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(Video) Percents and Equivalent Fractions

by Math Antics

Here's a video by mathantics that explains how to convert percentages to fractions.

$\large{\text{Summary}}$

A percent is just an equivalent fraction that tells you how many you would have IF it was out of $100$.

For example, $\frac12$ is an equivalent fraction of $\frac{50}{100}$. Since $100$ is in the denominator, we know this is $50\%$. In other words, $50\%$ of something is the same as half of that thing.

To turn a fraction into a percent, just multiply it by $100$. To turn a percent into a fraction, just divide the percent by $100$. Be sure to remove the percent sign at the end!

$$\frac38\times100\%=\frac{3\times100}{8}\%=\frac{3\times25}{2}\%=\frac{75}2\%=37.5\%\\ \\ \\ 37.5\%\div100=\frac{37.5}{100}=\frac{375}{1000}=\frac{15}{40}=\boxed{\frac{3}{8}}$$

In the above example, the fraction we started with was a proper fraction because it was between $0$ and $1$. If the fraction is proper, then the percent will be between $0\%$ and $100\%$. If the fraction we start with is a mixed number or an improper fraction, we use the same technique. However, the percentage will be greater than $100$. Let's demonstrate by turning $\frac{11}{8}$ into a percentage.

$$\frac{11}{8}\times100\%=\frac{11\times100}{8}\%=\frac{11\times25}{2}\%=\frac{275}2\%=137.5\%\\ \\ \\ 137.5\%\div100=\frac{137.5}{100}=\frac{1375}{1000}=\frac{55}{40}=\boxed{\frac{11}{8}}$$

As we can see, an improper fraction converts to a percentage greater than $100$, and vice versa. This is a good, fast way to check your work.

Converting Between Fractions and Percent

To convert a fraction to a percent, multiply the fraction by $100$ and simplify.

To convert a percent to a fraction, divide the percent by $100$ and reduce the fraction.

The diagram below might be referred to as a fractions calculator or a percentage calculator. But, both of those names are a bit misleading. Calling it a fraction calculator is confusing because that name doesn't suggest what we're turning fractions into. The name "percentage calculator" has the same issue. Thus it is more clear to call this a conversion between fractions and percentages.

To convert $\frac{3}{5}$ to a percent, all we have to do is multiply the fraction by $100$ and simplify. $$\begin{align} \dfrac{3}{5} \cdot 100 &= \dfrac{3 \cdot 100}{5}\\ &=\dfrac{300}{5}\\ &=60 \end{align}$$ So, $\frac{3}{5}$ in the form of a percent is $60\%.$

To convert $94\%$ to a fraction, all we have to do is divide $94$ by $100$ and simplify. $$\dfrac{94}{100}=\dfrac{47}{50}$$ So $94\%$ in the form of a fraction is $\boxed{\frac{47}{50}}$.

(Video) Fractions to Percentages

by tecmath

This video by tecmath explains how to convert fractions to percentages.

$\large{\text{Summary}}$

One way to turn a fraction into a percent is to multiply the fraction by $100$. For example, let's convert $\frac34$ into a percent. $$\frac34\times100=\frac3{\cancel{4}}\times\frac{\cancel{100}}1=\frac{3}{1}\times\frac{25}{1}=75$$ Attach a percent sign and you're good to go. $$\boxed{75\%}$$

Another way is to first do the division that the fraction represents. Then, move the decimal place to the right by $2$. Suppose we wanted to convert $\frac58$ into a percent. Move the decimal in $0.625$ to the right two places. This is the same as multiplying by $100$. $$0{\color{3db917}{.}}625\rightarrow6{\color{3db917}{.}}25\rightarrow62{\color{3db917}{.}}5$$

In each movement, the digits do not change their order; only the decimal point moves.

Converting Fractions to Percents

The key idea is that percents are always fractions with one hundred in the denominator. The $x\%$ means $\frac{x}{100}$. As a result, we often want to convert between fractions and percents.

So the plan is to convert the fraction's denominator into $100$ by multiplying the numerator and denominator by the same number. That number is chosen to turn the denominator into $100$.

Let's try this with $\frac{4}{5}$: $$\frac{4}{5}\times\frac{20}{20}=\frac{80}{100}=\boxed{80\%}$$

In summary, we took the top number ($4$) and the bottom number ($5$) and multiplied them both by $20$. We chose $20$ because $5 \times 20 = 100$.

Additional Example: Fraction to Percent

In some cases we will be able to reduce the fraction first, before converting to a percent. If we're able to reduce first, that makes converting to a percent easier afterwards.

For example, we can use this technique on the fraction $\frac{9}{12}$. The numerator and denominator have a common factor of $3$, so

$$\frac{9}{12} = \frac{3 \times 3}{3 \times 4} = \frac{\cancel{3} \times 3}{\cancel{3} \times 4} = \frac{3}{4}$$

Now, we can multiply the numerator and denominator by $25$. This turns the denominator into $100$, from which we get a percent:

$$\frac{3}{4} \times \frac{25}{25} = \frac{75}{100} = \boxed{75\%}$$

Using Proportions

If we can't figure out what number to multiply the numerator and denominator by, there's a trick for that. We know we want the denominator to be $100$, but don't know about the numerator. To find it, we set up a proportion:

$$\frac{3}{4} = \frac{x}{100}$$

The value of $x$ is the numerator we want to get. Multiply both sides of this equation by $100$:

$$\begin{align} \frac{3}{4} \times 100 &= \frac{x}{\cancel{100}} \times \cancel{100} \\ 75 &= x \end{align}$$

Finally, we can turn this into a percentage, just like before:

$$\frac{75}{100} = \boxed{75\%}$$

Luckily, we get the same numerator we did before. We did the same problem in two different ways, so we should get the same answer.

Converting Percents to Fractions

This appears to be even easier. Just put the percent on top of $100$, and remove the percent sign. $$37\%=\frac{37}{100}$$

However, you might end up with a situation where the fraction is not reduced in some way. In that case, act like it's a normal fraction problem and reduce it. Find the common factor between the numerator and denominator. Then cancel out that factor.

$$25\%=\frac{25}{100}= \frac{\cancel{25} \times 1}{\cancel{25} \times 4} = \boxed{\frac{1}{4}}$$

We can also convert between decimals and percents, or between decimals and fractions.