. Multiply \(5\) by \(3\) to get \(15\). Write down the \(5\) in the tens place and carry the \(1\).

2. Multiply \(5\) by \(2\) to get \(10\), and add the carried \(1\) to get \(11\).

\[\begin{align} 23& \\ \times \underline{\quad 56}& \\ 138& \\ 1150& \ \end{align}\]

Now, add the partial products:

\[\begin{align} 23& \\ \times \underline{\quad 56}& \\ 138& \\ \underline{1150}& \\ 1288 \end{align}\]

Thus, the product of \(23\) times \(56\) is \(1\text{,}288\).

Note: For simplicity, you can leave the comma out while doing the multiplication process.

Memorizing Multiplication Facts

Like the memorization of addition facts, knowing the multiplication facts for numbers between \(0 \times 0\) and \(12 \times 12\) will save you time and help you as the math concepts get more involved. When you are calculating longer, involved problems, you won’t have time to stop and figure out these simple problems. You should just have the products in your head.

One helpful tool to use is a list of the multiplication facts or a table like the one below. It’s used in the same way that we use the addition fact table above, but applying multiplication.

Division

You can think of division as the opposite of multiplication. The point of division is to find out how many of one number it takes to equal another number. When we write \(12 \div 4 = 3\), we are saying that four goes into \(12\) three times. In that example, \(12\) is the dividend, \(4\) is the divisor, and \(3\) is the quotient.

The most common division symbol is \(\div\). Sometimes division is represented as a fraction:

\[32/8 = 4 \text{ or } \frac{32}{8} = 4\]

In a word problem, there are key words and phrases that tell you to divide, including quotient, into, each, per, how many times, and goes into.

Models of Division

To visualize the division process, a drawing can be helpful. The one below shows the problem \(12 \div 3\). Simply draw the number of objects to be divided, then draw rings around the number of objects you are dividing by until all the objects are within rings. Then count the rings for your answer:

For another example, remember the rectangle model we used to show that \(7 \times 12 = 84\)?

Let’s think about that in a different way. How many rows of \(12\) squares did it take to give us \(84\) squares? Seven. So, we can just reverse the multiplication and say that \(84\) divided by \(12\) equals \(7\).

We could have also asked how many columns of seven squares did it take to give us \(84\) squares. That is, \(84\) divided by \(7\) equals \(12\).

Long Division

Division of some numbers may simply require a quick recall from memory, such as \(100 \div 4 = 25\) or \(36 \div 6 = 6\). If you know your multiplication tables well, such basic division is simple. However, division in the real world can involve multiple steps and result in decimal numbers or remainders. To handle more complex problems, you need to be familiar with the steps of long division.

Let’s practice with the problem \(8\text{,}528 \div 41\):

Step \(1\): Write the dividend inside a box and the divisor outside of that box, like so:

\[\require{enclose} \begin{array}{r} 41 \enclose{longdiv}{8528} \\[-3pt] \end{array}\]

Step \(2\): Divide every digit of the dividend by the divisor and write the answer on top of the digit being divided. Start by determining if the first digit of the dividend is large enough to be divided by the full divisor. In this example, it’s not (\(8\) is smaller than \(41\)), so divide the first two digits (\(85\)) by \(41\), which gives a result of \(2\). We write \(2\) on top of the symbol and align it with the \(5\):

\[\require{enclose} \begin{array}{r} 2\phantom{.28}\\[-3pt] 41 \enclose{longdiv}{8528} \\[-3pt] \end{array}\]

Step \(3\): Multiply the divisor (\(41\) in this example) by the number above the box (\(2\)), and write the product below the relevant portion of the dividend (\(85\)):

\[\require{enclose} \begin{array}{r} 2\phantom{.28}\\[-3pt] 41 \enclose{longdiv}{8528} \\[-3pt] \underline{82}\phantom{.28} \\[-3pt] \end{array}\]

Step \(4\): Subtract that product (\(82\)) from the first two digits of the dividend (\(85\)) and write down the difference (\(3\)). Then, bring down the next number from the dividend (\(2\) in this example):

\[\require{enclose} \begin{array}{r} 2\phantom{.28}\\[-3pt] 41 \enclose{longdiv}{8528} \\[-3pt] \underline{82}\phantom{.28} \\[-3pt] 32 \phantom{.8} \\[-3pt] \end{array}\]

Step \(5\): Divide the new dividend (\(32\)) by the divisor. In this case, \(41\) doesn’t go into \(32\), so we’ll write \(0\) and then repeat steps \(3\) and \(4\):

\[\require{enclose} \begin{array}{r} 20\phantom{.8}\\[-3pt] 41 \enclose{longdiv}{8528} \\[-3pt] \underline{82}\phantom{.28} \\[-3pt] 32 \phantom{.8} \\[-3pt] \underline{0} \phantom{.8} \\[-3pt] 328 \phantom{.} \\[-3pt] \end{array}\]

Step \(6\) Divide the new dividend (\(328\)) by the divisor (\(41\)), and write the result (\(8\)) in the quotient. Repeat steps \(3\) and \(4\):

\[\require{enclose} \begin{array}{r} 208\phantom{.}\\[-3pt] 41 \enclose{longdiv}{8528} \\[-3pt] \underline{82}\phantom{.28} \\[-3pt] 32 \phantom{.8} \\[-3pt] \underline{0} \phantom{.8} \\[-3pt] 328 \phantom{.} \\[-3pt] \underline{328} \phantom{.} \\[-3pt] 0 \phantom{.} \\[-3pt] \end{array}\]

Our final quotient is \(208\). If the dividend was a longer number, the procedure would simply be repeated until the last digit.

Remainders

The quotient won’t always be a whole number. When the result of subtraction in the last step of division is not \(0\), that leftover amount is known as a remainder. The remainder can be presented in three ways:

• as \(\text{R}\) followed by the remainder value

• as a fraction with the remainder as the numerator and the divisor as the denominator

• as digits after a decimal point when division is continued after the whole number part of the quotient (this involves adding zeros after the decimal point of the dividend)

For example, in the division below, we see a quotient of \(14\) with a remainder of \(2\):

\[\require{enclose} \begin{array}{r} 14\phantom{.}\\[-3pt] 3 \enclose{longdiv}{44} \\[-3pt] \underline{3}\phantom{.4} \\[-3pt] 14 \phantom{.} \\[-3pt] \underline{12} \phantom{.} \\[-3pt] 2 \phantom{.} \\[-3pt] \end{array}\]

We can write this result as \(14 \text{ R } 2\) or as \(14 \frac{2}{3}\). We could have also added zeros after the decimal point of the dividend and kept doing long division.

Note: If we had written our answer as a decimal number, we would have found that the number continued on forever. You’ll learn more about these types of non-terminating decimals later in this study guide.