Finding the common denomenator of \(\displaystyle 24\) yields a result of \(\displaystyle \frac{17}{24}\)

The least common multiple of 3, 10, and 12 is 60. 60 is divisible by all three numbers (60/3 = 20, 60/10 = 6, and 60/12 = 5). Therefore, you could convert these fractions to 20/60, 25/60, and 42/60.

Find the largest number that divides into both \(\displaystyle 24\) and \(\displaystyle 256\)

\(\displaystyle \frac{24}{8}=3\)

\(\displaystyle \frac{256}{8}=32\)

\(\displaystyle \frac{24}{256}=\frac{3}{32}\)

To simplify a fraction you need to find all the factors that the numerator and denominator have in common. You can see that both share 2 so when you divide both by 2 you get

\(\displaystyle \frac{308}{105}\)

this is close to the answer but it asks for no common factors, it is hard to see but both of these share 7 as a common factor. When you divide both by 7 you get

\(\displaystyle \frac{44}{15}\)

Finding the least common denominator in rational expressions follows the same procedue as finding the least common denominator in fractions.

The least common denominator for this rational exresspion will use all terms with the highest exponents of each.

The first fraction has \(\displaystyle e^2\) as the highest term, the second fraction has \(\displaystyle f^2\) as the highest term, and the third fraction has \(\displaystyle d^2\) as the highest term. Now we combine these and get the least common denominator to be:

\(\displaystyle d^2 \times e^2 \times f^2\)

To simplify a fraction, find the gcf (gcd) of the numerator and denominator and divide both by the gcf (gcd). the gcf of \(\displaystyle 96\) and \(\displaystyle 108\) is \(\displaystyle 12\) so:

\(\displaystyle \frac{96/12}{108/12}=\frac{8}{9}\)

The first and easiest simplification for this fraction is to divide numerator and denominator by \(\displaystyle \small 10\).  This gives you:

\(\displaystyle \small \frac{42}{539}\)

Next, notice that the two fractions do not both contain factors of \(\displaystyle \small 3\).  This is because the denominator's digits, when added up come to \(\displaystyle \small 17\), which is not divisible by \(\displaystyle \small 3\).  This means \(\displaystyle \small 539\) is not divisible by \(\displaystyle \small 3\).  Now, \(\displaystyle \small 42\) is \(\displaystyle \small 2*3*7\).  There is no shared \(\displaystyle \small 2\) between these numbers.  However, if you try, you will see that they are both divisible by \(\displaystyle \small 7\), which gives you:

\(\displaystyle \small \frac{6}{77}\)

This is simplest form.

First, begin by noticing that both numerator and denominator contain a \(\displaystyle \small 5\). Dividing this out gives you:

\(\displaystyle \small \small \frac{63}{165}\)

Now, \(\displaystyle \small 1+6+5=12\) and \(\displaystyle \small 6+3=9\).  Since each of these are true, we know that both numerator and denominator contain a \(\displaystyle \small 3\).  Dividing this out, you get:

\(\displaystyle \small \small \small \frac{21}{55}\)

This is the simplest possible form of the fraction.

Divide the total money spent by the cost of each painting.  

\(\displaystyle \frac{\$1850}{\$150}=12.33333\) 

Therefore, to make a profit, she needs to sell more than this amount.  Since she can't sell a portion of a painting, the answer has to be the next nearest whole number (\(\displaystyle 13\)).  

Add the ones digits:

\(\displaystyle 1+3+5=9\)

Since there is no tens digit to carry over, proceed to add the tens digits:

\(\displaystyle 1+1+1=3\)

The answer is \(\displaystyle 39\).