2.1: Multiplication of Whole Numbers
- understand the process of multiplication
- be able to multiply whole numbers
- be able to simplify multiplications with numbers ending in zero
- be able to use a calculator to multiply one whole number by another
Multiplication
Multiplication is a description of repeated addition.
In the addition of
\(5 + 5 + 5\)
the number 5 is repeated 3 times. Therefore, we say we have three times five and describe it by writing
\(3 \times 5\)
Thus,
\(3 \times 5 = 5 + 5 + 5\)
In a multiplication, the repeated addend (number being added) is called the multiplicand . In \(3 \times 5\), the 5 is the multiplicand.
Also, in a multiplication, the number that records the number of times the multiplicand is used is called the multiplier . In \(3 \times 5\), the 3 is the multiplier.
Express each repeated addition as a multiplication. In each case, specify the multiplier and the multiplicand.
\(7 + 7 + 7 + 7 + 7 + 7\)
Solution
\(6 \times 7\), Multiplier is 6. Multiplicand is 7.
\(18 + 18 + 18\)
Solution
\(3 \times 18\). Multiplier is 3. Multiplicand is 18.
Practice Set A
Express each repeated addition as a multiplication. In each case, specify the multiplier and the multiplicand.
\(12 + 12 + 12 + 12\)
. Multiplier is . Multiplicand is .
- Answer
-
\(4 \times 12\). Multiplier is 4. Multiplicand is 12.
Practice Set A
\(36 + 36 + 36 + 36 + 36 + 36 + 36 + 36\)
. Multiplier is . Multiplicand is .
- Answer
-
\(8 \times 36\). Multiplier is 8. Multiplicand is 36.
Practice Set A
\(0 + 0 + 0 + 0 + 0\)
. Multiplier is . Multiplicand is .
- Answer
-
\(5 \times 0\). Multiplier is 5. Multiplicand is 0.
Practice Set A
\(\underbrace{1847 + 1847 + \cdots + 1847}_{12,000 \text{ times}}\)
s. Multiplier is . Multiplicand is .me
- Answer
-
\(12,000 \times 1,847\). Multiplier is 12,000. Multiplicand is 1,847.
In a multiplication, the numbers being multiplied are also called factors .
The result of a multiplication is called the product . In \(3 \times 5 = 15\), the 3 and 5 are not only called the multiplier and multiplicand, but they are also called factors. The product is 15.
The multiplication symbol (\(\times\)) is not the only symbol used to indicate multiplication. Other symbols include the dot (\(\cdot\)) and pairs of parentheses ( ). The expressions
\(3 \times 5, 3 \cdot 5, 3(5), (3)5, (3)(5)\)
all represent the same product.
The Multiplication Process With a Single Digit Multiplier
Since multiplication is repeated addition, we should not be surprised to notice that carrying can occur. Carrying occurs when we find the product of 38 and 7:
First, we compute \(7 \times 8 = 56\). Write the 6 in the ones column. Carry the 5. Then take \(7 \times 3 = 21\). Add to 21 the 5 that was carried: \(21 + 5 = 26\). The product is 266.
Find the following products.
Solution
\(\begin{array} {lcl} {3 \times 4 = 12} &\ \ \ \ & {\text{Write the 2, carry the 1.}} \\ {3 \times 6 = 18} &\ \ \ \ & {\text{Add to 18 the 1 that was carried: 18 + 1 = 19.}} \end{array}\)
The product is 192.
Solution
\(\begin{array} {lcl} {5 \times 6 = 30} & \ \ \ \ & {\text{Write the 0, carry the 3.}} \\ {5 \times 2 = 10} & \ \ \ \ & {\text{Add to 10 the 3 that was carried: 10 + 3 = 13. Write the 3, carry the 1.}} \\ {5 \times 5 = 25} & \ \ \ \ & {\text{Add to 25 the 1 that was carried: 25 + 1 = 6.}} \end{array}\)
The product is 2,630.
