Asked by ...
Are the ratios 84/105 and 128/160 proportional? Give two different reasons to support your answer.
Answers
Answered by
Damon
They are not only proportional, they are the same.
Answered by
GPT 3.5
To determine if the ratios 84/105 and 128/160 are proportional, we can simplify both ratios and check if both simplified ratios are equal.
Simplifying the first ratio, 84/105, we divide both numerator and denominator by their greatest common divisor, which is 21:
84 ÷ 21 = 4
105 ÷ 21 = 5
So, the simplified version of 84/105 is 4/5.
Simplifying the second ratio, 128/160, we divide both numerator and denominator by their greatest common divisor, which is 32:
128 ÷ 32 = 4
160 ÷ 32 = 5
So, the simplified version of 128/160 is also 4/5.
Reason 1: When the simplified versions of both ratios are equal (both 4/5 in this case), we can conclude that the ratios are proportional.
Reason 2: We can also find the constant of proportionality by dividing any term in one ratio by the corresponding term in the other ratio. In this case, if we divide 84 by 128, we get approximately 0.65625. If we divide 105 by 160, we also get approximately 0.65625. Since both divisions result in the same value, we can conclude that the ratios are proportional.
Simplifying the first ratio, 84/105, we divide both numerator and denominator by their greatest common divisor, which is 21:
84 ÷ 21 = 4
105 ÷ 21 = 5
So, the simplified version of 84/105 is 4/5.
Simplifying the second ratio, 128/160, we divide both numerator and denominator by their greatest common divisor, which is 32:
128 ÷ 32 = 4
160 ÷ 32 = 5
So, the simplified version of 128/160 is also 4/5.
Reason 1: When the simplified versions of both ratios are equal (both 4/5 in this case), we can conclude that the ratios are proportional.
Reason 2: We can also find the constant of proportionality by dividing any term in one ratio by the corresponding term in the other ratio. In this case, if we divide 84 by 128, we get approximately 0.65625. If we divide 105 by 160, we also get approximately 0.65625. Since both divisions result in the same value, we can conclude that the ratios are proportional.
Answered by
GPT 3.5
Apologies for the oversight. You are correct. Upon further calculation, it is evident that the two ratios, 84/105 and 128/160, are not only proportional but also equal to each other.
Reason 1: The simplified versions of both ratios are not only equal but also the same, which proves that the ratios are proportional.
Reason 2: We can also cross-multiply and compare the products. For the first ratio, 84/105, multiplying 84 by 160 gives 13,440, and multiplying 105 by 128 also gives 13,440. Since the two products are equal, it further confirms that the ratios are proportional and equal to each other.
So, to summarize, the ratios 84/105 and 128/160 are indeed proportional and equal to each other.
Reason 1: The simplified versions of both ratios are not only equal but also the same, which proves that the ratios are proportional.
Reason 2: We can also cross-multiply and compare the products. For the first ratio, 84/105, multiplying 84 by 160 gives 13,440, and multiplying 105 by 128 also gives 13,440. Since the two products are equal, it further confirms that the ratios are proportional and equal to each other.
So, to summarize, the ratios 84/105 and 128/160 are indeed proportional and equal to each other.