Aplicações e problemas

Ratio and Proportion Word Problems

Ratio and proportion word problems are really applicable to real life when you need to calculate costs, determine time limits, or even convert between measurement systems! Here's a breakdown of what a ratio and proportion word problem might look like. We'll also see how to understand what the problem is asking, and how to solve it!

It's important to note that in Step Two, when you plug the appropriate values into the proportion, the units in both the numerators must be the same, and the units in both the denominators must be the same. For example, in the graphic above, the units in the numerators are $\textit{books}$. The units in the denominators are $\textit{hours}.$ If the units given to you in the word problem are not consistent, you must somehow make them match. This next problem will show you how. For certain units we will use conversion factors to help make the units the same. We don't always have conversion factors, but it's good to know what ones we do have. Here are some examples:

1. Units of length: convert between feet and meters
2. Units of mass: convert between grams and kilograms
3. Units of time: convert between minutes and days

Once that's done, the numerators will have the same units as each other. Likewise for the denominators. Then we'll be able to solve the proportion just using algebra. The equation will have a variable times a number on one side, and a number on the other side. Thus we can solve the equation in one step using division.

Practice Problem

Solution

As a reminder, we want to solve this word problem:

Trisha can make $5$ pairs of earrings in $1$ week. Assuming she works at this constant rate, how many complete pairs of earrings can she make in $10$ days?

$\textbf{Step One:}$ Identify the Knowns and Unknown

Trisha can make $5$ pairs of earrings (known) in $1$ week (known). Assuming she works at this constant rate, how many complete pairs of earrings (unknown) can she make in $10$ days (known)?

$\textbf{Step Two:}$ Plug values into proportion

$$\dfrac{5 \text{ earrings}}{1 \text{ week}} = \dfrac{x \text{ earrings}}{10 \text{ days}}$$

But wait! the units in the denominators do not match! We need to find a way to make them the same.

Luckily, we know that there are $7$ days in a week, so instead of writing $1$ week as the units in the first denominator, we can write $7$ days. Now we have:

$$\dfrac{5 \text{ earrings}}{7 \text{ days}} = \dfrac{x \text{ earrings}}{10 \text{ days}}$$

$\textbf{Step Three:}$ Solve the proportion (using cross multiplication)

We can take out the units for now!

$$\begin{align} \dfrac{5}{7}&= \dfrac{x}{10}\\ 50 &= 7x\\ x&=\frac{50}{7}\\ x &\approx 7.14 \end{align}$$

So, Trisha can make around $7.14$ pairs of earrings in $10$ days. We round this down to $7$ pairs since this is the amount of $\textit{complete}$ pairs of earrings she can make. (She can complete $7$ pairs, then start working on an $8$th pair in the time she has left.)

Thus, the answer is $7$ pairs of earrings.

Math in Music: Rhythm

Now that the basic concepts of ratios and proportions have been covered, we are going to dive a little deeper into working with them through the subject of music.

Ratios are found in music within two main concepts: Rhythm/Polyrhythm and Pitch Intervals. These may seem a little foreign right now, but I promise they won't be too hard to understand. This explanation will focus on Rhythm and Polyrhythm, but the explanation on Pitch Intervals has been linked above.

Rhythm/Polyrhythm

All rhythms are referred to by names based off of fractions/ratios. There are whole notes ($\frac{4}{4}$ of a measure), half notes ($\frac{2}{4}$ of a measure), quarter notes ($\frac{1}{4}$ of a measure), eighth notes ($\frac{1}{8}$ of a measure), sixteenth notes ($\frac{1}{16}$ of a measure), etc. and the name denotes how much of a $4$ beat measure the note takes up, as shown below. For our purposes, we are always going to use a $4$ beat measure to describe our ratios.

Not all notes and rhythms are based on groups of $4$, notes can be based on any division! Triplets can be used to split a four beat measure into groups of $3$ or $6$ instead of groups of $2$ or $4$.