Solution
\(\begin{array} {lcl} {9 \times 4 = 36} & \ \ \ \ & {\text{Write the 6, carry the 3.}} \\ {9 \times 0 = 0} & \ \ \ \ & {\text{Add to 0 the 3 that was carried: 0 + 3 = 13. Write the 3.}} \\ {9 \times 8 = 72} & \ \ \ \ & {\text{Write the 2, carry the 7.}} \\ {} & \ \ \ \ & {\text{Since there are no more multiplications to perform, write both the 1 and 6.}} \end{array}\)
The product is 16,236.
Practice Set B
Find the following products.
\(\begin{array} {r} {37} \\ {\underline{\times \ \ 5}} \end{array}\)
- Answer
-
185
Practice Set B
Find the following products.
\(\begin{array} {r} {78} \\ {\underline{\times \ \ 8}} \end{array}\)
- Answer
-
624
Practice Set B
Find the following products.
\(\begin{array} {r} {537} \\ {\underline{\times \ \ \ \ 7}} \end{array}\)
- Answer
-
3,752
Practice Set B
Find the following products.
\(\begin{array} {r} {40,019} \\ {\underline{\times \ \ \ \ \ \ \ \ \ 8}} \end{array}\)
- Answer
-
320,152
Practice Set B
Find the following products.
\(\begin{array} {r} {301,599} \\ {\underline{\times \ \ \ \ \ \ \ \ \ \ \ 3}} \end{array}\)
- Answer
-
904,797
The Multiplication Process With a Multiple Digit Multiplier
In a multiplication in which the multiplier is composed of two or more digits, the multiplication must take place in parts . The process is as follows:
- First Partial Product Multiply the multiplicand by the ones digit of the multiplier. This product is called the first partial product .
- Second Partial Product Multiply the multiplicand by the tens digit of the multiplier. This product is called the second partial product . Since the tens digit is used as a factor, the second partial product is written below the first partial product so that its rightmost digit appears in the tens column.
- If necessary, continue this way finding partial products. Write each one below the previous one so that the rightmost digit appears in the column directly below the digit that was used as a factor.
- Total Product Add the partial products to obtain the total product .
It may be necessary to carry when finding each partial product.
Multiply 326 by 48.
Solution
Part 1:
Part 2:
Part 3: This step is unnecessary since all of the digits in the multiplier have been used.
Part 4: Add the partial products to obtain the total product.
The product is 15,648.
Multiply 5,369 by 842.
Solution
Part 1:
Part 2:
Part 3:
The product is 4,520,698.
Multiply 1,508 by 206.
Solution
Part 1:
Part 2:
Since 0 times 1508 is 0, the partial product will not change the identity of the total product (which is obtained by addition). Go to the next partial product.
Part 3:
The product is 310,648
Practice Set C
Multiply 73 by 14.
- Answer
-
1,022
Practice Set C
Multiply 86 by 52.
- Answer
-
4,472
Practice Set C
Multiply 419 by 85.
- Answer
-
35,615
Practice Set C
Multiply 2,376 by 613.
- Answer
-
1,456,488
Practice Set C
Multiply 8,107 by 304.
- Answer
-
2,464,528
Practice Set C
Multiply 66,260 by 1,008.
- Answer
-
66,790,080
Practice Set C
Multiply 209 by 501.
- Answer
-
104,709
Practice Set C
Multiply 24 by 10.
- Answer
-
240
Practice Set C
Multiply 3,809 by 1,000.
- Answer
-
3,809,000
Practice Set C
Multiply 813 by 10,000.
- Answer
-
8,130,000
Multiplications With Numbers Ending in Zero
Often, when performing a multiplication, one or both of the factors will end in zeros. Such multiplications can be done quickly by aligning the numbers so that the rightmost nonzero digits are in the same column.
Perform the multiplication (49,000)(1,200).
\((49,000)(1,200)\) = \(\begin{array} {r} {49000} \\ {\underline{\times \ \ 1200}} \end{array}\)
Since 9 and 2 are the rightmost nonzero digits, put them in the same column.
Draw (perhaps mentally) a vertical line to separate the zeros from the nonzeros.
Multiply the numbers to the left of the vertical line as usual, then attach to the right end of this product the total number of zeros.
The product is 58,800,000
Practice Set D
Multiply 1,800 by 90.
- Answer
-
162,000
Practice Set D
Multiply 420,000 by 300.
