Question
A fraction \( \frac{a}{b} \) can be written in simplest terms (that is, as \( \frac{c}{d} \), where \( c \) and \( d \) have no common factor other than 1) by dividing both a and b by \( G C D(a, b) \). (a) Compute \( G C D(30,228) \). (b) Determine a fraction in simplest terms equivalent to \( \frac{30}{228} \). (a) \( G C D(30,228)=6 \) (b) \( \frac{30}{228}=\square \)
Ask by Boone Hanson. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \frac{30}{228} = \frac{5}{38} \)
Solution
**Step 1. Compute \( GCD(30,228) \):**
- We divide both numerator and denominator by their greatest common divisor.
- Given that \( GCD(30,228)=6 \).
**Step 2. Simplify the fraction \( \frac{30}{228} \):**
- Divide the numerator by 6:
\[
\frac{30}{6}=5
\]
- Divide the denominator by 6:
\[
\frac{228}{6}=38
\]
Thus, the fraction in simplest terms is:
\[
\frac{30}{228}=\frac{5}{38}
\]
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Simplify this solution Mind Expander
To compute the GCD of 30 and 228, we can use the Euclidean algorithm. Dividing 228 by 30 gives a quotient of 7 and a remainder of 18. Then, we apply the same process to 30 and 18: 30 divided by 18 gives a quotient of 1 and a remainder of 12. Next, we take 18 and 12: 18 divided by 12 gives a quotient of 1 and a remainder of 6. Finally, taking 12 and 6 gives us a quotient of 2 and a remainder of 0. Hence, the GCD is indeed 6! Now, simplifying the fraction \( \frac{30}{228} \): both the numerator and denominator can be divided by the GCD (6). Hence, \( \frac{30 \div 6}{228 \div 6} = \frac{5}{38} \). So, \( \frac{30}{228}=\frac{5}{38} \).
