Question 10

The decimal expansion of the rational number $\frac{\mathrm{14587/}}{1250}$ will terminate after

Solution:

To determine how many decimal places the decimal expansion of the rational number $$\frac{\mathrm{14587/}}{1250}$$ will terminate after, we follow these steps:

Step 1: Factor the denominator
First, we need to factor the denominator $$1250$$.

$$1250=125\times 10=5^3\times (2\times 5)=5^4\times 2^1$$

Step 2: Check the prime factors of the denominator
A rational number has a terminating decimal expansion if the prime factorization of its denominator (after simplification) contains only the prime factors 2 and/or 5.

From our factorization, we see that:

$$1250=2^1\times 5^4$$

Step 3: Determine the maximum power of 10
The maximum power of 10 that can be formed from the factors of 2 and 5 is determined by the smaller of the two powers:

– The power of 2 is 1.
– The power of 5 is 4.

The limiting factor here is $$2^1$$. Therefore, the maximum power of 10 that can be formed is $${10}^1$$.

Step 4: Calculate the number of decimal places
The number of decimal places in the terminating decimal expansion is equal to the maximum power of 10 that can be formed, which is 1.

Thus, the decimal expansion of $$\frac{\mathrm{14587/}}{1250}$$ will terminate after 1 decimal place.

Final Answer
The decimal expansion of $$\frac{\mathrm{14587/}}{1250}$$ will terminate after 1 decimal place.