The Correct Answer and Explanation is:
The correct options are 40:28 and 10:7.
To determine which ratios are equivalent to 30:21, we must find the options that represent the same proportional relationship. The most straightforward method is to simplify the original ratio to its lowest terms and then compare it with the simplified form of each option.
First, let’s simplify the given ratio, 30:21. To do this, we find the greatest common divisor (GCD), which is the largest number that can divide both 30 and 21 without leaving a remainder. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 21 are 1, 3, 7, and 21. The greatest common divisor for both numbers is 3. Now, we divide both parts of the ratio by 3.
30 ÷ 3 = 10
21 ÷ 3 = 7
So, the simplest form of the ratio 30:21 is 10:7. Any ratio that also simplifies to 10:7 will be equivalent.
Next, we will check each of the options provided:
- 6:3: The GCD of 6 and 3 is 3. When we simplify this ratio, we get:
6 ÷ 3 = 2
3 ÷ 3 = 1
The simplified ratio is 2:1. Since 2:1 is not the same as 10:7, this option is not equivalent. - 40:28: The GCD of 40 and 28 is 4. Simplifying this ratio gives us:
40 ÷ 4 = 10
28 ÷ 4 = 7
The simplified ratio is 10:7. This matches the simplified form of the original ratio, which means 40:28 is an equivalent ratio. - 10:7: This ratio is already in its simplest form because the only common factor of 10 and 7 is 1. It is identical to the simplified form of 30:21. Therefore, 10:7 is also an equivalent ratio.
Based on this analysis, the two ratios that are equivalent to 30:21 are 40:28 and 10:7.
