Decompose 8/9
To decompose 8/9 means to express it as a sum of (or difference between) simpler fractions. In this case, we need to find fractions with a common denominator that add up to 8/9.
One approach is to try dividing the numerator (8) by the denominator (9) to check if there is an exact division with no remainder:
8 ÷ 9 = 0 remainder 8.
Since there is a remainder, we can't find fractions with a denominator of 9 that add up to 8/9. However, we can still express 8/9 as a sum or difference of fractions with a different denominator.
Let's try different possible denominators:
1. Denominator of 2:
To get a denominator of 2, we can multiply both the numerator and denominator by 2/2:
8/9 = (8 × 2) / (9 × 2) = 16/18.
2. Denominator of 3:
Multiplying numerator and denominator by 3/3 gives:
8/9 = (8 × 3) / (9 × 3) = 24/27.
3. Denominator of 4:
Multiplying numerator and denominator by 4/4 results in:
8/9 = (8 × 4) / (9 × 4) = 32/36.
We can continue this process for other denominators until we find a fraction that fully simplifies. In this case, we observe that all the fractions obtained above can be simplified.
For example, let's simplify the fraction 16/18:
The greatest common divisor (GCD) of 16 and 18 is 2.
Dividing both the numerator and denominator by 2 gives us:
16 ÷ 2 / 18 ÷ 2 = 8/9.
Similarly, the fractions 24/27 and 32/36 can be simplified into 8/9 as well. Therefore, we can conclude that 8/9 can be decomposed as any of the following:
8/9 = 16/18 = 24/27 = 32/36 = ... (and so on).