# How do you insert a pair of brackets to make each statement true 1/2+ 2/3 - 1/4 * 2/5 ÷ 1/6 = 6 2/5?

Jul 4, 2015

(1/2+ 2/3 - 1/4 * 2/5) ÷ 1/6 = 6 2/5

#### Explanation:

1/2+ 2/3 - 1/4 * 2/5 ÷ 1/6 = 6 2/5

I first rewrote $\div \frac{1}{6}$ as $\cdot \frac{6}{1}$

$\frac{1}{2} + \frac{2}{3} - \frac{1}{4} \cdot \frac{2}{5} \cdot \frac{6}{1} = 6 \frac{2}{5}$

Now that mixed number on the right confuses me, so write:

$\frac{1}{2} + \frac{2}{3} - \frac{1}{4} \cdot \frac{2}{5} \cdot \frac{6}{1} = \frac{32}{5}$

That looks better but there are too many denominators especially in the addition and subtraction. So let's write:

$\frac{30}{60} + \frac{40}{60} - \frac{15}{60} \cdot \frac{2}{5} \cdot 6 = \frac{384}{60}$

Keeping the factors $\frac{15}{60} \cdot \frac{2}{5} \cdot 6$ together means we'll need to do something with
$- \frac{15}{60} \cdot \frac{2}{5} \cdot 6 = - \frac{3}{60} \cdot \frac{2}{3} \cdot 6 = - \frac{36}{60}$

But the first two only sum to $\frac{70}{60}$, so that won't work.

Next, I divided $384$ by $6$ to see what we'd need $\frac{1}{2} + \frac{2}{3} - \frac{1}{4} \cdot \frac{2}{5}$ to be, if multiply by $6$ was the last step.

$384 \div 6 = 64$

And, (Bonus)

$\frac{30}{60} + \frac{40}{60} - \frac{15}{60} \cdot \frac{2}{5} = \frac{30}{60} + \frac{40}{60} - \left(\frac{3}{60} \cdot \frac{2}{1}\right)$

$= \frac{30}{60} + \frac{40}{60} - \frac{6}{60} = \frac{64}{60}$

Last check:

(1/2+ 2/3 - 1/4 * 2/5) ÷ 1/6

$= \left(\frac{1}{2} + \frac{2}{3} - \left(\frac{1}{4} \cdot \frac{2}{5}\right)\right) \div \frac{1}{6}$

$= \left(\frac{1}{2} + \frac{2}{3} - \frac{1}{10}\right) \div \frac{1}{6}$

$= \left(\frac{15}{30} + \frac{20}{30} - \frac{3}{30}\right) \div \frac{1}{6}$

$= \left(\frac{35}{30} - \frac{3}{30}\right) \div \frac{1}{6}$

$= \frac{32}{30} \div \frac{1}{6}$

$= \frac{32}{30} \cdot \frac{6}{1} = \frac{32}{5} = 6 \frac{2}{5}$

It works!