Write down the decimal expansions of `13/6250`.

Options

A. 0.028
B. 0.00208
C. `0.005/2`
D. `0.004/6`

Answer

To find the decimal expansion of \( \frac{13}{6250} \), we can follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the denominator \( 6250 \).

\[ 6250 = 625 \times 10 \]
\[ 625 = 25 \times 25 = 5^4 \]
\[ 10 = 2 \times 5 \]

So, we can write:
\[ 6250 = 5^4 \times 2^1 \]

### Step 2: Rewrite the Fraction
Now, we can rewrite the fraction \( \frac{13}{6250} \) using the factored form of the denominator:

\[ \frac{13}{6250} = \frac{13}{5^4 \times 2^1} \]

### Step 3: Make the Denominator a Power of 10
To convert this fraction into decimal form, we want the denominator to be a power of 10. The highest power of 10 we can use is \( 10^5 \) (since \( 10 = 2 \times 5 \)), which means we need to balance the powers of 2 and 5.

We have:
- \( 5^4 \) in the denominator
- \( 2^1 \) in the denominator

To make the denominator \( 10^5 \), we can multiply both the numerator and the denominator by \( 2^4 \) (which is \( 16 \)):

\[ \frac{13 \times 16}{5^4 \times 2^1 \times 2^4} = \frac{208}{5^4 \times 2^5} \]

### Step 4: Simplify the Denominator
Now, the denominator becomes:

\[ 5^4 \times 2^5 = 10^5 \]

So we have:

\[ \frac{208}{10^5} \]

### Step 5: Convert to Decimal
Now, we can convert this fraction into decimal form:

\[ \frac{208}{10^5} = 0.00208 \]

### Final Answer
Thus, the decimal expansion of \( \frac{13}{6250} \) is:

\[ \boxed{0.00208} \]

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