Are the ratios 84/105 and 128/160 proportional? Give two different reasons to support your answer.
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.
They are not only proportional, they are the same.
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.
To determine whether the ratios 84/105 and 128/160 are proportional, we need to check if their cross products are equal.
The cross product of the first ratio (84/105) is calculated by multiplying the numerator of one ratio with the denominator of the other ratio:
84 * 160 = 13,440
The cross product of the second ratio (128/160) is calculated in the same way:
128 * 105 = 13,440
Comparing the cross products, we find that both are equal to 13,440.
First reason: Since the cross products are equal, we can conclude that the ratios are proportional.
Second reason: We can also check if the simplified forms of the ratios are equal. To simplify the ratios, we divide the numerator and denominator of each ratio by their greatest common divisor (GCD).
For the ratio 84/105, the GCD of 84 and 105 is 21. Dividing both numerator and denominator by 21, we get:
84 ÷ 21 = 4
105 ÷ 21 = 5
So the simplified form of 84/105 is 4/5.
For the ratio 128/160, the GCD of 128 and 160 is 16. Dividing both numerator and denominator by 16, we get:
128 ÷ 16 = 8
160 ÷ 16 = 10
So the simplified form of 128/160 is 8/10, which can be further simplified to 4/5.
Since the simplified forms are the same, this also shows that the ratios 84/105 and 128/160 are proportional.