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# Animal Populations

## Task

Suppose $P$ and $Q$ give the sizes of two different animal populations, where $Q > P$. In (a)–(f), say which of the given pair of expressions is larger. Brieﬂy explain your reasoning in terms of the two populations.

$P + Q$ and $2P$

$\displaystyle{\frac{P}{P + Q}}$ and $\displaystyle{ \frac{P + Q}{2}}$

$(Q - P)/2$ and $Q - P/2$

$P + 50t$ and $Q + 50t$

$\displaystyle{\frac{P}{P + Q}}$ and $0.5 $

$\displaystyle{\frac{P}{Q}}$ and $\displaystyle{\frac{Q}{P}}$

## IM Commentary

In this task students have to interpret expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations. For example, $ \frac{P}{P+Q}$ is the fraction that population $P$ makes up of the combined population $P+Q$.

Although the context is quite thin, posing the question in terms of populations rather than bare numbers encourages students to think about the variables as numbers and provides avenues for them to use their common sense in explaining their reasoning. This encourages them to see expressions as having meaning in terms of operations, rather than seeing them as abstract arrangements of symbols.

(Task adapted from Algebra: Form and Function, McCallum et al., Wiley 2010. Solutions for (e)-(f) were written by Ashley Trujillo , Benjamin Rivera, James Stoddard, Jian Feng Huang, Lisa Brown, Lisa Orton, Moses Baca, Ruth Ramirez, and Saul Goldblatt.)

## Solution

The expression $P+Q$ is larger.

The expression $P+Q$ gives the total size of the two populations put together.

The expression $2P$ gives the size of a population twice as large as $P$.

Putting the smaller population together with the larger yields more animals than merely doubling the smaller.

Another way to see this is to notice that $2P=P+P$, which is smaller than $P+Q$ because adding $P$ to $P$ is less than adding $Q$ to $P$.

The expression $\frac{P+Q}2$ is larger.

The total size of the two populations put together is $P+Q$, so the expression $\frac P{P+Q}$ gives the fraction of this total belonging to $P$. Since $P \lt P+Q$, this will be a number less than 1. For instance, if $P=100$ and $Q=150$, this fraction equals $100/(100+150)=0.4=40\%$.

The average or mean size of the two populations is their sum divided by two, or $\frac{P+Q}2$. This will be a number between $P$ and $Q$, so it is larger than 1 (since $P$ and $Q$ describe animal populations). For instance, if $P=100$ and $Q=150$, the average is $(100+150)/2=125$.

The expression $Q-P/2$ is larger.

The expression $(Q-P)/2$ gives half the difference between $P$ and $Q$. For instance, if $Q=150$ and $P=100$, half the difference is $(150-100)/2=25$.

The expression $Q-P/2$ gives the difference between $Q$ and a population half the size of $P$. For instance, if $Q=150$ and $P=100$, this difference equals $150-100/2=100$.

To see why the second of these is bigger, write \begin{equation*} (Q-P)/2=Q/2-P/2 \end{equation*} In the expression $Q-P/2$, we subtract $P/2$ from $Q$. But in $(Q-P)/2$, we subtract the same value, $P/2$, from a smaller amount, $Q/2$.

The expression $Q+50t$ is larger.

In both expressions, the same value, $50t$, is added to the population.

Since $P \lt Q$, adding $50t$ to $P$ results in a smaller value than adding the same amount to $Q$.

The expression $0.5$ is larger.

The total size of the two populations put together is $P+Q$, so the expression $\frac P{P+Q}$ gives the fraction of this total population belonging to $P$. Since there are fewer animals in population $P$ than $Q$, this fraction is less than $\frac12$. For instance, if $P=100$ and $Q=150$, this fraction equals $100/(100+150)=0.4$.

$\frac{P}{Q}$ and $\frac{Q}{P}$ can be interpreted in two different ways.

$\frac{P}{Q} $ can be interpreted as a unit rate, namely, the number of animals in population $P$ for every 1 animal in population $Q$. Similarly, $\frac{Q}{P} $ can be interpreted as the number of animals in population $Q$ for every 1 animal in population $P$. Since there are more animals in population $Q$, the unit rate $\frac{Q}{P}$ will be greater than the unit rate $\frac{P}{Q}$.

