Simplify the ratio 120 : 148 to its lowest Terms
lowest terms.
A) 13: 37
B) 30:37
C) 31:37
D) 33:37
The correct answer and explanation is:
To simplify the ratio 120:148 to its lowest terms, the first step is to find the greatest common divisor (GCD) of 120 and 148. The GCD is the largest number that divides both numbers without leaving a remainder.
Step 1: Find the GCD of 120 and 148
To find the GCD, we can use the prime factorization method or the Euclidean algorithm.
Prime Factorization of 120:
120 = 2 × 2 × 2 × 3 × 5 = 23×3×52^3 \times 3 \times 5
Prime Factorization of 148:
148 = 2 × 2 × 37 = 22×372^2 \times 37
The common factor between 120 and 148 is 222^2 (or 4), since both numbers have at least two 2s in their factorization.
So, the GCD of 120 and 148 is 4.
Step 2: Simplify the ratio
Now that we know the GCD is 4, we divide both terms of the ratio by 4. 1204=30and1484=37\frac{120}{4} = 30 \quad \text{and} \quad \frac{148}{4} = 37
Thus, the simplified ratio is 30:37.
Conclusion
The correct answer is:
B) 30:37
By simplifying the ratio in this way, we ensure that it is in its lowest terms, with no further common factors between the two numbers. This process of simplification is key when working with ratios, as it makes comparisons easier and more straightforward.