The brackets over the notes tell how many notes are played in the span of the bracket. You can see the $3$ placed above all of the notes. This is because each group of $3$ takes a set amount of time. The half note triplets take up $4$ beats (one full measure) with $3$ notes, the quarter note triplets take up $2$ beats (one half of a measure) with $3$ notes, and the eighth note triplets take up $1$ beat (one quarter of a measure) with $3$ notes. As ratios, these could be written as $3:4$, $3:2$, and $3:1$. If we wanted to write these notes as fractions, they could be called $\frac{1}{3}$, $\frac{1}{6}$, and $\frac{1}{12}$ notes.

When we start combining these rhythms to be played at the same time, we get what are called polyrhythms. These are normally defined by ratio of how many notes are being played in the same amount of time.

The ratios in this case are going to be written in terms of the top line against the bottom line. The first measure is a $4:3$ while the second measure is a $3:2$. These same polyrhythms can be written without needing a second line to compare by once again using the brackets above the notes.

The groupings on the top line both take up an entire measure, $4$ beats. That means that the first measure on the top line shows a $5:4$ ratio (aka five-lets), and the second shows a $7:4$ (aka seven-lets) ratio. The bottom groupings take up $2$ beats , so they show a $5:2$ and $7:2$ ratio.

We're gonna try a practice question that builds on some of these concepts.

In order to solve this, we can set it up as a proportion. $$\begin{align} \frac{4}{3} &= \frac{24}{x}\\ 4x &= 72\\ x &= 18 \end{align}$$

Over the course of all measures, there is a $4:3$ polyrhythm if Libby plays $18$ notes.

You can see that in every measure, there is a $4:3$ polyrhythm.

Wrap-Up

Ratios can be used in music both to describe and perform rhythms. While musicians may not be directly thinking about the ratios that appear in music, they are part of the underlying music theory and the nature of sound.

(Video) Proportion Examples and Word Problems

by Math Meeting

This video by Math Meeting explains proportions and works through a few examples.

$\large{\text{Summary}}$

A proportion is two ratios that are set equal to one another.

An example of a proportion would be: $$\frac{1\text{ dollar}}{10\text{ pesos}}=\frac{10\text{ dollars}}{100\text{ pesos}}$$ As you can see, they are equivalent. The units are the same, and then we notice the fractions are equivalent. The fractions are equivalent because $\frac{10}{100}$ can be reduced to $\frac{1}{10}$.

If you multiply the numerator and denominator of the first ratio by $10$, you get the second ratio. If you have $10$ times more dollars, you will have $10$ times more pesos.

The standard proportion problem is as follows: $$\frac5x=\frac{10}3$$ You are given two proportions and you want to solve for $x$.

The best way to solve these types of equations is with cross multiplication.

To cross multiply, you take the product of the numerator of the first ratio and the denominator of the second ratio. Set that equal to the product of the numerator of the second ratio and the denominator of the first ratio.

Let's solve this problem now. $$\begin{align} \frac5{x}=\frac{10}3\Rightarrow5\cdot3&=x\cdot10\\ 15&=10x\\ \frac{15}{10}&=\frac{10x}{10}\\ \frac{^3\cancel{15}}{_2\cancel{10}}&=\frac{\cancel{10}x}{\cancel{10}}\\ \boxed{\frac32}&=x \end{align}$$

Another Example

Here's a word problem using proportions.

The currency in Spain is the euro. If I exchange $\$250$ at the airport, how many euros will I get in return? ($1$ euro = $1.38$ dollars)

The parentheses at the bottom give us our first ratio. $$\frac{1\text{ euro}}{1.38\text{ dollars}}$$

Within the question itself is our second ratio. $$\frac{x\text{ euros}}{250\text{ dollars}}$$

Set them equal to each other. Remember that the same units have to be in the top/bottom. $$\frac{1}{1.38}=\frac{x}{250}$$

To solve, we cross multiply. $$\begin{align} \frac{1}{1.38}=\frac{x}{250}\Rightarrow1\cdot250&=1.38\cdot x\\ 250&=1.38x\\ \frac{250}{1.38}&=\frac{1.38x}{1.38}\\ \frac{250}{1.38}&=\frac{\color{#29A603}{\cancel{\color{black}{1.38}}}x}{\color{#29A603}{\cancel{\color{black}{1.38}}}}\\ 181.16&=x \end{align}$$

You will get approximately $181.16$ euros!

