Let's analyze the problem and explain why Kate's method is incorrect and how to properly find a common denominator for the fractions [tex]\(\frac{3}{5}\)[/tex], [tex]\(\frac{3}{4}\)[/tex], and [tex]\(\frac{2}{3}\)[/tex].
### Misstep in Choosing 20 as the Common Denominator
Firstly, Kate chose 20 as the common denominator. To understand why this choice is incorrect, we need to find a suitable common denominator that works for all given fractions. A common denominator must be a common multiple of all the denominators in the fractions.
The fractions given are:
[tex]\[
\frac{3}{5}, \frac{3}{4}, \frac{2}{3}
\][/tex]
The denominators are 5, 4, and 3. To compare these fractions using a common denominator, we need the least common multiple (LCM) of these denominators.
### Finding the Least Common Multiple (LCM)
To find the LCM of 5, 4, and 3:
- The prime factorization of 5 is [tex]\(5\)[/tex].
- The prime factorization of 4 is [tex]\(2^2\)[/tex].
- The prime factorization of 3 is [tex]\(3\)[/tex].
To find the LCM, we take the highest power of each prime number that appears in these factorizations:
[tex]\[
\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60
\][/tex]
Thus, the least common multiple of 5, 4, and 3 is 60, not 20. Therefore, 60 should be used as the common denominator.
### Converting Fractions to the Common Denominator
Now, we need to convert each fraction to have the common denominator of 60.
1. Convert [tex]\(\frac{3}{5}\)[/tex]:
[tex]\[
\frac{3}{5} = \frac{3 \times 12}{5 \times 12} = \frac{36}{60}
\][/tex]
2. Convert [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\frac{3}{4} = \frac{3 \times 15}{4 \times 15} = \frac{45}{60}
\][/tex]
3. Convert [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}
\][/tex]
So, the fractions converted to have a common denominator of 60 are:
[tex]\[
\frac{36}{60}, \frac{45}{60}, \frac{40}{60}
\][/tex]
### Summary
Kate's method was incorrect because she used 20 as a common denominator, which is not a multiple of all the denominators involved. The correct common denominator should be 60, as it is the least common multiple of 5, 4, and 3. Using 60, the fractions [tex]\(\frac{3}{5}\)[/tex], [tex]\(\frac{3}{4}\)[/tex], and [tex]\(\frac{2}{3}\)[/tex] convert to [tex]\(\frac{36}{60}\)[/tex], [tex]\(\frac{45}{60}\)[/tex], and [tex]\(\frac{40}{60}\)[/tex] respectively.
