Dividing Fractions - Fast And Easy Math Learning Videos
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If you prefer to read, rather than watch this video, here’s the transcript:
Title: Dividing Fractions
You know that the multiplication of both whole numbers and fractions can be represented using an array that looks like this. This example shows four-fifths multiplied by two-thirds.
The product of multiplying the fractions is eight-fifteenths.
You also know that division is just another way of writing multiplication when one of the factors is unknown. That’s easy to see using this array. Let’s assume that you know the total size of the array, which is eight-fifteenths, and you know one of the factors in the multiplication, four-fifths.
But you don’t know the other factor, which we’ll now call X.
X is the number you need to multiply by four-fifths to get eight-fifteenths.
We can represent this situation with two different number sentences.
The first number sentence is written as multiplication. We know that four-fifths is multiplied by some number on this side called X.
And the result is eight-fifteenths.
We can also describe exactly the same situation with a division number sentence like this.
We know that we have eight-fifteenths, which is divided by four-fifths.
But we don’t know the length of this side of the array, which is the value for X.
So, these equations tell us exactly the same thing because they represent the same situation shown by the array.
The X in both equations represents the same number – it’s the missing value for the length of one side of the array.
While this array is helpful in seeing what a division problem represents, it’s not very helpful in finding out the number we need on this side, because we don’t know how many parts to divide each whole unit into. So how do we find this number? The best way is to use what we know about algebraic reasoning, using these two equations to describe the array.
We’ll start by finding the value of X in the multiplication equation.
This will help us later in understanding how to solve the division equation.
You know that you can multiply both sides of an equation by the same number and it doesn’t change the equation. We can use this to our advantage if we choose the right number to multiply by. We want a number that will give us one X instead of four-fifths X on the left side of the equation.
You know that multiplying any number by its reciprocal gives you 1. Let’s multiply fourth-fifths by its reciprocal like this.
Five-fourths times four-fifths is one, so the left side of the equation now becomes 1 times X, which we write as just X. You also know that if you multiply one side of an equation by a number you have to multiply the other side by the same number.
So, we multiply the right side of the equation by five-fourths.
When we multiply, we get forty-sixtieths.
We can divide both terms by 20, to see that X equals two thirds.
When we compare this to the multiplication problem we started with, we see that two-thirds is the correct answer, because two-thirds times fourth-fifths is indeed eight-fifteenths.
But how does this help us understand how to divide fractions? It’s actually very simple. Let’s look at these two statements from the equations.
Remember that X represents the same number in both equations. It’s the length of this line in the aray.
The equation on the right says you can find X by dividing eight-fifteenths by four-fifths. The eqution on the left says you can find X by multiplying eight-fifteenths by the reciprocal of four-fifths.
Using what you already know about algebra, you can prove that this same approach always works, but we won’t go through that proof here. It means that dividing any number by a fraction and multiplying that number by its reciprocal gives you the same result.
We’ll just look at a couple of examples to verify that this works.
We’ll start with this multiplication statement. Three-fourths times five-eighths equals fifteen-thirty-seconds.
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