Question 19

Without actually performing the long divison, find if $\frac{\mathrm{987/}}{10500}$ will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer

Solution:

To determine whether the fraction $$\frac{\mathrm{987/}}{10500}$$ has a terminating or non-terminating (repeating) decimal expansion, we can follow these steps:

Step 1: Simplify the Fraction
First, we need to simplify the fraction $$\frac{\mathrm{987/}}{10500}$$. We can do this by finding the greatest common divisor (GCD) of the numerator and the denominator.

1. Find the GCD of 987 and 10500:
– The prime factorization of 987 is $$3\times 7\times 47$$.
– The prime factorization of 10500 can be calculated as follows:
– $$10500=105\times 100=(3\times 5\times 7)\times \left({10}^2\right)=(3\times 5\times 7)\times (2^2\times 5^2)$$.
– Therefore, $$10500=2^2\times 3^1\times 5^3\times 7^1$$.

The common factors between 987 and 10500 are $$3$$ and $$7$$. Thus, the GCD is $$21$$.

2. Divide both numerator and denominator by the GCD:
– $$\frac{987}{÷}=\frac{47}{500}$$.

Step 2: Factor the Denominator
Next, we need to analyze the denominator $$500$$ to determine its prime factors.

1. Find the prime factorization of 500:
– $$500=5^3\times 2^2$$.

Step 3: Check the Prime Factors
For a fraction to have a terminating decimal expansion, the denominator (in its simplest form) must only have the prime factors $$2$$ and/or $$5$$.

1. Analyze the factors of the denominator:
– The denominator $$500$$ has the prime factors $$2$$ and $$5$$ only.

Conclusion
Since the denominator $$500$$ only contains the prime factors $$2$$ and $$5$$, the fraction $$\frac{\mathrm{987/}}{10500}$$ will have a terminating decimal expansion.