Distributive Property of Multiplication Over Addition

The distributive property of multiplication over addition states that, given three numbers \( a \), \( b \), and \( c \): $$ a \times (b + c) = (a \times b) + (a \times c) $$

In other words, multiplying a number by the sum of two other numbers is the same as multiplying that number by each of them separately and then adding the results.

Think of it as a “shortcut” to make calculations easier and to simplify algebraic expressions that might initially seem more complex than they are, by distributing the multiplication across each term in the addition.

Remember, the distributive property of multiplication also applies when using subtraction. $$ a \times (b - c) = (a \times b) - (a \times c) $$

Example

Imagine you’re at the market, and you need to buy some apples and pears.

Let’s say apples cost 2 euros each, and pears cost 3 euros each.

You want to buy 4 apples and 4 pears. To find out the total cost, you can multiply the price of the apples (2 euros times 4) and then the price of the pears (3 euros times 4) and add those results together.

$$ 2 \times 4 + 3 \times 4 $$

But there’s another way to think about this problem: you can combine the calculations.

First, add the price of one apple (2 euros) and one pear (3 euros), then multiply that sum by the 4 units you plan to buy.

$$ (2 + 3) \times 4 $$

Why are these two methods equivalent? Because the distributive property tells us that:

$$ a \times (b + c) = a \times b + a \times c $$

In this case, "a" is the number 4, while "b" and "c" are the prices of apples and pears.

The distributive property allows you to apply the multiplication to each term in the addition before summing them up.

If you’re still unsure, try calculating it step by step. If you compute \((2 + 3) \times 4\), you get 20: $$ 5 \times 4 = 20 $$. Now, calculate the terms separately, \((2 \times 4) + (3 \times 4)\): $$ 8 + 12 = 20 $$. As you can see, both methods lead to the same result: 20! This illustrates how the distributive property works, no matter which approach you take.

Example 2

Let’s look at another example involving subtraction. Suppose \( a = 3 \), \( b = 8 \), and \( c = 5 \).

$$ 3 \times (8 - 5) = 3 \times 3 = 9 $$

Now, we apply the distributive property:

$$ 3 \times (8 - 5) = (3 \times 8) - (3 \times 5) = 24 - 15 = 9 $$

Both results are the same, confirming that the distributive property applies to subtraction as well.

The Geometric Explanation

The distributive property of multiplication over addition can also be understood geometrically by looking at the area of a rectangle.

Imagine a rectangle with a total area of \( a \cdot (b + c) \).

This means the base of the rectangle is \( a \), and the height is \( b + c \), so the total area is calculated as \( a \cdot (b + c) \).

You can divide this rectangle into two smaller ones.

One has a height of \( b \) and an area of \( a \cdot b \), while the other has a height of \( c \) and an area of \( a \cdot c \).

The sum of the areas of these two smaller rectangles is \( a \cdot b + a \cdot c \).

Since the overall area of the rectangle hasn't changed, it’s clear that both expressions are equal.

$$ a \cdot (b + c) = a \cdot b + a \cdot c $$

This visual breakdown shows how the distributive property of multiplication works: the total area of the original rectangle, \( a \cdot (b + c) \), is the same as the sum of the areas of the two smaller rectangles, proving that \( a \cdot (b + c) = a \cdot b + a \cdot c \).

But why does this matter?

Using the distributive property can make calculations simpler, especially when dealing with larger numbers. It’s like having a shortcut that helps you avoid more complex steps, solving problems faster.

You can use it in either direction, depending on what makes the calculation easier.

Here’s a practical example:

Imagine you need to calculate this expression:

$$ 17 \times 3 + 23 \times 3 $$

You could find the total by performing two separate calculations: \( 17 \times 3 = 51 \quad \text{and} \quad 23 \times 3 = 69 \). Then, add the results:

$$ 51 + 69 = 120 $$

But you can simplify this by using the distributive property.

Since both numbers (17 and 23) are multiplied by the same number (3), you can add them first and then multiply:

$$ (17 + 23) \times 3 $$

$$ 40 \times 3 = 120 $$

This way, instead of doing two multiplications and then an addition, you perform one addition and just one multiplication.

The sum gives you a nice round number (40), making the multiplication by 3 much easier.

In practice, the distributive property often helps you calculate more efficiently and reduces the chance of errors, especially when the numbers are larger.