Sure, let's break this down step-by-step.
### Part 1: Understanding the Problem
You're given fractions and you need to determine equivalence and convert them into like fractions. Then, you are to find three equivalent fractions for each of the given fractions.
### Part 2: Definitions and Clarifications
1. Like Fractions: Fractions that have the same denominator.
2. Equivalent Fractions: Different fractions that represent the same part of a whole.
### Part 3: Converting Given Fractions into Like Fractions
Let's list the original fractions:
- \(\frac{2}{5}\)
- \(\frac{3}{4}\)
- \(\frac{4}{7}\)
- \(2 \frac{8}{9} \rightarrow \frac{2 \cdot 9 + 8}{9} = \frac{26}{9}\)
To convert these fractions into like fractions, we need to find a common denominator. The denominators are 5, 4, 7, and 9. The least common multiple (LCM) of these numbers is 1260.
Now, convert each fraction:
1. \(\frac{2}{5}\):
[tex]\[
\frac{2}{5} \times \frac{252}{252} = \frac{504}{1260}
\][/tex]
2. \(\frac{3}{4}\):
[tex]\[
\frac{3}{4} \times \frac{315}{315} = \frac{945}{1260}
\][/tex]
3. \(\frac{4}{7}\):
[tex]\[
\frac{4}{7} \times \frac{180}{180} = \frac{720}{1260}
\][/tex]
4. \(\frac{26}{9}\):
[tex]\[
\frac{26}{9} \times \frac{140}{140} = \frac{3640}{1260}
\][/tex]
### Part 4: Finding Three Equivalent Fractions for Each
Now, let's find three equivalent fractions for each of the given fractions using smaller multipliers:
1. For \(\frac{2}{5}\):
[tex]\[
\frac{2}{5} \times 1 = 0.4
\][/tex]
[tex]\[
\frac{2}{5} \times 2 = 0.8
\][/tex]
[tex]\[
\frac{2}{5} \times 3 = 1.2
\][/tex]
2. For \(\frac{3}{4}\):
[tex]\[
\frac{3}{4} \times 1 = 0.75
\][/tex]
[tex]\[
\frac{3}{4} \times 2 = 1.5
\][/tex]
[tex]\[
\frac{3}{4} \times 3 = 2.25
\][/tex]
3. For \(\frac{4}{7}\):
[tex]\[
\frac{4}{7} \times 1 = 0.5714
\][/tex]
[tex]\[
\frac{4}{7} \times 2 = 1.1429
\][/tex]
[tex]\[
\frac{4}{7} \times 3 = 1.7143
\][/tex]
4. For \(\frac{26}{9}\):
[tex]\[
\frac{26}{9} \times 1 = 2.8889
\][/tex]
[tex]\[
\frac{26}{9} \times 2 = 5.7778
\][/tex]
[tex]\[
\frac{26}{9} \times 3 = 8.6667
\][/tex]
### Summary
The equivalent fractions were found to be:
1. For \(\frac{2}{5}\):
- 0.4, 0.8, 1.2
2. For \(\frac{3}{4}\):
- 0.75, 1.5, 2.25
3. For \(\frac{4}{7}\):
- 0.5714, 1.1429, 1.7143
4. For \(\(\frac{26}{9}\)\):
- 2.8889, 5.7778, 8.6667
These are their step-by-step equivalent fractions, and the process involved finding the least common denominators and calculating equivalent fractions by multiplication.
