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 express the denominator \( 6250 \) in terms of its prime factors.

\[ 6250 = 625 \times 10 = 625 \times (2 \times 5) \]
Next, we factor \( 625 \):
\[ 625 = 25 \times 25 = (5 \times 5) \times (5 \times 5) = 5^4 \]
Thus,
\[ 6250 = 5^4 \times 2^1 \]

### Step 2: Convert to a Power of 10
To convert \( \frac{13}{6250} \) into a decimal, we need the denominator to be a power of 10. We can achieve this by multiplying both the numerator and the denominator by \( 2^4 \) (which is \( 16 \)) to balance the powers of 2 and 5.

\[ \frac{13}{6250} \times \frac{16}{16} = \frac{13 \times 16}{6250 \times 16} \]

### Step 3: Calculate the New Numerator and Denominator
Calculating the new numerator:
\[ 13 \times 16 = 208 \]

Calculating the new denominator:
\[ 6250 \times 16 = 100000 \quad \text{(since \( 6250 = 625 \times 10 \) and \( 625 \times 16 = 10000 \))} \]

### Step 4: Write the Fraction
Now we have:
\[ \frac{208}{100000} \]

### Step 5: Convert to Decimal
To convert \( \frac{208}{100000} \) into decimal form, we can simply move the decimal point 5 places to the left (since the denominator is \( 10^5 \)):
\[ \frac{208}{100000} = 0.00208 \]

### Final Answer
The decimal expansion of \( \frac{13}{6250} \) is:
\[ \boxed{0.00208} \]

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