Adding fractions with unlike denominators six tenths

Key concept: The fraction sum $\frac{5}{6}+\frac{3}{10}$ becomes easy once both terms share a common denominator.

Step by step

Step 1: Find a common denominator

For $\frac{5}{6}+\frac{3}{10}$, the denominators are 6 and 10. The least common denominator is 30, because both 6 and 10 divide into 30 evenly.

Rewrite each fraction with denominator 30:

$\frac{5}{6}=\frac{25}{30}, \qquad \frac{3}{10}=\frac{9}{30}.$

Now the fractions have the same unit size, so the numerators can be added directly.

Step 2: Add the numerators

With the common denominator in place, the sum is

$\frac{25}{30}+\frac{9}{30}=\frac{34}{30}.$

This fraction is improper, so it is better to simplify it before finishing.

Step 3: Simplify the result

The fraction $\frac{34}{30}$ can be reduced by dividing numerator and denominator by 2:

$\frac{34}{30}=\frac{17}{15}.$

If you want a mixed number, that is $1\frac{2}{15}$.

Check your work

The exact sum of $\frac{5}{6}$ and $\frac{3}{10}$ is $\frac{17}{15}$. A quick decimal check gives about 0.8333 + 0.3 = 1.1333, and $\frac{17}{15}\approx 1.1333$, so the answer is consistent.

Pitfall alert

A common mistake with $\frac{5}{6}+\frac{3}{10}$ is to add the denominators directly and write something like $\frac{8}{16}$, which does not work for fractions. The denominator tells you the size of the pieces, so it must stay consistent before adding. Another error is to use 60 as the common denominator and then forget to simplify at the end. That is not wrong, but it often leads to careless arithmetic: $\frac{50}{60}+\frac{18}{60}=\frac{68}{60}$, and then students stop too early. Always check whether the answer can be reduced. One more trap is mixing up the scaling factors: when converting $\frac{5}{6}$ to thirtieths, multiply by 5, not by 6, because $6\times 5=30$.

Try different conditions

If the problem changed to $\frac{5}{6}+\frac{7}{10}$, the same method would still apply, but the final number would be different. The least common denominator would still be 30, so $\frac{5}{6}=\frac{25}{30}$ and $\frac{7}{10}=\frac{21}{30}$. Adding gives $\frac{46}{30}$, which simplifies to $\frac{23}{15}$ or $1\frac{8}{15}$. This variant is useful because it checks whether you can keep the denominator fixed, convert each fraction correctly, and simplify the final improper fraction. If you can do both the original and this modified version, you understand the method rather than memorizing one answer.

Further reading

least common denominator, equivalent fractions, improper fraction