Select all the ratios that are equivalent to the ratio 12 : 3. Explain how you know.
A. 6:1
B. 1:4
C. 4:1
D. 24 : 6
E. 15:6
F. 1,200 : 300
G. 112: 13
The Correct Answer and Explanation is:
The correct equivalent ratios to 12:3 are A. 6:1, D. 24:6, and F. 1,200:300.
Explanation:
A ratio compares two quantities. To determine if two ratios are equivalent, we can simplify both ratios and check if they reduce to the same value.
- Starting with the ratio 12:3, let’s simplify it:
- The greatest common divisor (GCD) of 12 and 3 is 3.
- Dividing both parts of the ratio by 3, we get:
[
\frac{12}{3} : \frac{3}{3} = 4 : 1
]
So, the simplified form of 12:3 is 4:1.
Now, let’s check each option.
Option A: 6:1
- To check if this ratio is equivalent to 12:3, simplify it by dividing both parts by their GCD, which is 6:
[
\frac{6}{6} : \frac{1}{6} = 1 : 1
]
Thus the ratio is indeed simplified, and we can see that it is equivalent to 12:3. So, the answer is “yes.”
Option B: 1:4
- In this case, the ratio of 1:4 is different from 12:3, as simplifying 12:3 equals 4:1, not 1:4. Therefore, this ratio is not equivalent.
Option C: 4:1
- As explained earlier, 12:3 simplifies to 4:1. So, this ratio is equivalent to 12:3.
Option D: 24:6
- Let’s simplify 24:6 by dividing both parts by their GCD, which is 6:
[
\frac{24}{6} : \frac{6}{6} = 4:1
]
Thus, 24:6 simplifies to 4:1, which is equivalent to 12:3.
Option E: 15:6
- Simplifying 15:6 by dividing both parts by their GCD (3):
[
\frac{15}{3} : \frac{6}{3} = 5:2
]
This ratio simplifies to 5:2, not 4:1, so it is not equivalent to 12:3.
Option F: 1,200:300
- Simplifying 1,200:300 by dividing both parts by their GCD (300):
[
\frac{1200}{300} : \frac{300}{300} = 4:1
]
Thus, 1,200:300 simplifies to 4:1, which is equivalent to 12:3.
Option G: 112:13
- Simplifying 112:13 shows that there is no common divisor between 112 and 13, so the ratio does not simplify to 4:1. Therefore, this ratio is not equivalent to 12:3.
Final Answer:
The ratios that are equivalent to 12:3 are A. 6:1, D. 24:6, and F. 1,200:300.