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7.4 Whole numbers: multiplication  (Page 2/3)

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Vertical multiplication. 1,804 times 9 is 16,236. The 3 is carried on top of the 0, and the 7 is carried on top of the 1.

9 × 4 = 36 Write the 6, carry the 3. 9 × 0 = 0 Add to the 0 the 3 that was carried: 0 + 3 = 3 . Write the 3. 9 × 8 = 72 Write the 2, carry the 7. 9 × 1 = 9 Add to the 9 the 7 that was carried:  9 + 7 = 16 . Since there are no more multiplications to perform,write both the 1 and 6.

The product is 16,236.

Practice set b

Find the following products.

37 ×   5 ̲

185

78 ×   8 ̲

624

536 ×     7 ̲

3,752

40,019 ×         8 ̲

320,152

301,599 ×           3 ̲

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.

Sample set c

Multiply 326 by 48.

  • Vertical multiplication. 326 times 48 is 2608. The 4 is carried on top of the 2, and the 2 is carried on top of the 3. The product is labeled, first partial product.

  • Vertical multiplication. 326 times 48, with the first part of the product, 2608, in the first line of the product space, and the second part of the product, 1304, in the second line of the product space. This number is naturally aligned with the tens column of the number above it. The second round of numbers are carried, with a 2 in the tens column and a 1 in the hundreds column. 1304 is labeled, second partial product.

  • This step is unnecessary since all of the digits in the multiplier have been used.
  • Add the partial products to obtain the total product.

    Vertical multiplication. 326 times 48, with the first part of the product, 2608, in the first line of the product space, and the second part of the product, 1304, in the second line of the product space. The two lines of the product space are added together to make a total product of 15648.

  • The product is 15,648.

Multiply 5,369 by 842.

  • Vertical multiplication. 5369 times 842, with the first part of the product, 10738, in the first line of the product space. A 1 is carried above the 6, and a 1 is carried above the 3. 10738 is labeled, first partial product.

  • Vertical multiplication. 5639 times 842, with the first part of the product, 10738, in the first line of the product space, and the second part of the product, 21476, in the second line of the product space. This number is aligned with the tens column of the number above it. A second round of numbers are carried, with a 3 in the tens column, a 2 in the hundreds column, and a 1 in the thousands column. 21476 is labeled, second partial product.

  • Vertical multiplication. 5639 times 842, with the first part of the product, 10738, in the first line of the product space, and the second part of the product, 21476, in the second line of the product space. This number is aligned with the tens column of the number above it. The third partial of the product, 42952, is below it, and is aligned with the hundreds column. A third round of numbers are carried, with a 7 in the tens column, a 5 in the hundreds column, and a 2 in the thousands column. Adding the partial products together makes a total product of 4520698, labeled Part 4.

  • The product is 4,520,698.

Multiply 1,508 by 206.

  • Vertical multiplication. 1508 times 206, with the first part of the product, 9048, in the first line of the product space. A 4 is carried in the tens column, and a 3 is carried in the thousands column. 9048 is labeled, first partial product.

  • Vertical multiplication. 1508 times 206, with the first part of the product, 9048, in the first line of the product space. A 4 is carried in the tens column, and a 3 is carried in the thousands column.
    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.

  • Vertical multiplication. 1508 times 206, with the first part of the product, 9048, in the first line of the product space, and the third part of the product, 3016, which is aligned in the hundreds column. A second round of numbers are carried, with a 1 in the tens column and a 1 in the thousands column. Adding the partial products together makes a total product of 310648, labeled Part 4.

  • The product is 310,648

Practice set c

Multiply 73 by 14.

1,022

Multiply 86 by 52.

4,472

Multiply 419 by 85.

35,615

Multiply 2,376 by 613.

1,456,488

Multiply 8,107 by 304.

2,464,528

Multiply 66,260 by 1,008.

66,790,080

Multiply 209 by 501.

104,709

Multiply 24 by 10.

240

Multiply 3,809 by 1,000.

3,809,000

Multiply 813 by 10,000.

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.

Sample set d

Perform the multiplication ( 49 , 000 ) ( 1, 200 ) size 12{ \( "49","000" \) \( 1,"200" \) } {} .

(49,000)(1,200) = 49000 ×    1200 ̲

Since 9 and 2 are the rightmost nonzero digits, put them in the same column.

49000 times 1200, with the 1200 aligned one space to the left.

Draw (perhaps mentally) a vertical line to separate the zeros from the nonzeros.

49000 times 1200, with the 1200 aligned one space to the left. A vertical line is drawn to separate the zeros in both numbers from the nonzero digits.

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.

49000 times 1200, with the 1200 aligned one space to the left. A vertical line is drawn to separate the zeros in both numbers from the nonzero digits. 98 is the first partial product, and 49 is the second partial product. The final product is 588, and the 5 zeros are then attached to the end of the product, making a total product of 58800000.

The product is 58,800,000

Practice set d

Multiply 1,800 by 90.

162,000

Multiply 420,000 by 300.

126,000,000

Multiply 20,500,000 by 140,000.

2,870,000,000,000

Calculators

Most multiplications are performed using a calculator.
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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