The problem asks us to select all ratios that are equivalent to the ratio 12:15. The given options are 84:105, 20:25, and 2:5.

1. Problem Description

The problem asks us to select all ratios that are equivalent to the ratio 12:

1

5. The given options are 84:105, 20:25, and 2:

5.

2. Solution Steps

First, we simplify the given ratio 12:15 by finding the greatest common divisor (GCD) of 12 and

1

5. The GCD of 12 and 15 is

3. Dividing both parts of the ratio by 3, we get:

$12/3 : 15/3 = 4:5$.

So, the simplified ratio is 4:

5.

Now we check the given options:

a) 84:

1

0

5. We want to see if $84/105$ simplifies to $4/5$. We can find the GCD of 84 and

1

0

5. The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42,

8

4. The factors of 105 are 1, 3, 5, 7, 15, 21, 35,

1

0

5. The GCD of 84 and 105 is

2

1. Dividing both parts of the ratio by 21, we get:

$84/21 : 105/21 = 4:5$.

So, 84:105 is equivalent to 4:5 and hence equivalent to 12:

1

5.

b) 20:

2

5. We want to see if $20/25$ simplifies to $4/5$. We can find the GCD of 20 and

2

5. The factors of 20 are 1, 2, 4, 5, 10,

2

0. The factors of 25 are 1, 5,

2

5. The GCD of 20 and 25 is

5. Dividing both parts of the ratio by 5, we get:

$20/5 : 25/5 = 4:5$.

So, 20:25 is equivalent to 4:5 and hence equivalent to 12:

1

5.

c) 2:

5. This ratio is already in its simplest form. Since 2:5 is not equal to 4:5, it is not equivalent to 12:

1

5.

3. Final Answer

The ratios equivalent to 12:15 are 84:105 and 20:25.