# How do you add 3\frac { 7} { 10} + 4\frac { 1} { 15} + 2\frac { 2} { 13}?

Jan 5, 2017

$9 \setminus \frac{359}{390}$

#### Explanation:

Okay, so first you put all the fractions into a Common Denominator what this means is that all the bottom numbers have to be equal.

So let's add the first two numbers first

$3 \setminus \frac{7}{10} + 4 \setminus \frac{1}{15}$

What is a common denominator for $15$ and $10$: the answer is $30$.

A brief overview of a common denominator: to find the common denominator list the multiples of $15$ and $10$. For 15 it would be:

$15 \left(15 \cdot 1 = 15\right)$

$30 \left(15 \cdot 2 = 30\right)$

Therefore, we could conclude that since the multiples of $10$ are pretty simple.

$10 \cdot 1 = 10$

$10 \cdot 2 = 20$

$10 \cdot 3 = 30$

We found a common denominator: $30$!

So next we multiply each number the top and the bottom the same so if we multiple $10$ by $3$ to equal $30$; we do the same for the top number so

$7 \cdot 3 = 21$

Same goes for the other number we multiplied $15$ by $2$ to find $30$ and we do the same for the top

$1 \cdot 2 = 2$

But we don't do anything to the whole number because it's not a part of the fraction! Therefore the numbers are going to look like

$3 \setminus \frac{21}{30} + 4 \setminus \frac{2}{30}$

Which equals this is all just addition which I shouldn't be explaining

7\frac { 23} { 30

Now we do the next part

$7 \setminus \frac{23}{30} + 2 \setminus \frac{2}{13}$

A common multiple of $13$ and $30$ would be $390$.

Which sounds like a lot but is just a multiple of $9$ for $13$ and a multiple of $13$ for $30$.

So we do the same thing we did above.

$9 \setminus \frac{359}{390}$

Which cannot be simplified!

Remember always simplify during tests or quizzes if you don't you will definite loose points; which is a frugal way to lose points after all that hard work.

Jan 5, 2017

$9 \textcolor{w h i t e}{.} \frac{359}{390}$

#### Explanation:

Split the numbers so that we have:
$\left(3 + 4 + 2\right) + \left(\frac{7}{10} + \frac{1}{15} + \frac{2}{13}\right)$

The brackets are only there to highlight the grouping of numbers.

This gives: $9 + \left(\frac{7}{10} + \frac{1}{15} + \frac{2}{13}\right)$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Dealing with the fractional group of numbers}}$

A fractions structure is: $\left(\text{count")/("size indicator")->("numerator")/("denominator}\right)$

You can not directly add the 'counts' (numerators) unless the
'size indicators' (denominators) are all the same.

$\textcolor{b r o w n}{\text{A sort of cheat method for a common denominator}}$

$\textcolor{w h i t e}{3} 15$
$\underline{\textcolor{w h i t e}{3} 13} \leftarrow \text{ Multiply}$
$150$
$\underline{\textcolor{w h i t e}{1} 45} \leftarrow \text{ Add}$
$195 \leftarrow \text{ both 15 and 13 are factors of this number}$

The last digit of 195 is 5 so 195 can not have 10 as a whole number factor. So lets try changing the 5 into 0

$195$
$\underline{\textcolor{w h i t e}{19} 2} \leftarrow \text{ multiply}$
$390 \leftarrow \text{ all of 10, 15 and 13 will divide into this number}$
.........................................................................................

$390 \div 10 = 39$
$390 \div 15 = 26$
$390 \div 13 = 30$

color(green)([7/10color(red)(xx1)]color(white)(..)+color(white)(..)[1/15color(red)(xx1)]color(white)(..)+color(white)(..)[2/13color(red)(xx1)]

color(green)([7/10color(red)(xx39/39)]+color(white)(.)[1/15color(red)(xx26/26)]color(white)(.)+color(white)(.)[2/13color(red)(xx30/30)]

$\textcolor{g r e e n}{\text{ "[273/390]" "+" "[26/390]" "+" } \left[\frac{60}{390}\right]}$

$\textcolor{w h i t e}{.}$

$\textcolor{g r e e n}{\text{ } \frac{359}{390}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Putting it all together}}$

$\text{ } \textcolor{b l u e}{9 \frac{359}{390}}$