To determine what number must be added to [tex]\(\frac{7}{13}\)[/tex] to obtain [tex]\(\frac{-5}{6}\)[/tex], we'll follow these steps:
1. Let [tex]\( x \)[/tex] be the number we need to find. We can write the equation as:
[tex]\[
\frac{7}{13} + x = \frac{-5}{6}
\][/tex]
2. Isolate [tex]\( x \)[/tex] by subtracting [tex]\(\frac{7}{13} \)[/tex] from both sides of the equation:
[tex]\[
x = \frac{-5}{6} - \frac{7}{13}
\][/tex]
3. To subtract the fractions, we need a common denominator. The denominators here are 6 and 13. The least common multiple (LCM) of 6 and 13 is 78.
4. Convert both fractions to have a common denominator of 78:
[tex]\[
\frac{7}{13} = \frac{7 \times 6}{13 \times 6} = \frac{42}{78}
\][/tex]
[tex]\[
\frac{-5}{6} = \frac{-5 \times 13}{6 \times 13} = \frac{-65}{78}
\][/tex]
5. Now we can subtract these two fractions:
[tex]\[
x = \frac{-65}{78} - \frac{42}{78}
\][/tex]
6. Combine the numerators over the common denominator:
[tex]\[
x = \frac{-65 - 42}{78} = \frac{-107}{78}
\][/tex]
7. Simplifying the fraction, we get:
[tex]\[
x = -\frac{107}{78}
\][/tex]
8. Converting [tex]\(-\frac{107}{78}\)[/tex] to a decimal, we get approximately:
[tex]\[
x \approx -1.3717948717948718
\][/tex]
So, the number [tex]\( x \)[/tex] that must be added to [tex]\(\frac{7}{13}\)[/tex] to get [tex]\(\frac{-5}{6}\)[/tex] is [tex]\(-\frac{107}{78}\)[/tex], or approximately [tex]\(-1.3717948717948718\)[/tex].
