By what number should `6 2/9` be multiplied to get 40?

Options

A. `7 3/7`
B. `9 3/7`
C. `6 3/7`
D. `5 3/7`

Answer

To solve the problem "By what number should \( 6 \frac{2}{9} \) be multiplied to get 40?", we can follow these steps:

### Step 1: Convert the mixed number to an improper fraction
The mixed number \( 6 \frac{2}{9} \) can be converted into an improper fraction.

\[ 6 \frac{2}{9} = \frac{6 \times 9 + 2}{9} = \frac{54 + 2}{9} = \frac{56}{9} \]

### Step 2: Set up the equation
Let the unknown number be \( x \). According to the problem, we need to find \( x \) such that:

\[ \frac{56}{9} \times x = 40 \]

### Step 3: Solve for \( x \)
To isolate \( x \), we can rearrange the equation:

\[ x = \frac{40}{\frac{56}{9}} = 40 \times \frac{9}{56} \]

### Step 4: Simplify the expression
Now, we can simplify \( 40 \times \frac{9}{56} \):

1. First, simplify \( \frac{40}{56} \):
- Both 40 and 56 can be divided by 4:
\[ \frac{40 \div 4}{56 \div 4} = \frac{10}{14} \]
- Now, simplify \( \frac{10}{14} \):
- Both 10 and 14 can be divided by 2:
\[ \frac{10 \div 2}{14 \div 2} = \frac{5}{7} \]

2. Now substitute back:
\[ x = \frac{5}{7} \times 9 = \frac{5 \times 9}{7} = \frac{45}{7} \]

### Step 5: Convert to mixed number
To convert \( \frac{45}{7} \) into a mixed number:

1. Divide 45 by 7:
- 7 goes into 45 six times (since \( 7 \times 6 = 42 \)).
- The remainder is \( 45 - 42 = 3 \).

Thus, we can write:
\[ \frac{45}{7} = 6 \frac{3}{7} \]

### Final Answer
Therefore, the number that \( 6 \frac{2}{9} \) should be multiplied by to get 40 is:

\[ \boxed{6 \frac{3}{7}} \]

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