For example, if $P=100$ and $Q=150$, then $\frac{100}{150} = \frac23$, so there would be $\frac23$ of an animal in population $P$ for every 1 animal in population $Q$, while $\frac{150}{100} = \frac32$, so there would be $\frac32$ of an animal in population $Q$ for every 1 animal in population $P$.

$\frac{P}{Q}$ can also be interpreted as the fraction that population $P$ is of population $Q$. Since there are fewer animals in population $P$, as a fraction of the population of $Q$ it will be less than 1. Similarly, $\frac{Q}{P}$ can also be interpreted as the fraction that population $Q$ is of population $P$. Since there are more animals in population $Q$, as a fraction of the population of $P$ it will be greater than 1.

For example, if $P=100$ and $Q=150$, this fraction equals $\frac{100}{150} = \frac23$, so there are $\frac23$ as many animals in population $P$ as there are in population $Q$, while $\frac{150}{100} = \frac32$, so there are $\frac32$ as many animals in population $Q$ as there are in population $P$.

Some people think it is awkward to talk about fractions of animals, so here is another way to think about it:

## Animal Populations

Suppose $P$ and $Q$ give the sizes of two different animal populations, where $Q > P$. In (a)–(f), say which of the given pair of expressions is larger. Brieﬂy explain your reasoning in terms of the two populations.

$P + Q$ and $2P$

$\displaystyle{\frac{P}{P + Q}}$ and $\displaystyle{ \frac{P + Q}{2}}$

$(Q - P)/2$ and $Q - P/2$

$P + 50t$ and $Q + 50t$

$\displaystyle{\frac{P}{P + Q}}$ and $0.5 $

$\displaystyle{\frac{P}{Q}}$ and $\displaystyle{\frac{Q}{P}}$

## Comments

Log in to comment## Ellen Whitesides says:

over 5 yearsAnother way to see the comparison in part c without doing any algebraic manipulations is to use the difference between the populations (Q - P) as a base number and compare both expressions in part c to that difference. The first expression is half the difference between the populations so that will always be less than the difference between the populations. The second expression is the difference between population Q and half of population P. Since half of population P is less then P the difference between that smaller number and population Q will be more than the difference between the two populations. So we are comparing an expression that is less than the difference between the populations (the first) and an expression that is more than the difference between the populations (the second). So the first expression will always be less than the second expression.

This is a similar way that we teach students to compare fractions in younger grades. Think of a base number (like 1/2 or 1) and compare each of the fractions to that base number.

## Noah Heller says:

over 5 yearsPosing the question in terms of populations implies that both P and Q have associated units (number of animals). Without specifying the units of t, doesn't (d) become problematic? If the idea is to contextualize the variables in a real world example, the units, or lack of units seem important in building a conceptual understanding of relative magnitudes.

## Kristin says:

over 5 yearsThat's an interesting question. I think that coefficients in equations often have implicit units. For example, if you have a a car moving at a constant velocity and someone tells you that the equation for the distance it has traveled as a function of time is

d=60t

then the units for d and t define the units for 60, don't they? In other words, if t is the number of hours and d is the number of miles, then the car must be moving at 60 mph, yet the units are not specified anywhere. Those are the only units 60 can have if the equation is true.

So I'm thinking the same thing applies here: whatever the units are for the variable t, the 50 must have a unit "number of animals per unit associated with t" for the expression even to make sense. So you don't really need to know what the units for t (or 50) are to know that P+50t < Q+50t if you assume that the expressions are well-defined in the first place.

## Jason says:

over 6 yearsOn my browser, the answer to part (b) has a Tex compile error or something. I see a dollar sign and then the sentence ends.

BTW, not suggesting these be added, but the task made me think of a couple of additional comparison ideas:

P/(P+Q) and 0.5

P/Q and Q/P

## Cam says:

about 5 yearsTwo new parts added as well (rather belatedly!)

## Bill says:

over 6 yearsThanks, fixed now.