Question

Write the rational expression with the indicated denominator.
   StartFraction 2 t Over 7 t minus 28 EndFraction
equalsStartFraction question mark Over 21 t minus 84 EndFraction

Answer 1

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:

1. For \(7t - 28\):
\[
7t - 28 = 7(t - 4)
\]

2. 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:

\[
\frac{6t}{21t - 84}
\]