# Marvin has 4 books to read for homework this week. If he reads 2/3 of a book each night, how many nights will it take him to read all four books?

Apr 5, 2018

$6$ nights

#### Explanation:

Since it takes Marvin one night to $\frac{2}{3}$ of a book, then you can think of it as

Book:Night
$\frac{2}{3} : 1$

$4$: $x$

$x = \frac{4 \times 1}{\frac{2}{3}}$

$x = 4 \times \frac{3}{2}$

$x = 6$

therefore, it would take 6 nights for Marvin to finish 4 books

Apr 5, 2018

If he reads $\frac{2}{3}$ of a book in $1$ night, then he'll read $\frac{4}{3}$ of a book in 2 days.

$\frac{2}{3} x = y$

$\textcolor{w h i t e}{\ldots . .} x \textcolor{w h i t e}{. .} | \textcolor{w h i t e}{. .} y$
$\textcolor{w h i t e}{.} \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$
$\textcolor{w h i t e}{\ldots . .} 1 \textcolor{w h i t e}{. .} | \textcolor{w h i t e}{. .} \frac{2}{3}$

$\textcolor{w h i t e}{\ldots . .} 2 \textcolor{w h i t e}{. .} | \textcolor{w h i t e}{. .} \frac{4}{3}$

$\textcolor{w h i t e}{\ldots . .} 3 \textcolor{w h i t e}{. .} | \textcolor{w h i t e}{. .} 2 \textcolor{w h i t e}{/ x}$

$\textcolor{w h i t e}{\ldots . .} 4 \textcolor{w h i t e}{. .} | \textcolor{w h i t e}{. .} \frac{8}{3}$

$\textcolor{w h i t e}{\ldots . .} 5 \textcolor{w h i t e}{. .} | \textcolor{w h i t e}{. .} \frac{10}{3}$

$\textcolor{w h i t e}{\ldots . .} 6 \textcolor{w h i t e}{. .} | \textcolor{w h i t e}{. .} 4 \textcolor{w h i t e}{/ x}$

So, according to this table, if he reads $\frac{2}{3}$ of a book a night, he will finish $4$ books in $6$ days

We could also have solved this using an equation:

$\frac{2}{3} x = 4$

$x = 4 \times \frac{3}{2}$

$x = \frac{12}{2}$

$x = 6$

Apr 5, 2018

6 nights

#### Explanation:

Using ratio but in fraction format (this is not a fraction!!!!!)

Let the unknown time measured in nights be $t$

$\left(\text{count of books read")/("time in nights") color(white)("dd")-> color(white)("dd}\right) \frac{\textcolor{w h i t e}{. .} \frac{2}{3} \textcolor{w h i t e}{. .}}{1} = \frac{4}{t}$

There is a mathematical way of doing this but I am just going to say: turn EVERYTHING up the other way giving:

("time in nights")/("count of books read") color(white)("dd")-> color(white)("dd")(color(white)(..)1color(white)(..))/ (2/3) = t/4

Multiply both sides by 4

$\frac{\textcolor{w h i t e}{. .} \left(1 \times 4\right) \textcolor{w h i t e}{. .}}{\frac{2}{3}} = t$

$t = 4 \times \frac{3}{2}$

$t = \frac{4}{2} \times 3$

$t = 2 \times 3 = 6 \text{ nights}$