# How do you solve 5 2/7 + k = 2 27/70?

Apr 24, 2016

$k = - 2 \frac{9}{10}$

#### Explanation:

$\textcolor{red}{\text{Using shortcut method}}$

Move $5 \frac{2}{7}$ to the other side

$k = \textcolor{b r o w n}{2 \frac{27}{70}} \textcolor{g r e e n}{- 5 \frac{2}{7}}$

By reversing the numbers I can subtract the lesser value from the greater. However, I have to multiply the whole by (-1) to maintain the original values.
Write as $k = - 1 \times \left(\textcolor{g r e e n}{4 \frac{9}{7}} \textcolor{b r o w n}{- 2 \frac{2.7}{7}}\right) = - 2 \frac{6.3}{7} = - 2 \frac{63}{70} = - 2 \frac{9}{10}$

$\textcolor{b l u e}{\text{ } k = - 2 \frac{9}{10}}$

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$\textcolor{red}{\text{Using first principles method}}$

I have elected to show this process in a lot of detail

Given:$\text{ } 5 \frac{2}{7} + k = 2 \frac{27}{70}$

Subtract $5 \frac{2}{7}$ from both sides

$k = 2 \frac{27}{70} - 5 \frac{2}{7}$

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$\textcolor{b l u e}{\text{Consider } 2 \frac{27}{70}}$

Write as $2 + \frac{27}{70}$

Multiply the 2 by 1 but in the form $1 = \frac{70}{70}$

$\left(\frac{2}{1} \times \frac{70}{70}\right) + \frac{27}{70} \text{ "->" } \frac{140}{70} + \frac{27}{70} \textcolor{b l u e}{= \frac{167}{70}}$
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$\textcolor{b l u e}{\text{Consider } 5 \frac{2}{7}}$

Write as $5 + \frac{2}{7}$

Multiply 5 by 1 but in the form $1 = \frac{7}{7}$

$\left(\frac{5}{1} \times \frac{7}{7}\right) + \frac{2}{7} \text{ "->" } \frac{35}{7} + \frac{2}{7} = \frac{37}{7}$

Multiply by 1 but in the form $\frac{10}{10}$

$\frac{37}{7} \times \frac{10}{10} \textcolor{b l u e}{= \frac{370}{70}}$

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$\textcolor{b l u e}{\text{Putting it all together}}$

color(brown)(" "k=2 27/70-5 2/7" "->" "color(green)(k=167/70-370/70)

$\textcolor{b l u e}{\text{ } k = - \frac{203}{70} = - 2 \frac{9}{10}}$