# A farm has 76 acres of wheat. The farmer can harvest the wheat from 12 acres per day. In how many days will all of the fields be harvested?

Oct 22, 2016

$6 \frac{1}{3}$ days

#### Explanation:

Let the unknown number of days be $x$
$\textcolor{w h i t e}{.}$
$\textcolor{b l u e}{\text{Using ratio to show why the shortcut method works.}}$

$\left(76 \text{ acres")/(12" acres") =(x" days")/(1" day}\right)$

so $x = \frac{76}{12} \to \frac{76 \div 4}{12 \div 4} = \frac{19}{3} = 6 \frac{1}{3} \text{ days}$
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$\textcolor{b l u e}{\text{Using ratios differently}}$

The clue is in the wording of 'acres per day' which translates to:
$\left(\text{acres")/("days}\right)$

$\left(12 \text{ acres")/(1" day}\right) = \frac{76}{x}$

To maintain the correct ration, if you multiply the top number of a fraction or ratio you must do the same to the bottom number
Note that $6 \frac{1}{3} \times 12 = 76$

Multiply by 1 but in the form of $1 = \frac{6 \frac{1}{3}}{6 \frac{1}{3}}$

$\left[\left(12 \text{ acres")/(1" day}\right) \times 1\right] = \frac{76}{x}$

$\left[\left(12 \text{ acres")/(1" day}\right) \times \frac{6 \frac{1}{3}}{6 \frac{1}{3}}\right] = \frac{76}{x}$

$\frac{76}{6 \frac{1}{3}} = \frac{76}{x}$

$x = 6 \frac{1}{3} \text{ days}$