I'm getting insanely frustrated with this.
The average of $6$ numbers is $4$. A seventh number is added and the new average is $5$. What is the seventh number?
How do I solve this algebraically?
$6 \cdot 4 = 24$. $\dfrac{24 + x}{7} = 5$
Why am I getting a negative number?
You solved for $x$ wrong. Here is for $x$ correctly:
We first multiply by $7$ to both sides:
$$\frac{24+x}{7}=5,$$
then subtract $24$ to both sides:
$$24+x=35,$$
therefore $x$ is left alone and we get:
$$x=11.$$
There are several ways to solve this. However, I've provided a comprehensive walkthrough that I think will help you understand why the methods other people will list actually work.
$$\frac{a+b+c+d+e+f}{6}=4$$
$$a+b+c+d+e+f=24$$
$$\frac{24+g}{7}=5$$
$$24+g=35$$
$$g=35-24$$
$$g=11$$
${x_1+...+ x_6 \over 6} = 4$. ${x_1+...+ x_6 +x_7\over 7} = 5$. So $x_1+...+ x_6 = 24, x_1+...+ x_6 + x_7 = 35$. What is $x_7$?
Hint: The trick is to realize that, because $$Average =\frac{ Sum }{ Count} $$ for a given collection of numbers, you also have that $$Sum = Average \cdot Count$$ for that collection.
If you augment the original collection by one number, then you will increase the sum by this number and the count by one, so you have that, for the augmented collection,
$$New Average = \frac{New Sum}{New Count}=\frac{OldSum +NewNumber}{OldCount +1}$$
Shortly after I posted this, I realized what my mistake was.
I divided 5 by 7, but I should have multiplied 5 by 7. I needed to do the inverse operation.
