# In college, summer vacation is five days more than 5 times the length of winter break. Spring break is one-third as long as winter break. If winter break is r days long, how many days off do college students receive for the three breaks?

Apr 27, 2017

$\frac{19}{3} r + 5$ days

#### Explanation:

If r equals the number of days for winter break then the length of summer break is
$5 r + 5$

and the length of spring break is
$\frac{1}{3} r$

To get the total number of days, add all three together to get
$r + \left(5 r + 5\right) + \frac{1}{3} r$
$\frac{19}{3} r + 5$

Since there are no numerical values expressed the value in days can not be found.

Apr 27, 2017

$\textcolor{p u r p \le}{\text{A demonstration on how to build an equation(s) from a word question.}}$

You can only express this in terms of $r \to 6 \frac{1}{3} r + 5$

This is the same as: $\frac{19}{3} r + 5$

#### Explanation:

Breaking the question down into its component parts:

Let summer break be $. {S}_{u} \leftarrow$ (my own permanent notation)
Let spring break be $\ldots . {S}_{p} \leftarrow$ (my own permanent notation)
Let winter break be $\ldots . w \textcolor{w h i t e}{.} \leftarrow$ (my temporary notation)

Summer vacation is ...........->S_u=?
5 day more than..................->S_u=?+5
5 times .............................->S_u=5?+5
the length of winter break $. . \to {S}_{u} = 5 w + 5 \text{ } \ldots . . E q u a t i o n \left(1\right)$

Spring break is ->S_p=?
one third $\ldots \ldots . . \to {S}_{p} = \frac{1}{3}$
as long as .....->S_p=1/3?
winter break $. . \to {S}_{p} = \frac{1}{3} w \text{ } \ldots \ldots \ldots \ldots \ldots \ldots E q u a t i o n \left(2\right)$

If winter break is $r$ days long $\implies$ substitute $r$ for $w$

${S}_{u} = 5 r + 5 \text{ } \ldots \ldots \ldots . . E q u a t i o n \left({1}_{a}\right)$
${S}_{p} = \frac{1}{3} r \text{ } \ldots \ldots \ldots \ldots \ldots \ldots E q u a t i o n \left({2}_{a}\right)$

If declared in the question you must use the notation they define.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The total days of expressed in terms of $r$

Winter $\ldots \ldots w \to \textcolor{w h i t e}{\ldots . .} r$
Spring $\ldots \ldots {S}_{p} \to \textcolor{w h i t e}{. .} \frac{1}{3} r$
Summer $. . {S}_{s} \to \textcolor{w h i t e}{\ldots .} \underline{5 r + 5} \leftarrow \text{ add}$
$\text{ } 6 \frac{1}{3} r + 5$