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 compare their simplified forms.
1. Simplification of ratios:
84/105 = 4/5
128/160 = 4/5
As both simplified forms are equal to 4/5, it indicates that the ratios are proportional.
2. Cross-multiplication:
For two ratios to be proportional, the cross-products must be equal. Let's check:
84 × 160 = 13,440
105 × 128 = 13,440
Since the cross-products are equal, it confirms that the ratios are proportional.
Both methods show that the ratios 84/105 and 128/160 are proportional.
Are there any other ways?
Yes, there is another way to determine if the ratios 84/105 and 128/160 are proportional. We can check if their cross-products are proportional by comparing their reduced forms:
84/105 = 4/5
128/160 = 4/5
To compare these ratios, we can find a common multiple of both denominators (105 and 160). In this case, the least common multiple is 840.
Now, we can multiply each ratio by the corresponding factor needed to achieve the common multiple:
84/105 = (84/105) * (8/8) = 672/840
128/160 = (128/160) * (5/5) = 640/800
As we can see, both ratios reduce to 4/5 when multiplied by the necessary factors to achieve the common multiple of 840. Therefore, the ratios 84/105 and 128/160 are proportional.
To determine if the ratios 84/105 and 128/160 are proportional, we can compare the two ratios by simplifying them.
First, let's simplify the ratio 84/105:
- We can divide both the numerator (84) and the denominator (105) by their greatest common divisor (GCD), which is 21.
- Dividing 84 by 21 gives us 4, and dividing 105 by 21 gives us 5.
- So, the simplified form of 84/105 is 4/5.
Next, let's simplify the ratio 128/160:
- We can divide both the numerator (128) and the denominator (160) by their GCD, which is 32.
- Dividing 128 by 32 gives us 4, and dividing 160 by 32 gives us 5.
- Hence, the simplified form of 128/160 is also 4/5.
Now that we have simplified both ratios, we can determine if they are proportional by comparing the two simplified forms:
- The ratios 4/5 and 4/5 are equivalent.
- This indicates that the original ratios 84/105 and 128/160 are indeed proportional.
Two different reasons to support this conclusion are:
1. Simplified form: By simplifying both ratios, we obtained the same simplified form, 4/5. This shows that when reduced to their simplest terms, the ratios have the same value, indicating proportionality.
2. Common ratio: Another way to determine proportionality is by comparing the ratios' cross products. If we multiply the numerator of one ratio by the denominator of the other ratio, we should get the same result. Let's try it:
- Cross-multiplying for the ratios 84/105 and 128/160:
84 * 160 = 13440
128 * 105 = 13440
- The cross products are equal, again indicating proportionality.
Therefore, both the simplified form and the equal cross products confirm that the ratios 84/105 and 128/160 are proportional.