# Question Video: The Distributive Property of Multiplication

The diagram shows the number 𝑎 + 𝑏. By dividing it into four parts, as shown by the dashed lines, one gets (𝑎 + 𝑏) ÷ 4. Using the diagram, find another expression for (𝑎 + 𝑏) ÷ 4. Given that (𝑎 + 𝑏) × 𝑓 is a multiplication equivalent to this division, what is the multiplication factor 𝑓. What property of multiplication was used to rewrite (𝑎 + 𝑏) × (1/4) as (1/4) × (𝑎 + 𝑏)? What property of multiplication has been used to rewrite (1/4) (𝑎 + 𝑏) as (𝑎/4) + (𝑏/4)?

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### Video Transcript

The diagram shows the number 𝑎 plus 𝑏. By dividing it into four parts, as shown by the dashed lines, one gets 𝑎 plus 𝑏 divided by four. Using the diagram, find another expression for 𝑎 plus 𝑏 divided by four. Given that 𝑎 plus 𝑏 times 𝑓 is a multiplication equivalent to this division, what is the multiplication factor 𝑓? What property of multiplication was used to rewrite 𝑎 plus 𝑏 times one-fourth as one-fourth times 𝑎 plus 𝑏? What property of multiplication has been used to rewrite one-fourth times 𝑎 plus 𝑏 as 𝑎 over four plus 𝑏 over four?

Let’s begin with the diagram. It says the diagram shows the number 𝑎 plus 𝑏. So, again, this entire diagram represents 𝑎 plus 𝑏. And then, it says by dividing it into four parts, as shown by the dashed line here, here, and here, we get 𝑎 plus 𝑏, so the entire diagram divided by four because there are one, two, three, four parts now. So, 𝑎 plus 𝑏 divided by four.

So, the first question says, using the diagram, find another expression for 𝑎 plus 𝑏 divided by four. Well, instead of saying the entire diagram has been divided by four, we could say the 𝑎 portion was divided into four pieces. And the 𝑏 portion was divided into four pieces, and together makes the whole diagram. So, we have the 𝑎-divided-by-four portion plus the 𝑏-divided-by-four portion. So, here we’ve used the diagram to find another expression for 𝑎 plus 𝑏 divided by four.

Now we could double-check our answer. Because if we would expand this dividing by four, we would need to take 𝑎 divided by four and 𝑏 divided by four. So, we would still get 𝑎 divided by four plus 𝑏 divided by four.

Our next question states: given that 𝑎 plus 𝑏 times 𝑓 is a multiplication equivalent to this division, what is the multiplication factor 𝑓? So, it’s saying that 𝑎 plus 𝑏 times 𝑓 is a multiplication equivalent to this division. So, here we’ve written the division, the divided by four, and we wanna know an equivalent multiplication to this. So, writing 𝑎 plus 𝑏 times 𝑓, we have the 𝑎 plus 𝑏s and we wanna know what can we multiply by to make it the exact same as dividing by four. Well, dividing something by four is the same as multiplying by one-fourth. So, the multiplication equivalent to this division would be the multiplication factor 𝑓 of one-fourth.

Next, we’re asked what property of multiplication was used to rewrite 𝑎 plus 𝑏 times one-fourth as one-fourth times 𝑎 plus 𝑏. So, we chose to rewrite these. So, they’re equivalent to each other. And there’s a property that tells us that this is okay to do. Well, notice on both sides the 𝑎 and the 𝑏 are still being added together. And on both sides, we’re still multiplying the 𝑎 plus 𝑏 times a one-fourth. However, the order at which we’re writing these has switched. So, say we called 𝑎 plus 𝑏 𝑥 and one-fourth 𝑦. 𝑥 times 𝑦 is equal to 𝑦 times 𝑥. And this is true because of the property called the commutative property.

Lastly, it says what property of multiplication has been used to rewrite one-fourth times 𝑎 plus 𝑏 as 𝑎 over four plus 𝑏 over four. So, what exactly happened here? Taking one-fourth times 𝑎, we get 𝑎 over four, or 𝑎 divided by four. And taking one-fourth times 𝑏, we get 𝑏 over four, or 𝑏 divided by four. And then we keep the addition sign between. This is called distributing. So, we use the distributive property. And now we’ve completed this question.