## Proportions

#### Common Questions

#### How is a proportion different from a ratio?

Stellar question! A proportion is an equation that simply lets us know that two ratios are equal. For example, we know that $\frac{16}{8}$ and $\frac{4}{2}$ are equivalent because dividing each ratio gives us the same value of $2$.

Read below to learn more.

#### Common Questions

Stellar question! A proportion is an equation that simply lets us know that two ratios are equal. For example, we know that $\frac{16}{8}$ and $\frac{4}{2}$ are equivalent because dividing each ratio gives us the same value of $2$.

Read below to learn more.

### Ratio vs Proportion… What's the difference?

A ratio is a comparison of two quantities, whereas a proportion is an equation that lets us know two ratios are equal to each other. For example, we know that $\frac{16}{8}$ and $\frac{4}{2}$ are equivalent because dividing each ratio gives us the same value of $2$.

If you're not quite sure what ratios are, click below for a refresher!

If you do know your ratios, match the ratios that are equivalent to each other:

We can express proportions in two different ways:

Format | Example |
---|---|

$a:b = c:d$ | $16:8 = 4:2$ |

$\frac{a}{b} = \frac{c}{d}$ | $\frac{16}{8} = \frac{4}{2}$ |

But, why does understanding ratios and proportions matter?

### Why does understanding ratios and proportions matter?

Why does what I'm learning matter? That's always an important question to ask yourself when learning something new. Understanding ratios and proportions matters because we can use them to analyze data and calculate unknown variables.

One of which is identifying underrepresented minorities. An underrepresented minority can be defined as a group whose percentage of the population in a given group is lower than their percentage of the population in the country.

In this lesson we are going to use proportions to analyze underrepresentation in America's medical workforce. To do this, we'll use data from the Association of American Medical Colleges and the US Census Bureau.

### Using Ratios to Analyze Data

Full disclosure: since we're working with national populations, we're going to see some pretty large numbers. But don't sweat it! We’ve put all the numbers in thousands and rounded them to the nearest whole number to make things a little bit easier for you. We’re also going to take care of the calculations for you! All you have to do is trust the process.

Race and Ethnicity | Number of Active Doctors in America (2018), in thousands | Number of People in America (2018), in thousands |
---|---|---|

White | 516 | 197,535 |

Asian | 157 | 18,545 |

Hispanic | 54 | 59,640 |

Black or African | 46 | 40,861 |

American Indian or Alaska Native | 3 | 2,420 |

Other | 17 | 7,100 |

Unknown | 127 | 794 |

Total | 920 | 326,895 |

If we're trying to see if Black doctors are an underrepresented minority, which type of ratio should we use?

Remember, a part-to-part ratio compares one part of a group to another part of the same group. A part-to-whole ratio compares one part of a group to the total parts in the group

Perfect! Looks like we're ready to start. To help you out with each step of the process, we'll first walk through an example that analyzes the representation of Hispanic doctors in America, and then you'll do the same thing for Black doctors.

Step 1: Ratio of Doctors

Here's an example of how we calculate the percentage of Hispanic doctors using a part-to-whole ratio:

$\frac{54}{920} \approx 0.06 = 6\%$Remember, the above numbers are in thousands. So, there are really about 54,000 Hispanic doctors and about 920,000 total doctors in the US.

Your turn! Make a part-to-whole ratio that compares the number of Black doctors to the total number of doctors:

Great! We can now convert this ratio into a decimal and a percentage:

$\frac{46}{920} \approx 0.05 = 5\%$Step 2: Ratio of Population

Here's an example of how we calculate the percentage of Hispanic people in the US population using a part-to-whole ratio:

$\frac{59,640}{326,895} \approx 0.18 = 18\%$Your turn! Make a part-to-whole ratio that compares the number of Black people to the total number of people:

Great! We can now convert this ratio into a decimal and a percentage.

$\frac{40,861}{326,895} \approx 0.13 = 13\%$What do these numbers mean?

Black people make up 13% of the US population, but only 5% of US doctors, so we can clearly see that the ratio of Black doctors to total doctors is significantly LESS THAN the ratio of Black people to total people in America.

This means that we do not have a proportion here because Black people are underrepresented in the field of medicine.

In this scenario, we worked with really big numbers, which can be really annoying to deal with when doing algebra. This is where pie charts could come in super handy since they help us visualize part-to-whole relationships.

White | |

Asian | |

Hispanic | |

Black or African | |

American Indian or Alaska Native | |

Other | |

Unknown |

US Doctors

US Population

US Doctors

US Population

Hispanic |

Black or African |

If you compare the pie charts for US Doctors and the US Population, you can see that the “slice” for Black US doctors is significantly smaller than the “slice” for the Black population.

