To determine which fraction is larger between [tex]\(\frac{2}{9}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex], let's follow step-by-step:
### Step 1: Identify the denominators
The denominators of the given fractions are 9 and 8.
### Step 2: Find the Least Common Multiple (LCM) of the denominators
To compare fractions or plot them on the same number line, we need to have a common denominator. To do this, we find the least common multiple (LCM) of 9 and 8.
The least common multiple of 9 and 8 is 72. This is the smallest number that both 9 and 8 can divide into without leaving a remainder.
### Step 3: Convert the fractions to have the common denominator
We will convert [tex]\(\frac{2}{9}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex] to equivalent fractions with the denominator of 72.
1. Convert [tex]\(\frac{2}{9}\)[/tex] into a fraction with the denominator 72:
[tex]\[
\frac{2 \times 8}{9 \times 8} = \frac{16}{72}
\][/tex]
2. Convert [tex]\(\frac{3}{8}\)[/tex] into a fraction with the denominator 72:
[tex]\[
\frac{3 \times 9}{8 \times 9} = \frac{27}{72}
\][/tex]
Now, we have the equivalent fractions:
[tex]\[
\frac{2}{9} = \frac{16}{72}
\][/tex]
[tex]\[
\frac{3}{8} = \frac{27}{72}
\][/tex]
### Step 4: Compare the equivalent fractions
Since both fractions now have the same denominator, we can compare their numerators directly:
[tex]\[
\frac{16}{72} \quad \text{and} \quad \frac{27}{72}
\][/tex]
Clearly, 16 is less than 27, therefore:
[tex]\[
\frac{2}{9} < \frac{3}{8}
\][/tex]
### Step 5: Determine the smallest number of pieces to partition the number line
To plot both fractions on the number line accurately, the number line should be partitioned into segments corresponding to the least common multiple of the denominators, which is 72. Thus, we need to partition the segment from 0 to 1 into 72 equal pieces.
### Conclusion
The smallest number of pieces you need to partition, or break, the segment from 0 to 1 into, to accurately plot both [tex]\(\frac{2}{9}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex] on the number line is [tex]\( 72 \)[/tex].