- Answer
-
126,000,000
Practice Set D
Multiply 20,500,000 by 140,000.
- Answer
-
2,870,000,000,000
Calculators
Most multiplications are performed using a calculator.
Multiply 75,891 by 263.
Solution
| Display Reads | ||
| Type | 75891 | 75891 |
| Press | × | 75891 |
| Type | 263 | 263 |
| Press | = | 19959333 |
The product is 19,959,333.
Multiply 4,510,000,000,000 by 1,700.
Solution
| Display Reads | ||
| Type | 451 | 451 |
| Press | × | 451 |
| Type | 17 | 17 |
| Press | = | 7667 |
The display now reads 7667. We'll have to add the zeros ourselves. There are a total of 12 zeros. Attaching 12 zeros to 7667, we get 7,667,000,000,000,000.
The product is 7,667,000,000,000,000.
Multiply 57,847,298 by 38,976.
Solution
| Display Reads | ||
| Type | 57847298 | 57847298 |
| Press | × | 57847298 |
| Type | 38976 | 38976 |
| Press | = | 2.2546563 12 |
The display now reads 2.2546563 12. What kind of number is this? This is an example of a whole number written in scientific notation . We'll study this concept when we get to decimal numbers.
Practice Set E
Use a calculator to perform each multiplication.
\(52 \times 27\)
- Answer
-
1,404
Practice Set E
\(1,448 \times 6,155\)
- Answer
-
8,912,440
Practice Set E
\(8,940,000 \times 205,000\)
- Answer
-
1,832,700,000,000
Exercises
For the following problems, perform the multiplications. You may check each product with a calculator.
Exercise \(\PageIndex{1}\)
\(\begin{array} {r} {8} \\ {\underline{\times 3}} \end{array}\)
- Answer
-
24
Exercise \(\PageIndex{2}\)
\(\begin{array} {r} {3} \\ {\underline{\times 5}} \end{array}\)
Exercise \(\PageIndex{3}\)
\(\begin{array} {r} {8} \\ {\underline{\times 6}} \end{array}\)
- Answer
-
48
Exercise \(\PageIndex{4}\)
\(\begin{array} {r} {5} \\ {\underline{\times 7}} \end{array}\)
Exercise \(\PageIndex{5}\)
\(6 \times 1\)
- Answer
-
6
Exercise \(\PageIndex{6}\)
\(4 \times 5\)
Exercise \(\PageIndex{7}\)
\(75 \times 3\)
- Answer
-
225
Exercise \(\PageIndex{8}\)
\(35 \times 5\)
Exercise \(\PageIndex{9}\)
\(\begin{array} {r} {45} \\ {\underline{\times \ \ 6}} \end{array}\)
- Answer
-
270
Exercise \(\PageIndex{10}\)
\(\begin{array} {r} {31} \\ {\underline{\times \ \ 7}} \end{array}\)
Exercise \(\PageIndex{11}\)
\(\begin{array} {r} {97} \\ {\underline{\times \ \ 6}} \end{array}\)
- Answer
-
582
Exercise \(\PageIndex{12}\)
\(\begin{array} {r} {75} \\ {\underline{\times 57}} \end{array}\)
Exercise \(\PageIndex{13}\)
\(\begin{array} {r} {64} \\ {\underline{\times 15}} \end{array}\)
- Answer
-
960
Exercise \(\PageIndex{14}\)
\(\begin{array} {r} {73} \\ {\underline{\times 15}} \end{array}\)
Exercise \(\PageIndex{15}\)
\(\begin{array} {r} {81} \\ {\underline{\times 95}} \end{array}\)
- Answer
-
7,695
Exercise \(\PageIndex{16}\)
\(\begin{array} {r} {31} \\ {\underline{\times 33}} \end{array}\)
Exercise \(\PageIndex{17}\)
\(57 \times 64\)
- Answer
-
3,648
Exercise \(\PageIndex{18}\)
\(76 \times 42\)
Exercise \(\PageIndex{19}\)
\(894 \times 52\)
- Answer
-
46,488
Exercise \(\PageIndex{20}\)
\(684 \times 38\)
Exercise \(\PageIndex{21}\)
\(\begin{array} {r} {115} \\ {\underline{\times \ \ 22}} \end{array}\)
- Answer
-
2,530
Exercise \(\PageIndex{22}\)
