Why are 2/3, 4/6, and 6/9 equivalent fractions?
Simplify 4/6 and 6/9. What do you get?
To determine if two fractions are equivalent, we need to check if they represent the same value. In this case, we will compare the fractions 2/3, 4/6, and 6/9 to see if they are equivalent.
Step 1: Simplify the fractions
The simplest form of a fraction is when the numerator and denominator have no common factors other than 1. Let's simplify the fractions one by one:
For 2/3:
2 and 3 have no common factors other than 1. Therefore, 2/3 is already in its simplest form.
For 4/6:
To simplify 4/6, we need to find the greatest common divisor (GCD) of 4 and 6, which is 2. We will divide both the numerator and the denominator by 2:
4 ÷ 2 = 2
6 ÷ 2 = 3
So, 4/6 simplifies to 2/3.
For 6/9:
Let's find the GCD of 6 and 9, which is 3. Dividing both the numerator and the denominator by 3:
6 ÷ 3 = 2
9 ÷ 3 = 3
Hence, 6/9 simplifies to 2/3.
Step 2: Comparing the simplified fractions
Now that we have simplified all three fractions, we can see that they have the same value, 2/3. Thus, 2/3, 4/6, and 6/9 are equivalent fractions.
In summary:
2/3, 4/6, and 6/9 are equivalent fractions because, when simplified, they all simplify to 2/3.
To understand why 2/3, 4/6, and 6/9 are equivalent fractions, we need to remember that equivalent fractions are fractions that represent the same amount or value. In other words, although they may be written differently, they represent the same proportion of a whole.
To determine if two fractions are equivalent, we need to simplify or reduce them to their simplest form. One way to do this is to find the greatest common divisor (GCD) of the numerator and denominator and divide both by this common factor.
Let's look at each fraction:
1. 2/3: The GCD of 2 and 3 is 1. Since there are no common factors other than 1, 2/3 is already in its simplest form.
2. 4/6: The GCD of 4 and 6 is 2. We can divide both the numerator and denominator by 2 to simplify it: 4 ÷ 2 = 2 and 6 ÷ 2 = 3. So, 4/6 simplifies to 2/3.
3. 6/9: The GCD of 6 and 9 is 3. Dividing both the numerator and denominator by 3 gives us: 6 ÷ 3 = 2 and 9 ÷ 3 = 3. Thus, 6/9 is equivalent to 2/3.
Therefore, 2/3, 4/6, and 6/9 are equivalent fractions because, after simplification, they all represent the same value of two-thirds (2 parts out of 3).