This shows us that Black doctors are an underrepresented group, which aligns with the calculations we made above. If you compare the slices and calculated values for Hispanic doctors with the corresponding slices and values for the Hispanic population, you can see that they, too, are an underrepresented minority in the medicinal field.

#### Why don't we have more BIPOC doctors?

Many factors contribute to racial disparities in the medical workforce. Research points to complex reasons including economic barriers, lack of role models, limited access to educational opportunities, and implicit and explicit biases.

But while racial disparity in education continues, doctors in the U.S. are caring for an increasingly diverse group of patients, and it is more important than ever to reflect that same diversity in medicine.

Research has shown that having more physicians of color can help improve health outcomes for patients (source). See the photo below for the receipts 👀👀👀.

Bottom line is this: diversity matters.

### Using Ratios to Solve for Unknown Variables

Now, we're going to use ratios and proportions to figure out how many Black doctors there would need to be in order to reflect the representative numbers of the Black population in America. Let's think about what we know and what's unknown:

Number of Doctors, in thousands | Number of People, in thousands | |
---|---|---|

Black | $\boldsymbol\color{#00bbff}x$ | 40,861 |

Total | 920 | 326,895 |

Solve for $\boldsymbol\color{#00bbff}x$

In order for the number of Black doctors to reflect the representative numbers of the Black population in America, the following proportion must exist:

$\frac{{\color{#00bbff}\text{Black doctors}}}{\text{total doctors}}=\frac{\text{Black people}}{\text{total people}}$Since we're trying to figure out the number of Black doctors needed, we're going to...

Replace Black doctors with an $\boldsymbol\color{#00bbff}x$ to represent the unknown number we're solving for:

$\frac{{\boldsymbol\color{#00bbff}x}}{\text{total doctors}}=\frac{\text{Black people}}{\text{total people}}$And replace the categories with their appropriate values from the table:

$\frac{{\boldsymbol\color{#00bbff}x}}{920}=\frac{40{,}861}{326{,}895}$

Perfect, now we have an equation that we can solve. One way we can solve this proportion is by using cross multiplication and division:

Step 1: Multiply across the corners

$\frac{{\color{#00bbff}\colorbox{yellow}{$\boldsymbol x$}}}{\colorbox{lightgreen}{920}}=\frac{\colorbox{lightgreen}{40{,}861}}{\colorbox{yellow}{326{,}895}}$$\colorbox{yellow}{326{,}895{\color{#00bbff}$\boldsymbol x$}}=\colorbox{lightgreen}{40{,}861}\times\colorbox{lightgreen}{920}$Step 2: Divide both sides of the equation to isolate $\boldsymbol\color{#00bbff}x$

$\frac{\colorbox{yellow}{326{,}895{\color{#00bbff}$\boldsymbol x$}}}{\colorbox{yellow}{326{,}895}}=\frac{\colorbox{lightgreen}{40{,}861}\times\colorbox{lightgreen}{920}}{\colorbox{yellow}{326{,}895}}$$\frac{\sout{\colorbox{yellow}{326{,}895}}{\colorbox{yellow}{\color{#00bbff}$\boldsymbol x$}}}{\sout{\colorbox{yellow}{326{,}895}}}=\frac{\colorbox{lightgreen}{40{,}861}\times\colorbox{lightgreen}{920}}{\colorbox{yellow}{326{,}895}}$${\boldsymbol\color{#00bbff}x}=\frac{\colorbox{lightgreen}{40{,}861}\times\colorbox{lightgreen}{920}}{\colorbox{yellow}{326{,}895}}$${\boldsymbol\color{#00bbff}x}\approx 115$Remember, the above numbers are in thousands. This means that the US would need to have about $\color{#00bbff}115{,}000$ Black doctors to make up a representative portion of total doctors in the US.

Absolutely! That's the beauty of math. There's almost always more than one way to get to the correct answer. Another approach you can take to solve this proportion is to...

Step 1: Multiply the two known values that are diagonally across from each other

$\frac{{\color{#00bbff}\boldsymbol x}}{\colorbox{lightgreen}{920}}=\frac{\colorbox{lightgreen}{40{,}861}}{326{,}895}$$920\times40{,}861=\colorbox{lightgreen}{37{,}551{,}259}$Step 2: Divide the product by the number diagonally across from $\boldsymbol\color{#00bbff}x$

$\frac{\colorbox{lightgreen}{37{,}551{,}259}}{\colorbox{yellow}{326{,}895}}\approx 115$(which, as a quick reminder, represents about $\color{#00bbff}115{,}000$ Black doctors)

Tomato, tomata. They both get us to the same result, so you can choose your own adventure!

👌🏿👌🏾👌🏼👌🏼👌🏻

#### Proportion Practice

If $\frac{{\boldsymbol\color{#00bbff}x}}{2}=\frac{9}{3}$,

then ${\boldsymbol\color{#00bbff}x}=$