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Title: Equivalent Fractions: Expanding Fractions



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Title: Equivalent Fractions: Least Common Multiple (LCM)

You already know one way to find a common denominator for any two fractions and one way to find a common denominator for some fractions. In this and the next lesson, you’ll learn another way to find a common denominator for any two fractions. This method uses what is called the least common multiple. In this lesson, you’ll learn to find the least common multiple for two numbers. In the next lesson, you’ll learn to use the least common multiple as the common denominator for two fractions.
Consider these two numbers, 6 and 4. We want to find the smallest number that’s a multiple of each of these numbers. Let’s look at these numbers on a number line.
Let’s start with the multiples of 6. Six times one is 6.
Six times 2 is 12.
Six times 3 is 18.
Six times 4 is 24.
Now let’s look at the multiples of 4. Four times one is 4.
Four times 2 is 8.
Four times 3 is 12.
Four times 4 is 16.
Four times 5 is 20.
Four times 6 is 24.
Notice that there are two places where the multiples of 4 and 6 are the same. The first one is here, at 12.
Twelve is a multiple of both 6 and 4.
The second place where multiples of 4 and 6 are the same is here, at 24.
For our purposes, what we want is the smallest number that’s a multiple of both 6 and 4. That number is 12.
We call this the “least common multiple” of 6 and 4.
We can also find the least common multiple of two numbers without using a number line. Here’s an example, 5 and 4. First, we list several multiples of 5 in order:
Now we list several multiples of 4 in order.
We can see that there’s a common multiple, which is 20.
Since this is the smallest number that’s a multiple of both 5 and 4, this is the least common multiple.
Another example, 6 and 9. First, we list several multiples of 6
Next, we list several multiples of 9.
The smallest number that’s a multiple of both 6 and 9 is 18.
Last example, 2 and 9. First, we list multiples of 2.
Then we list multiples of 9.
The smallest number that’s a multiple of both 2 and 9 is 18.
In the remainder of this lesson, you’ll find the least common multiple of two numbers like these.

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Title: Equivalent Fractions: Rewriting With Smaller Denominator - Fast And Easy Math Learning Videos

You already know that when you find an equivalent fraction with a larger denominator it’s called expanding the fraction. In this lesson you’re going to learn how to find an equivalent fraction with a smaller denominator.
This is called reducing the fraction.
For example, suppose we want to find a fraction that’s equivalent to six-eighths but has a denominator of 4. We can do this using the number line.
First we find six-eighths on the number line.
There are eight parts in each whole unit, and six parts are counted. So this fraction represents six eighths on the number line. Since we want our equivalent fraction to have a denominator of 4, we want each whole unit to have 4 parts instead of 8. We’ll show this on another number line, like this.
Each unit is divided into four parts.
When we compare the number line we see that each fourth here is the same size as two eighths here.
So each fourth of a unit represents 2 eighths of a unit. Now let’s look at the parts that are counted in the fraction six eighths.
When we show the same fraction on the line for fourths, we have only 3 total parts representing the fraction.
Since there are half as many parts in each whole, there are half as many parts counted. In other words, we have divided the numerator and denominator of our original fraction by the same number. In this case, we divided by 2.
This gives us four in the denominator and 3 in the numerator. Since the point on the number line is the same after dividing by 2 the fractions are equivalent. So, six-eights can be reduced and rewritten as the equivalent fraction three-fourths.
Now let’s look at an example without using the number line. In this case, we want to reduce the fraction nine-fifteenths by writing it with a denominator of 5.
First, we want to determine which number we need to divide the original denominator, 15, by to get the new denominator, 5
We know that 15 divided by 3 is 5.
So, we also divide the numerator by 3 to get an equivalent fraction.
9 divided by 3 equals 3.
So the fraction nine-fifteenths can be rewritten as the equivalent fraction - three-fifths.
In this case, we say that the fraction nine-fifteenths is reduced to three fifths. The number represented by the fraction is still the same, but the numerator and denominator are both smaller numbers.
One last example. Here we want to reduce three-twelfths to an equivalent fraction with a denominator of 4.
First, we find the number we need to divide 12 by to get 4.
We know that 12 divided by 3 is 4, so we divide the numerator by 3 also.
Three divided by 3 equals 1, so the numerator of the reduced fraction is 1.
The fraction three-twelfths is reduced to the equivalent fraction one-fourth.
In the remainder of this lesson you’ll practice reducing fractions like these.

