leveled practice for 5-10, decompose each fraction or mixed number in two different ways. use a tool if needed.
5. $\frac{4}{6}=$ $\frac{4}{6}=$ 6. $\frac{7}{8}=$ $\frac{7}{8}=$
7. $1\frac{3}{5}=$ $1\frac{3}{5}=$ 8. $2\frac{1}{2}=$ $2\frac{1}{2}=$
9. $\frac{9}{12}=$ $\frac{9}{12}=$ 10. $1\frac{1}{3}=$ $1\frac{1}{3}=$
topic 9 | lesson 9-2

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Accepted Answer

### Question 5: Decompose $\boldsymbol{\frac{4}{6}}$
# Explanation:
## Step 1: Decompose into unit fractions
A unit fraction has a numerator of 1. So, we can write $\frac{4}{6}$ as the sum of 4 unit fractions with denominator 6.
$\frac{4}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}$
## Step 2: Decompose into two fractions
We can split the numerator 4 into two parts, say 1 and 3. Then,
$\frac{4}{6} = \frac{1}{6} + \frac{3}{6}$ (and $\frac{3}{6}$ can be simplified to $\frac{1}{2}$, but we can leave it as $\frac{3}{6}$ for decomposition)
# Answer:
$\frac{4}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}$; $\frac{4}{6} = \frac{1}{6} + \frac{3}{6}$

### Question 6: Decompose $\boldsymbol{\frac{7}{8}}$
# Explanation:
## Step 1: Decompose into unit fractions
Using unit fractions (numerator 1), we sum 7 unit fractions with denominator 8.
$\frac{7}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}$
## Step 2: Decompose into two fractions
Split the numerator 7 into, for example, 3 and 4. Then,
$\frac{7}{8} = \frac{3}{8} + \frac{4}{8}$
# Answer:
$\frac{7}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}$; $\frac{7}{8} = \frac{3}{8} + \frac{4}{8}$

### Question 7: Decompose $\boldsymbol{1\frac{3}{5}}$
# Explanation:
## Step 1: Decompose into a whole number and a fraction
A mixed number $1\frac{3}{5}$ can be written as the sum of the whole number part and the fractional part: $1 + \frac{3}{5}$
## Step 2: Decompose the fractional part further (or decompose into improper fraction and then split)
First, convert the mixed number to an improper fraction: $1\frac{3}{5} = \frac{5 + 3}{5} = \frac{8}{5}$. Now, split $\frac{8}{5}$ into two fractions, e.g., $\frac{5}{5} + \frac{3}{5}$, and $\frac{5}{5} = 1$, so $1\frac{3}{5} = 1 + \frac{3}{5}$ (wait, no, another way: split $\frac{8}{5}$ into $\frac{1}{5} + \frac{7}{5}$, but maybe better to split the improper fraction into a whole number and a fraction or two fractions. Wait, alternatively, $1\frac{3}{5} = \frac{5}{5} + \frac{3}{5}$ (since $\frac{5}{5}=1$), and also, we can write $\frac{8}{5} = \frac{2}{5} + \frac{6}{5}$, but maybe simpler: $1\frac{3}{5} = 1 + \frac{1}{5} + \frac{2}{5}$ (decomposing the fractional part into two unit fractions or other fractions). Wait, let's correct:
First way: $1\frac{3}{5} = 1 + \frac{3}{5}$
Second way: Convert to improper fraction $\frac{8}{5}$, then split numerator 8 into 5 and 3: $\frac{8}{5} = \frac{5}{5} + \frac{3}{5} = 1 + \frac{3}{5}$ (no, that's same as first way). Wait, split 8 into 1 and 7: $\frac{8}{5} = \frac{1}{5} + \frac{7}{5}$, and $\frac{7}{5} = 1\frac{2}{5}$, so $1\frac{3}{5} = \frac{1}{5} + 1\frac{2}{5}$. But maybe better: $1\frac{3}{5} = \frac{5}{5} + \frac{3}{5}$ (which is $1 + \frac{3}{5}$) and $1\frac{3}{5} = \frac{1}{5} + \frac{2}{5} + \frac{5}{5}$ (but that's three parts). Wait, the problem says "two different ways". So first way: $1 + \frac{3}{5}$. Second way: Decompose the fractional part $\frac{3}{5}$ into two fractions, e.g., $\frac{1}{5} + \frac{2}{5}$, so $1\frac{3}{5} = 1 + \frac{1}{5} + \frac{2}{5}$. But maybe the problem allows decomposing into a whole number and a sum of fractions, or into two fractions (including improper). Let's do:
First way: $1\frac{3}{5} = 1 + \frac{3}{5}$
Second way: $1\frac{3}{5} = \frac{5}{5} + \frac{3}{5}$ (same as first, since $\frac{5}{5}=1$). Wait, maybe I made a mistake. Let's try again. The mixed number $1\frac{3}{5}$ can be decomposed as:
1. $1 + \frac{3}{5}$
2. $\frac{5}{5} + \frac{3}{5}$ (which is the same as above, but $\frac{5}{5}=1$). Alternatively, convert to improper fraction $\frac{8}{5}$, then split into $\frac{4}{5} + \frac{4}{5}$ (since $\frac{4}{5} + \frac{4}{5} = \frac{8}{5}$). Yes, that's a good way: $\frac{8}{5} = \frac{4}{5} + \frac{4}{5}$, so $1\frac{3}{5} = \frac{4}{5} + \frac{4}{5}$ (but wait, $\frac{4}{5} + \frac{4}{5} = \frac{8}{5} = 1\frac{3}{5}$, correct). So:
First way: $1 + \frac{3}{5}$
Second way: $\frac{4}{5} + \frac{4}{5}$
# Answer:
$1\frac{3}{5} = 1 + \frac{3}{5}$; $1\frac{3}{5} = \frac{4}{5} + \frac{4}{5}$ (or other valid decompositions like $1 + \frac{1}{5} + \frac{2}{5}$)

