Malin and Shawn begin with the same number, $t$. Malin subtracts $6$ from $t$ and then divides the result by $6$. Shawn adds $7$ to $t$ and then divides the result by $7$. If Malin's final answer is the same as Shawn's final answer, what was the number $t$

t=84

Step-by-step explanation:

Malin's answer is $\frac{t-6}{6}$. Shawn's answer is $\frac{t+7}{7}$. We know these are equal, so we have the equation

$$\frac{t-6}{6} = \frac{t+7}{7}.$$

To eliminate denominators from the problem, we multiply both sides by $6\cdot 7$:

$$\frac{6\cdot 7\cdot (t-6)}{6} = \frac{6\cdot 7\cdot (t+7)}{7},$$

then simplify to get

$$7\cdot (t-6) = 6\cdot (t+7).$$

The parentheses are important here! For example, the parentheses on the left side of the equation tell us that it is $t-6$, not just $t$, which is multiplied by $7$.

Now we expand using the distributive property:

7t - 7\cdot 6 &= 6t + 6\cdot 7;\\

7t - 42 &= 6t + 42.

Adding $42$ to both sides gives

$$7t = 6t + 84,$$

then subtracting $6t$ from both sides gives $t=\boxed{84}$.

(We can check that starting from $t=84$, Malin and Shawn do indeed get the same final answer -- namely, $13$.)