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Title: Improper Fractions and Mixed Numbers - Fast And Easy Math Learning Videos

In this lesson, you will learn about proper fractions, improper fractions, and mixed numbers. Before we begin this lesson, though, we will review fractions that are less than one, equal to one, and more than one. In a previous lesson, you learned that a fraction is less than one when the numerator is less than the denominator.
One-half is less than one.
You also learned that a fraction is equal to one when the numerator and denominator are the same.
Two-halves is equal to one.
And you learned that a fraction is more than one when the numerator is more than the denominator.
Three-halves is more than one.
Here's a new word: proper fraction.
A proper fraction is a fraction less than one. One-half is a proper fraction.
Two-halves is not a proper fraction because two-halves is not less than one
Three-halves is not a proper fraction because three halves is not less than one.
Here's a new word: Improper fraction.
An improper fraction is a fraction equal to one, or more than one.
Two-halves is an improper fraction and three-halves is an improper fraction.
Let's look at a few examples of proper and improper fractions: A proper fraction is a fraction less than one. Five-sixths is a proper fraction.
An improper fraction is a fraction equal to one, or more than one.
Six-sixths is an improper fraction and seven-sixths is an improper fraction.
Here’s an example. In this example, the correct answer is proper fraction. Three-fourths is less than one, so three-fourths is a proper fraction.
So far in this lesson we've covered proper fractions and improper fractions. Now, we will learn about mixed numbers. A mixed number has both a whole number and a fraction.
A mixed number has both a whole number and a fraction.
Now try a few on your own. For the problems that follow, click on the correct answer. In this example, the correct answer is mixed number. Three and one-half has both a whole number and a fraction, so three and one-half is a mixed number.

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Title: Mixed Numbers as Improper Fractions - Fast And Easy Math Learning Videos

To change a mixed number into an improper fraction, first, multiply the whole units by the number of parts in each whole unit.
Then add the left over part
Seven and one-third is equal to twenty-two thirds
Here's another one. First, multiply the whole units by the number of parts in each whole unit.
Then add the left over part
Three and two-fifths is equal to seventeen-fifths.
Last one. First, multiply the whole units by the number of parts in each whole unit.
Then add the left over part
One and three-sixths is equal to nine-sixths.

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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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Title: Fractions as Division - Fast And Easy Math Learning Videos

You already know that you can write the remainder in a division problem as a fraction. In this lesson you’ll learn that all division problems can be written as fractions. First let’s review writing remainders as fractions:
Thirteen divided by 3.
Remember: to solve this problem we make an array.
The array we made is 3 by 4, and there’s 1 left over.
Now we break each of our remaining blocks into 3 parts.
And we continue making the array with the parts.
So, now the array we’ve made is 3, by 4 and one-third.
So, 13 divided by 3 is equal to 4 and one-third.
Instead of just splitting the remainder blocks into thirds, we could have split all of the blocks into thirds before forming the array, like this.
Now we’ll make an array. Remember, since we split the blocks into thirds, each whole block will end up being 1 column in the array.
So the array we formed is 3 by thirteen-thirds.
So 13 divided by 3 is equal to thirteen-thirds.
We can do the same thing for all division problems. Any division problem can be rewritten as a fraction with the numerator as the dividend and the denominator as the divisor.
Let’s look at another example.
Three divided by 4.
Since 3 is less than 4 we cannot make an array without splitting each of the blocks into fourths, like this.
Now we can make an array.
So the array we formed is 4 by three-fourths
So 3 divided by 4 is equal to three-fourths.
Let’s look at another example: 56 divided by 9. We can rewrite this as a fraction with 56 as the numerator, and 9 as the denominator.
Let’s look at one last example
Four-sixths. Just as we can rewrite division problems as fractions, we can also rewrite fractions as division problems. So four-sixths is equal to 4 divided by 6.
In the remainder of this lesson you’ll rewrite division problems as fractions and fractions as division problems.

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