\(\begin{array} {r} {706} \\ {\underline{\times \ \ 81}} \end{array}\)
Exercise \(\PageIndex{23}\)
\(\begin{array} {r} {328} \\ {\underline{\times \ \ 21}} \end{array}\)
- Answer
-
6,888
Exercise \(\PageIndex{24}\)
\(\begin{array} {r} {550} \\ {\underline{\times \ \ 94}} \end{array}\)
Exercise \(\PageIndex{25}\)
\(930 \times 26\)
- Answer
-
24,180
Exercise \(\PageIndex{26}\)
\(318 \times 63\)
Exercise \(\PageIndex{27}\)
\(\begin{array} {r} {582} \\ {\underline{\times 127}} \end{array}\)
- Answer
-
73,914
Exercise \(\PageIndex{28}\)
\(\begin{array} {r} {247} \\ {\underline{\times 116}} \end{array}\)
Exercise \(\PageIndex{29}\)
\(\begin{array} {r} {305} \\ {\underline{\times 225}} \end{array}\)
- Answer
-
68,625
Exercise \(\PageIndex{30}\)
\(\begin{array} {r} {782} \\ {\underline{\times 547}} \end{array}\)
Exercise \(\PageIndex{31}\)
\(\begin{array} {r} {771} \\ {\underline{\times 663}} \end{array}\)
- Answer
-
511,173
Exercise \(\PageIndex{32}\)
\(\begin{array} {r} {638} \\ {\underline{\times 516}} \end{array}\)
Exercise \(\PageIndex{33}\)
\(1,905 \times 710\)
- Answer
-
1,352,550
Exercise \(\PageIndex{34}\)
\(5,757 \times 5,010\)
Exercise \(\PageIndex{35}\)
\(\begin{array} {r} {3,106} \\ {\underline{\times 1,752}} \end{array}\)
- Answer
-
5,441,712
Exercise \(\PageIndex{36}\)
\(\begin{array} {r} {9,300} \\ {\underline{\times 1,130}} \end{array}\)
Exercise \(\PageIndex{37}\)
\(\begin{array} {r} {7,057} \\ {\underline{\times 5,229}} \end{array}\)
- Answer
-
36,901,053
Exercise \(\PageIndex{38}\)
\(\begin{array} {r} {8,051} \\ {\underline{\times 5,580}} \end{array}\)
Exercise \(\PageIndex{39}\)
\(\begin{array} {r} {5,804} \\ {\underline{\times 4,300}} \end{array}\)
- Answer
-
24,957,200
Exercise \(\PageIndex{40}\)
\(\begin{array} {r} {357} \\ {\underline{\times \ \ 16}} \end{array}\)
Exercise \(\PageIndex{41}\)
\(\begin{array} {r} {724} \\ {\underline{\times \ \ \ \ 0}} \end{array}\)
- Answer
-
0
Exercise \(\PageIndex{42}\)
\(\begin{array} {r} {2,649} \\ {\underline{\times \ \ \ \ \ 41}} \end{array}\)
Exercise \(\PageIndex{43}\)
\(\begin{array} {r} {5,173} \\ {\underline{\times \ \ \ \ \ \ \ 8}} \end{array}\)
- Answer
-
41,384
Exercise \(\PageIndex{44}\)
\(\begin{array} {r} {1,999} \\ {\underline{\times \ \ \ \ \ \ \ 0}} \end{array}\)
Exercise \(\PageIndex{45}\)
\(\begin{array} {r} {1,666} \\ {\underline{\times \ \ \ \ \ \ \ 0}} \end{array}\)
- Answer
-
0
Exercise \(\PageIndex{46}\)
\(\begin{array} {r} {51,730} \\ {\underline{\times \ \ \ \ \ 142}} \end{array}\)
Exercise \(\PageIndex{47}\)
\(\begin{array} {r} {387} \\ {\underline{\times 190}} \end{array}\)
- Answer
-
73,530
Exercise \(\PageIndex{48}\)
\(\begin{array} {r} {3,400} \\ {\underline{\times \ \ \ \ \ 70}} \end{array}\)
Exercise \(\PageIndex{49}\)
\(\begin{array} {r} {460,000} \\ {\underline{\times \ \ 14,000}} \end{array}\)
- Answer
-
6,440,000,000
Exercise \(\PageIndex{50}\)
\(\begin{array} {r} {558,000,000} \\ {\underline{\times \ \ \ \ \ \ \ \ \ 81,000}} \end{array}\)
Exercise \(\PageIndex{51}\)
\(\begin{array} {r} {37,000} \\ {\underline{\times \ \ \ \ \ 120}} \end{array}\)
- Answer
-
4,440,000
Exercise \(\PageIndex{52}\)
