Question

The decimal expansion of \(\frac{147}{120}\) will terminate after how many places of the decimal ?

• A. 1
• B. 2
• C. 3
• D. Will not terminate

Hint:

To find out how many decimal places a terminating fraction has: 1. {Simplify the fraction} to its lowest terms. (e.g., \(\frac{147}{120} = \frac{49}{40}\)). 2. {Prime factorize the denominator} of the simplified fraction. (e.g., \(40 = 2^3 \times 5^1\)). 3. The fraction terminates if the only prime factors are 2s and 5s. 4. The number of decimal places it terminates after is the {highest power of 2 or 5} in this denominator. For \(2^3 \times 5^1\), the powers are 3 (for base 2) and 1 (for base 5). The highest power is 3. So, it terminates after 3 decimal places.

Answer 1

The Correct Option is C

Concept: A rational number (a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\)) has a terminating decimal expansion if and only if its denominator \(q\), when the fraction is in its simplest form, has prime factors only of 2 and/or 5. The number of decimal places after which it terminates is determined by the highest power of 2 or 5 in the prime factorization of the denominator of the simplified fraction. Step 1: Simplify the given fraction The given fraction is \(\frac{147}{120}\). First, find the greatest common divisor (GCD) of the numerator and the denominator. Prime factorization of 147: \(147 = 3 \times 49 = 3 \times 7^2\). Prime factorization of 120: \(120 = 10 \times 12 = (2 \times 5) \times (2^2 \times 3) = 2^3 \times 3 \times 5\). The GCD of 147 and 120 is 3. Divide both numerator and denominator by their GCD (3): \[ \frac{147 \div 3}{120 \div 3} = \frac{49}{40} \] So, the simplified fraction is \(\frac{49}{40}\). Step 2: Prime factorize the denominator of the simplified fraction The denominator of the simplified fraction is 40. Prime factorization of 40: \(40 = 8 \times 5 = 2^3 \times 5^1\). Step 3: Determine if the decimal expansion terminates Since the prime factors of the denominator (40) are only 2 and 5 (specifically \(2^3\) and \(5^1\)), the decimal expansion of \(\frac{49}{40}\) will terminate. Step 4: Determine the number of decimal places The decimal expansion will terminate after a number of places equal to the highest power of 2 or 5 in the prime factorization of the denominator. The powers are:
Power of 2 is 3 (from \(2^3\)).
Power of 5 is 1 (from \(5^1\)). The highest of these powers is 3. Therefore, the decimal expansion of \(\frac{49}{40}\) will terminate after 3 decimal places. To verify by actual division: \(\frac{49}{40} = \frac{49 \times 25}{40 \times 25} = \frac{1225}{1000} = 1.225\). The decimal 1.225 has 3 places after the decimal point. This matches option (3).