To find the numerator of the rational expression such that
\[ \frac{2t}{7t - 28} = \frac{?}{21t - 84} \]
we need to express both denominators in a compatible way.
First, let's factor the denominators:
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For \(7t - 28\): \[ 7t - 28 = 7(t - 4) \]
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For \(21t - 84\): \[ 21t - 84 = 21(t - 4) = 3 \cdot 7(t - 4) \]
This tells us that \(21t - 84\) is three times \(7t - 28\).
Next, we can then set up the relationship between the two fractions:
\[ \frac{2t}{7(t - 4)} = \frac{?}{3 \cdot 7(t - 4)} \]
To maintain equality between the two fractions, the numerators must relate as follows:
\[ \text{If } \frac{a}{b} = \frac{c}{k \cdot b}, \text{ then } c = k \cdot a \]
In our case, \(k = 3\) and \(a = 2t\), so:
\[ c = 3 \cdot 2t = 6t \]
Thus, the numerator we are looking for is \(6t\). Therefore, we have:
\[ \frac{2t}{7t - 28} = \frac{6t}{21t - 84} \]
So the final expression is:
\[ \frac{6t}{21t - 84} \]