\(\begin{array} {r} {498,000} \\ {\underline{\times \ \ \ \ \ \ \ \ \ \ \ 0}} \end{array}\)
Exercise \(\PageIndex{53}\)
\(\begin{array} {r} {4,585,000} \\ {\underline{\times \ \ \ \ \ \ \ \ \ \ 140}} \end{array}\)
- Answer
-
641,900,000
Exercise \(\PageIndex{54}\)
\(\begin{array} {r} {30,700,000} \\ {\underline{\times \ \ \ \ \ \ \ \ \ \ \ \ 180}} \end{array}\)
Exercise \(\PageIndex{55}\)
\(\begin{array} {r} {8,000} \\ {\underline{\times \ \ \ \ \ 10}} \end{array}\)
- Answer
-
80,000
Exercise \(\PageIndex{56}\)
Suppose a theater holds 426 people. If the theater charges $4 per ticket and sells every seat, how much money would they take in?
Exercise \(\PageIndex{57}\)
In an English class, a student is expected to read 12 novels during the semester and prepare a report on each one of them. If there are 32 students in the class, how many reports will be prepared?
- Answer
-
384 reports
Exercise \(\PageIndex{58}\)
In a mathematics class, a final exam consists of 65 problems. If this exam is given to 28 people, how many problems must the instructor grade?
Exercise \(\PageIndex{59}\)
A business law instructor gives a 45 problem exam to two of her classes. If each class has 37 people in it, how many problems will the instructor have to grade?
- Answer
-
3,330 problems
Exercise \(\PageIndex{60}\)
An algebra instructor gives an exam that consists of 43 problems to four of his classes. If the classes have 25, 28, 31, and 35 students in them, how many problems will the instructor have to grade?
Exercise \(\PageIndex{61}\)
In statistics, the term "standard deviation" refers to a number that is calculated from certain data. If the data indicate that one standard deviation is 38 units, how many units is three standard deviations?
- Answer
-
114 units
Exercise \(\PageIndex{62}\)
Soft drinks come in cases of 24 cans. If a supermarket sells 857 cases during one week, how many individual cans were sold?
Exercise \(\PageIndex{63}\)
There are 60 seconds in 1 minute and 60 minutes in 1 hour. How many seconds are there in 1 hour?
- Answer
-
3,600 seconds
Exercise \(\PageIndex{64}\)
There are 60 seconds in 1 minute, 60 minutes in one hour, 24 hours in one day, and 365 days in one year. How many seconds are there in 1 year?
Exercise \(\PageIndex{65}\)
Light travels 186,000 miles in one second. How many miles does light travel in one year? ( Hint: Can you use the result of the previous problem?)
- Answer
-
5,865,696,000,000 miles per year
Exercise \(\PageIndex{66}\)
An elementary school cafeteria sells 328 lunches every day. Each lunch costs $1. How much money does the cafeteria bring in in 2 weeks?
Exercise \(\PageIndex{67}\)
A computer company is selling stock for $23 a share. If 87 people each buy 55 shares, how much money would be brought in?
- Answer
-
$110,055
Exercises for Review
In the number 421,998, how may ten thousands are there?
Round 448,062,187 to the nearest hundred thousand.
- Answer
-
448,100,000
Find the sum. 22,451 + 18,976.
Subtract 2,289 from 3,001.
- Answer
-
712
Specify which property of addition justifies the fact that (a first whole number + a second whole number) = (the second whole number + the first whole number)