How do you convert a fraction to a specific denominator?

Relevant MM Lessons:

Overview: Converting Fractions

This question has its own dedicated lesson ». Make sure to check it out if you need a more thorough read, or want to become an expert on how this shows up on tests.Converting a fraction to an equivalent form is always a matter of multiplying by some form of the number $1$. We accomplish this by multiplying the original fraction by a fraction with an equal numerator and denominator (so that it's really equivalent to multiplying by $1$) and then proceding with standard fraction multiplication. For example, we could turn $1/2$ into an equivalent fraction like so:$$\frac{1}{2} \cdot \frac{3}{3} = \frac{3}{6}$$The most common reason that we have to do this is fraction addition and subtraction. We can't perform those operations on a pair of fractions by hand without both fractions first having a common denominator.

Equivalent Fractions - Bigger Numbers

If you want to turn a particular fraction into an equivalent fraction with a specific denominator, all you need to do is figure out the correct multiplier via some basic division.For example, if you have the fraction $2/5$ and you want to turn it into an equivalent fraction with a denominator of $35$, you need to figure out what number you multiply $5$ by to get $35$. In short, you're wondering what $35 \div 5$ is. Since $35 \div 5$ is $7$ in this case, we multiply by $7/7$ (which is really the same as multipying by $1$).$$\frac{2}{5} \cdot \frac{7}{7} = \frac{14}{35}$$

Equivalent Fractions - Smaller Numbers

If you want to turn a particular fraction into an equivalent one that has smaller integers, we often use the words "reducing" or "simplifying" to describe the process. We can do the same thing that we did above, but instead reverse the roles of multiplication and division.Let's say you want to turn the fraction $20/100$ into an equivalent one that has a denominator of $50$. In this case, you want to know how many times $50$ goes into $100$ (again, division is the key). But this time, we'll divide each the numerator and denominator by that number to obtain the result we want. Since $100 \div 50 = 2$, we can proceed as follows:$$\frac{20}{100} \longrightarrow \frac{20 \div 2}{100 \div 2} = \frac{10}{50}$$

Equivalent Fractions - Smallest Numbers

If you want to find the smallest possible integers that comprise an equivalent fraction that you can reduce your fraction to, you're asking how to fully simplify numeric fractions », which we've discussed on its own.