Cody an Aria are painting a mural. They decide they want part of the mural to have a purple background. Aria uses 7 tbsp. of blue to 2 tbsp. of red to make her purple paint. Cody is planning to use more and uses 21 tbsp. of blue to 6 tbsp. of red. Are their ratios of purple mixture equivalent to one another’s?
To determine if the ratios of the purple mixture are equivalent, we need to compare the ratios of blue to red in their mixtures.
Aria's ratio of blue to red is 7:2.
Cody's ratio of blue to red is 21:6.
To simplify both ratios, we can divide both sides by the greatest common divisor.
For Aria's ratio:
7 ÷ gcd(7,2) = 7 ÷ 1 = 7
2 ÷ gcd(7,2) = 2 ÷ 1 = 2
So Aria's simplified ratio is 7:2.
For Cody's ratio:
21 ÷ gcd(21,6) = 21 ÷ 3 = 7
6 ÷ gcd(21,6) = 6 ÷ 3 = 2
So Cody's simplified ratio is 7:2.
Since the simplified ratios for both Aria and Cody are the same (7:2), their ratios of the purple mixture are equivalent.
To determine if the ratios of purple mixture are equivalent, we need to compare the amount of blue to red in each ratio.
Aria's ratio: 7 tbsp. of blue to 2 tbsp. of red
Cody's ratio: 21 tbsp. of blue to 6 tbsp. of red
To simplify the ratios and make them comparable, we need to find a common denominator.
Aria's ratio: 7 tbsp. of blue to (2 * 3) tbsp. of red = 7 tbsp. of blue to 6 tbsp. of red
Cody's ratio: (21 * 1) tbsp. of blue to (6 * 3) tbsp. of red = 21 tbsp. of blue to 18 tbsp. of red
Now let's compare the simplified ratios:
Aria's ratio: 7 tbsp. of blue to 6 tbsp. of red
Cody's ratio: 21 tbsp. of blue to 18 tbsp. of red
Since both ratios have the same amount of blue (7 tbsp.), and Aria's ratio has fewer red (6 tbsp.) compared to Cody's ratio (18 tbsp.), the ratios are not equivalent.
Therefore, their ratios of purple mixture are not equivalent to one another's.
To determine if their ratios of purple mixture are equivalent, we need to compare the ratios of blue to red used by both Aria and Cody.
Aria used 7 tbsp. of blue and 2 tbsp. of red, so her ratio of blue to red can be written as 7:2.
Cody used 21 tbsp. of blue and 6 tbsp. of red, so his ratio of blue to red can be written as 21:6.
To determine if these ratios are equivalent, we can simplify them by dividing both numbers by their greatest common divisor.
For Aria's ratio:
Greatest common divisor of 7 and 2 is 1.
Dividing both numbers by 1 gives us the simplified ratio of 7:2.
For Cody's ratio:
Greatest common divisor of 21 and 6 is 3.
Dividing both numbers by 3 gives us the simplified ratio of 7:2.
As we can see, both Aria's and Cody's ratios simplify to 7:2. Therefore, their ratios of purple mixture are indeed equivalent to each other's.