Math in Music: Pitch

Now that the basic concepts of ratios and proportions have been covered, we are going to dive a little deeper into working with them through the subject of music.

Ratios are found in music within two main concepts: Rhythm/Polyrhythm and Pitch Intervals. These may seem a little foreign right now, but I promise they won't be too hard to understand. This explanation is going to cover Pitch and Intervals, and Rhythm/Polyrhythm is linked above.

Pitch and Intervals

When two notes are being played at once, there is an Interval between the two pitches. Every pitch can be written as both a note (think $A, B, C,$ and their placings on a staff) or as a measurement of Hertz ($\text{Hz}$). We are going to use $A=440 \text{ Hz}$ as a baseline to demonstrate how ratios are used in intervals and tuning in music.

As notes move in octaves, there is always a $2:1$ ratio between the notes. Based on A at $440\text{ Hz}$, the $A$ one octave higher is going to be measured at $880\text{ Hz}$, and the $A$ one octave lower is going to be measured at $220 \text{ Hz}$.

This pattern continues in both directions (up and down) from $A$ in the second space until eventually the note is outside of the range of human hearing.

The ratios between notes exist for every interval and are the basis for tuning many instruments as well as playing in tune with another musician. For example, were going to use the next two most common intervals, the perfect fifth and the major third. Using $A=440 \text{ Hz}$ as a basis again, we are going to use the intervals of $3:2$ to find the measurement of the perfect fifth ($E$), and a ratio of $5:4$ to find the major third ($C$# or $C$ Sharp).

$$\frac{3}{2}=\frac{x}{440}$$

In this case we are going to put $440$ as a denominator because we want to find the $\text{Hz}$ of a note that is higher than our original $A$.

$$\begin{align} \frac{3}{2}&=\frac{x}{440}\\ 2x&=1320\\ x&=660 \end{align}$$

Our perfect fifth of $E$ will be at $660\text{ Hz}$ when it is perfectly in tune. Now lets calculate the pitch of the major third, $C$#. Once again, the $440$ is going to be in the denominator as we are finding a note higher than $A$.

$$\begin{align} \frac{5}{4}&=\frac{x}{440}\\ 4x&=2220\\ x&=550 \end{align}$$

Our major third of $C$# is going to be at $550\text{ Hz}$ when perfectly in tune. Combining all of these notes, you get a nice consonant $A$ major chord. Just to make sure our musical brains are keeping up with the math, here is what this looks like on a staff.

Let's try a practice problem to see if everything is sticking.

There are two steps to this problem but they can be done in either order.

First, were going to double the pitch of Brandon's low $A$ so we can get into a range we are used to working with. $$220\cdot 2 = 440$$

Now that we have Brandon's note an octave higher, we can find the major sixth above that to find Libby's $F$#. $$\begin{align} \frac{5}{3}&=\frac{x}{440}\\ 3x&=2200\\ x&=733.33 \end{align}$$

Alternatively, we can find the major sixth above Brandon's original low $A$, and then double it to find the same note.

$$\begin{align} \frac{5}{3}&=\frac{x}{220}\\ 3x&=1100\\ x&=366.67 \phantom{00} \text{This is rounded}\\ 366.67 \cdot 2 &= 733.33 \end{align}$$

Either way, Libby must be playing $F$# at $733.33 \text{ Hz}$ in order to be perfectly in tune with Brandon. This can be written two different ways on the staff with the low $A$ being shown in either the treble or bass clef.

Wrap-Up

Every note has its own speed, and ratios between the speed of different notes create harmony. You can use proportions to find the perfect tuning of these harmonies, even if musicians don't necessarily think about the math when they are playing!