Adding Mixed Numbers with Unlike Denominators

Adding Mixed Numbers

$\mathsf{ 1\frac{1}{3} + 4\frac{6}{8} }$

Our first example is $\mathsf{ 1\frac{1}{3} + 4\frac{6}{8} }$. We first find the LCM of 3 and 8. The multiples of 3 are 3, 6, 9, 12, 15. The multiples of 8 are 8, 16, 24, 32, 40. There is no common multiple, so we keep going. 18, 21, 24. I know that 24 is also a multiple of 8. The LCM is 24. 24 will be our common denominator.

We re-write the addition expression. Now we find equivalent fractions. We multiplied 3 x 8 to get 24. We do the same to the numerator. $\mathsf{ \frac{1}{3} = \frac{8}{24} }$. We multiplied 8 x 3 to get 24. Now we do the same with the numerator. $\mathsf{ \frac{6}{8} = \frac{18}{24} }$.

$\mathsf{ 1\frac{8}{24} + 4\frac{18}{24} }$. Now we add. We add our whole numbers first: 1 + 4 = 5. Now we add our fractions. $\mathsf{ \frac{8}{24} + \frac{18}{24} = \frac{26}{24}}$. $\mathsf{ 5\frac{26}{24} }$. $\mathsf{ \frac{26}{24} }$ is an improper fraction, we have to convert this to a mixed number. $\mathsf{ 5\frac{26}{24} }$. 24 goes into 26 1 time with 2 left over. $\mathsf{ \frac{26}{24} }$ is equivalent to $\mathsf{ 1\frac{2}{24} }$. $\mathsf{ 5 + 1\frac{2}{24} = 6\frac{2}{24} }$. Our sum is $\mathsf{ 6\frac{2}{24} }$.

$\mathsf{ 2\frac{3}{4} + 3\frac{5}{6} }$

Our next example is $\mathsf{ 2\frac{3}{4} + 3\frac{5}{6} }$. We find the LCM of 4 and 6. The LCM of 4 and 6 is 12. We re-write our addition statement, and now we find equivalent fractions. We multiplied 4 by 3, so we do the same to the numerator. $\mathsf{ \frac{3}{4} }$ is equivalent to $\mathsf{ \frac{9}{12} }$ . We multiplied 6 by 2; we do the same to the numerator. $\mathsf{ \frac{5}{6} }$ is equivalent to $\mathsf{ \frac{10}{12} }$. We add our wholes first: 2 + 3 = 5. Now we add the fraction: $\mathsf{ \frac{9}{12} + \frac{10}{12} = \frac{19}{12} }$ . Since $\mathsf{ \frac{19}{12} }$ is an improper fraction, we need to convert this to a mixed number. 12 goes into 19 1 time with 7 left over. $\mathsf{ \frac{19}{12} }$ is equivalent to $\mathsf{ 1\frac{7}{12} }$. We combine this with the 5 wholes. Our sum is $\mathsf{ 6\frac{7}{12} }$.

$\mathsf{ 3\frac{2}{5} + 3\frac{1}{2} }$

We find the LCM of 5 and 2. The LCM of 5 and 2 is 10. We multiplied 5 by 2; we do the same to the numerator. $\mathsf{ \frac{2}{5} }$ is equivalent to $\mathsf{ \frac{4}{10} }$. We multiplied 2 by 5; we do the same to the numerator. $\mathsf{ \frac{1}{2} }$ is equivalent to $\mathsf{ \frac{5}{10} }$. Now we add our wholes first: 3 + 3 = 6. Now we add our fractions $\mathsf{ \frac{4}{10} + \frac{5}{10} = \frac{9}{10} }$. $\mathsf{ \frac{9}{10} }$ is a proper fraction. Our sum is $\mathsf{ 6\frac{9}{10} }$.