### Question 8: Decompose $\boldsymbol{2\frac{1}{2}}$
# Explanation:
## Step 1: Decompose into whole numbers and a fraction
The mixed number $2\frac{1}{2}$ can be written as the sum of the whole number part and the fractional part: $2 + \frac{1}{2}$
## Step 2: Decompose into two mixed numbers or two fractions (or decompose the improper fraction)
First, convert $2\frac{1}{2}$ to an improper fraction: $\frac{5}{2}$. Now, split $\frac{5}{2}$ into two fractions, e.g., $\frac{2}{2} + \frac{3}{2}$, where $\frac{2}{2} = 1$ and $\frac{3}{2} = 1\frac{1}{2}$. So $2\frac{1}{2} = 1 + 1\frac{1}{2}$
# Answer:
$2\frac{1}{2} = 2 + \frac{1}{2}$; $2\frac{1}{2} = 1 + 1\frac{1}{2}$ (or $\frac{5}{2} = \frac{1}{2} + \frac{4}{2}$ and $\frac{4}{2}=2$, so $2\frac{1}{2} = 2 + \frac{1}{2}$ and $2\frac{1}{2} = \frac{1}{2} + 2$ (same as first), or $1\frac{1}{2} + 1$ as above)

### Question 9: Decompose $\boldsymbol{\frac{9}{12}}$
# Explanation:
## Step 1: Decompose into unit fractions
A unit fraction has numerator 1. So, $\frac{9}{12}$ can be written as the sum of 9 unit fractions with denominator 12: $\frac{1}{12} + \frac{1}{12} + \dots + \frac{1}{12}$ (9 times)
## Step 2: Decompose into two fractions (simplify or split numerator)
First, simplify $\frac{9}{12}$ to $\frac{3}{4}$, but for decomposition, split the numerator 9 into two parts, e.g., 3 and 6. So $\frac{9}{12} = \frac{3}{12} + \frac{6}{12}$, and $\frac{3}{12} = \frac{1}{4}$, $\frac{6}{12} = \frac{1}{2}$, so $\frac{9}{12} = \frac{1}{4} + \frac{1}{2}$ (after simplifying the decomposed fractions)
# Answer:
$\frac{9}{12} = \frac{1}{12} + \frac{1}{12} + \dots + \frac{1}{12}$ (9 times); $\frac{9}{12} = \frac{3}{12} + \frac{6}{12}$ (or $\frac{1}{4} + \frac{1}{2}$ after simplifying)

### Question 10: Decompose $\boldsymbol{1\frac{1}{3}}$
# Explanation:
## Step 1: Decompose into a whole number and a fraction
The mixed number $1\frac{1}{3}$ can be written as the sum of the whole number part and the fractional part: $1 + \frac{1}{3}$
## Step 2: Decompose the improper fraction (or split the fractional part)
First, convert $1\frac{1}{3}$ to an improper fraction: $\frac{4}{3}$. Now, split $\frac{4}{3}$ into two fractions, e.g., $\frac{3}{3} + \frac{1}{3}$, where $\frac{3}{3} = 1$, so $1\frac{1}{3} = 1 + \frac{1}{3}$ (wait, no, that's same as first way). Alternatively, split $\frac{4}{3}$ into $\frac{1}{3} + \frac{3}{3}$, and $\frac{3}{3} = 1$, so again $1 + \frac{1}{3}$. Wait, another way: split $\frac{4}{3}$ into $\frac{2}{3} + \frac{2}{3}$, so $1\frac{1}{3} = \frac{2}{3} + \frac{2}{3}$ (since $\frac{4}{3} = \frac{2}{3} + \frac{2}{3}$)
# Answer:
$1\frac{1}{3} = 1 + \frac{1}{3}$; $1\frac{1}{3} = \frac{2}{3} + \frac{2}{3}$ (or other valid decompositions like $1 + \frac{1}{3} = 1 + \frac{1}{3}$, and $1\frac{1}{3} = \frac{1}{3} + 1$ (same as first), or $\frac{1}{3} + \frac{3}{3}$ (which is $\frac{1}{3} + 1$))

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Explanation:

Question 5: Decompose $\boldsymbol{\frac{4}{6}}$

Step 1: Decompose into unit fractions

Step 2: Decompose into two fractions

Step 1: Decompose into unit fractions

Step 2: Decompose into two fractions

Step 1: Decompose into a whole number and a fraction

Step 2: Decompose the fractional part further (or decompose into improper fraction and then split)

Answer:

Question 6: Decompose $\boldsymbol{\frac{7}{8}}$