Addition of Fractions: The common denominator is smaller than the product of the denominators

Let's try to find the lowest common denominator between 4 and 6

To find the lowest common denominator, we need to find a number that is divisible by both 4 and 6

In this case, the common denominator is 12

Now we'll multiply each fraction by the appropriate number to reach the denominator 12

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{1\times3}{4\times3}+\frac{3\times2}{6\times2}=\frac{3}{12}+\frac{6}{12}$

Now let's add them:

$\frac{6+3}{12}=\frac{9}{12}$

Let's try to find the lowest common denominator between 8 and 12

To find the lowest common denominator, we need to find a number that is divisible by both 8 and 12

In this case, the common denominator is 24

Now we'll multiply each fraction by the appropriate number to reach the denominator 24

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{2\times3}{8\times3}+\frac{5\times2}{12\times2}=\frac{6}{24}+\frac{10}{24}$

Now let's combine:

$\frac{6+10}{24}=\frac{16}{24}$

Let's try to find the lowest common denominator between 6 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 6 and 9

In this case, the common denominator is 18

Now we'll multiply each fraction by the appropriate number to reach the denominator 18

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{3\times3}{6\times3}+\frac{3\times2}{9\times2}=\frac{9}{18}+\frac{6}{18}$

Now let's combine:

$\frac{9+6}{18}=\frac{15}{18}$

Let's try to find the lowest common denominator between 10 and 12

To find the lowest common denominator, we need to find a number that is divisible by both 10 and 12

In this case, the common denominator is 60

Now we'll multiply each fraction by the appropriate number to reach the denominator 60

We'll multiply the first fraction by 6

We'll multiply the second fraction by 5

$\frac{4\times6}{10\times6}+\frac{5\times5}{12\times5}=\frac{24}{60}+\frac{25}{60}$

Now let's add:

$\frac{24+25}{60}=\frac{49}{60}$

Let's try to find the lowest common denominator between 8 and 12

To find the lowest common denominator, we need to find a number that is divisible by both 8 and 12

In this case, the common denominator is 24

Now we'll multiply each fraction by the appropriate number to reach the denominator 24

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{4\times3}{8\times3}+\frac{5\times2}{12\times2}=\frac{12}{24}+\frac{10}{24}$

Now let's add:

$\frac{12+10}{24}=\frac{22}{24}$

Let's try to find the lowest common multiple between 8 and 10

To find the lowest common multiple, we need to find a number that is divisible by both 8 and 10

In this case, the lowest common multiple is 40

Now, let's multiply each number in the appropriate multiples to reach the number 40

We will multiply the first number by 5

We will multiply the second number by 4

$\frac{4\times5}{8\times5}+\frac{4\times4}{10\times4}=\frac{20}{40}+\frac{16}{40}$

Now let's calculate:

$\frac{20+16}{40}=\frac{36}{40}$