Certainly! Let's solve the problem step-by-step.
We need to find the value of [tex]\( T \)[/tex] for the given expression:
[tex]\[
T = \frac{\frac{3}{4} + \frac{2}{3}}{\frac{2}{3} + T}
\][/tex]
### Step 1: Calculate the Numerator
First, we'll simplify the numerator:
[tex]\[
\frac{3}{4} + \frac{2}{3}
\][/tex]
To add these two fractions, we need a common denominator. The least common multiple (LCM) of 4 and 3 is 12.
Convert [tex]\(\frac{3}{4}\)[/tex] to a fraction with denominator 12:
[tex]\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\][/tex]
Convert [tex]\(\frac{2}{3}\)[/tex] to a fraction with denominator 12:
[tex]\[
\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}
\][/tex]
Now add the fractions:
[tex]\[
\frac{9}{12} + \frac{8}{12} = \frac{17}{12}
\][/tex]
So, the numerator is [tex]\( \frac{17}{12} \)[/tex].
### Step 2: Calculate the Denominator
Next, let's incorporate [tex]\( T \)[/tex] into the denominator:
[tex]\[
\frac{2}{3} + \frac{17}{12}
\][/tex]
First, let’s simplify [tex]\( \frac{2}{3} \)[/tex] to a fraction with denominator 12:
[tex]\[
\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}
\][/tex]
Now add [tex]\(\frac{8}{12}\)[/tex] and [tex]\(\frac{17}{12}\)[/tex]:
[tex]\[
\frac{8}{12} + \frac{17}{12} = \frac{25}{12}
\][/tex]
So, the denominator becomes [tex]\( \frac{25}{12} \)[/tex].
### Step 3: Simplify the Fraction
Now we have:
[tex]\[
T = \frac{\frac{17}{12}}{\frac{25}{12}}
\][/tex]
Since both the numerator and the denominator have the same denominator (12), we can simplify this by dividing the numerators directly:
[tex]\[
T = \frac{17}{25}
\][/tex]
### Step 4: Verify the Simplification
For a fraction [tex]\( \frac{a}{b} \)[/tex], if both the numerator and the denominator are divided by the same value, we get:
[tex]\[
\frac{\frac{a \cdot k}{d}}{\frac{b \cdot k}{d}} = \frac{a}{b}
\][/tex]
Applying this principle to our fraction, we get:
[tex]\[
T = \frac{17}{25}
\][/tex]
### Conclusion
Hence, the calculated value of [tex]\( T \)[/tex] is:
[tex]\[
T = 0.68
\][/tex]
