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A R I T H M E T I C

THE MEANING OF MULTIPLICATION

Mental Arithmetic

Lesson 8  Section 2

The distributive property

 3. What is the distributive property of multiplication? When multiplying a sum or a difference, we may multiply each term of the sum or difference, then add or subtract the partial products.

 2 × (100 + 5) = 2 × 100  +  2 × 5 = 200 + 10 = 210. 2 × (100 − 5) = 2 × 100  −  2 × 5 = 200 − 10 = 190.

In each case, we "distributed" 2 to both 100 and 5.

The distributive property is the most important property of multiplication.  It is a theorem that we can prove.  (That is the meaning of the word theorem:  it is a statement that we can prove.)

 4. How can we apply the distributive property to mental calculation? Expand the multiplicand mentally into its units. Distribute the multipier from left to right to each unit, then add the partial products.

 Example 1.    3 × 24 = 3 × 20  +  3 × 4 = 60  +  12 = 72.

We expanded 24 into 20 + 4, and then "distributed" 3 to each one.

Example 2.   Multiply  5 × 37  mentally.

Technique.  Expand 37 mentally into 30 + 7.  Then distribute 5:

 5 × (30 + 7) = 150 + 35 = 185.

Multiply the numbers as you read them, from left to right.  The last number you say is the answer.

Example 3.   Multiply mentally  8 × 46.

Say only,

"320 + 48 = 368."

Example 4.   800 × 460

Ignore the final 0's and multiply  8 × 46.  But we just saw that

8 × 46 = 368.

Therefore

800 × 460 = 368,000.

This is "368" with three 0's.

Example 5.   Multiply  6 × 7.30.  (Treat problems with decimal points as dollars and cents.)

Technique.  Expand 7.30 mentally into  7 + .30    Then

 6 × 7.30 = 42 + 1.80 = 43.80

Example 6.   What is the price of five items that cost \$3.25 each?

Answer.  Since  4 × \$.25 = \$1.00, then  5 × \$.25 = \$1.25.  Say,

"5 × 3.25 = 15 + 1.25 = 16.25"

Example 7.   Multiply  2 × 438  mentally.

 Solution. 2 × 438 = 2 × 400  +  2 × 30  +  2 × 8 = 800 + 60 + 16 = 876.

The point is to say each partial sum.  Say,

"860 + 16 is 876."

Example 8.   Multiply 4 × 709.

 Solution. 4 × 709 = 4 × 700  +  4 × 0  +  4 × 9 = 2800 + 0 + 36 = 2836.

Note:   Any number times 0, or 0 times any number, is 0.

Therefore, to calculate  4 × 709, simply ignore the 0 and say:

"2800 + 36 = 2836."

Example 9.   Multiply 8,000 × 4,310.

Technique.   Ignore the final 0's:

 8 × 431 = 3200 + 240 + 8 = 3440 + 8 = 3448.

Now replace the four 0's:

8,000 × 4,310 = 34,480,000

Example 10.   How much is 20% of \$68?

Solution.   10% of \$68 is \$6.80. (Lesson 3.)  Therefore, 20% is

2 × \$6.80 = \$12 + \$1.60 = \$13.60.

Example 11.   How many hours are there in one week?  How many minutes are there?

Solution.  There are 24 hours in one day, and there are 7 days in a week.  Therefore,

7 × 24 = 140 + 28 = 168 hours.

Now, in each hour there are 60 minutes. To multiply

60 × 168,

ignore the 0 and multiply

 6 × 168 = 600 + 360 + 48 = 960 + 48 = 1008.

Now replace the 0 that we ignored:

10080.

In one week, then, there are 10,080 minutes.

Two theorems

A theorem is a statement that can be proved.  One theorem is the distributive property of multiplication.  And from that follows the order property.

In algebra, distribution is an axiom. But in arithmetic we can define multiplication, which algebra cannot, and therefore we can prove it as a theorem.

Here is an example of the distributive property:

3 × (20 + 4) = 3 × 20  +  3 × 4.

Or, the other way around:

3 × 20  +  3 × 4 = 3 × (20 + 4).

When we look at it that way, it is the theorem of Adding the Same Multiple  (Euclid, V. 1):

Three 20's + Three 4's = Three 24's.

 If we add the same multiple of numbers,we will get that same multiple of the sum of those numbers.

Three 20's are the same multiple of 20 that Three 4's are of 4, namely the third; and Three 24's are also the third multiple of (20 + 4).

To see this, look:

 Three 20's + Three 4's = 20 + 20 + 20 + 4 + 4 + 4 = 20 + 4  +  20 + 4  +  20 + 4 = Three 24's.

From this theorem we can prove the order property of multiplication (Euclid, VII. 16):

 If two numbers multiply one another,the products will be equal to one another.

In other words, exchanging the multiplier and the multiplicand does not change the product.

We will show, for example, that

3 × 24 = 24 × 3.

That is, when we add 24 three times, we get the same number as when we add 3 twenty-four times:

24 + 24 + 24 = 3 + 3 + 3 + . . . + 3.

We have:

 Three 24's = 24  +  24  +  24 = Twenty-four 1's  +  Twenty-four 1's  +  Twenty-four 1's = Twenty-four (1 + 1+ 1)'s, on adding those same multiplesof 1, = Twenty-four 3's.

or

Continue on to Section 3:  Multiplying by rounding off

Section 1 of this Lesson

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