# Ky has three times more books as Grant, and Grant has 6 fewer books than Jaime. If the total combined number of books is 176, how many books does Jaime have?

Jun 18, 2018

See a solution process below:

#### Explanation:

Always location and name your variables first. So, let's call:
- The number of books Ky has: $k$
- The number of books Grant has: $g$
- The number of books Jamie has: $j$

Next, we can write the three equations from the information in the problem:

Equation 1: $k = 3 g$
Equation 2: $g = j - 6$
Equation 3: $k + g + j = 176$

First, solve Equation 2 for $j$:

$g = j - 6$

$g + \textcolor{red}{6} = j - 6 + \textcolor{red}{6}$

$g + 6 = j - 0$

$g + 6 = j$

$j = g + 6$

Next, using this result we can substitute $\left(g + 6\right)$ for $j$ in Equation 3. And using Equation 1 we can substitute $3 g$ for $k$ into Equation 3. Then we can solve Equation 3 for $g$:

$k + g + j = 176$ becomes:

$3 g + g + \left(g + 6\right) = 176$

$3 g + g + g + 6 = 176$

$3 g + g + g + 6 - \textcolor{red}{6} = 176 - \textcolor{red}{6}$

$3 g + g + g + 0 = 170$

$3 g + g + g = 170$

$3 g + 1 g + 1 g = 170$

$\left(3 + 1 + 1\right) g = 170$

$5 g = 170$

$\frac{5 g}{\textcolor{red}{5}} = \frac{170}{\textcolor{red}{5}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}} g}{\cancel{\textcolor{red}{5}}} = 34$

$g = 34$

Therefore, Grant has $\textcolor{red}{34}$ books.

Now, substitute $34$ for $g$ in the solution for $j$ we did previously and calculate the number of books Jamie has:

$j = g + 6$ becomes:

$j = 34 + 6 = 40$

Jamie has $\textcolor{red}{40}$ books

We can also calculate the number of books Ky has by substituting $34$ for $g$ in Equation 1 and calculating $k$:

$k = 3 g$ becomes:

$k = 3 \times 34 = 102$

Ky has $\textcolor{red}{102}$ books

$k + g + j = 102 + 34 + 40 = 176$