Rates and Unit Rates: Free Response

Take the first line. It puts $78$ tomato seedlings over $26$ pepper seedlings, and both quantities are counted in seedlings, so the same unit sits above and below and it cancels exactly as a common factor does:

$\frac{78 \text{ seedlings}}{26 \text{ seedlings}} = \frac{78}{26} = \frac{3}{1}.$

Dividing above and below by the greatest common factor $26$ leaves $3$ to $1$, and that answer is a bare number: three tomato seedlings for every pepper seedling, with no unit left attached, because the seedlings above and below have cancelled each other out.

The second line puts $78$ seedlings over $26$ liters. Seedlings and liters measure different things, so nothing cancels:

$\frac{78 \text{ seedlings}}{26 \text{ liters}}.$

The units survive the division and have to be spoken with the number. That is exactly what makes the second line a rate: it compares two quantities measured in different units, so its answer carries a compound unit, seedlings per liter, instead of being a pure number. The first line is a ratio in the ordinary sense, the same unit above and below, cancelling away.

The phrase seedlings per liter names liters as the denominator, so it asks a precise question: how many seedlings does one single liter account for? Splitting the $78$ seedlings into $26$ equal shares, one share for each liter, is exactly what division does, and the size of one share is the answer:

$78 \div 26 = 3.$

So the watering line reads

$\frac{78 \text{ seedlings}}{26 \text{ liters}} = 3 \text{ seedlings per liter},$

and the number $3$ counts seedlings, not liters: one liter of rainwater was enough for three seedlings.

This unit rate is an equivalent rate rather than a new one. Dividing both quantities of $78$ to $26$ by $26$ gives $3$ to $1$, which is the same scaling move that keeps a ratio equal, applied so that the second quantity lands on $1$.

Liters per seedling names seedlings as the denominator, so this time the seedlings are what the water is shared out among:

$26 \div 78 = 0.3333\ldots \approx 0.33.$

The quotient does not terminate, so it is rounded, and two decimal places is enough to read at a glance. Exactly, it is one third of a liter, which checks against the other version: if each seedling takes a third of a liter, three seedlings take a whole liter.

Both versions describe the same watering because both come from the same pair of quantities; only the direction of the division changed. What differs is the question each one answers. Three seedlings per liter answers how many seedlings one liter of rainwater covers, so it is what you want when the water is already collected and you are deciding how many seedlings it will support: multiply it by the liters you have. About $0.33$ liters per seedling answers how much water a single seedling takes, so it is what you want when the planting is already decided and you are working out how much water to collect: multiply it by the seedlings you intend to plant.

The word per is the signpost that tells the two apart. Whatever follows it names the quantity underneath, the quantity being divided by, and it is that quantity whose count is driven down to $1$. Read the phrase before you divide, and the order of the division is settled for you.

Dollars per meter names meters as the denominator, so the price is what gets shared out: divide the dollars by the meters.

$19 \div 6 = 3.1\overline{6} = 3.1666\ldots$

The division never terminates, because sixths repeat, so the exact price of one meter cannot be written down as a finite decimal at all. That leaves two choices: keep the price as the exact quotient of $19$ by $6$, or round it, which means any decimal price per meter is a rounded one. When you do round, the place to stop comes from the situation rather than from the digits.

The deciding digit for the hundredths place is the next one along, a $6$, which is past the halfway mark, so the hundredths digit goes up:

$3.1666\ldots \approx 3.17.$

Money is counted in whole cents, so two decimal places is where a price naturally stops. Nobody can hand over $3.1666\ldots$ dollars, and quoting more decimal places would claim a precision that a till cannot act on. Keep the unit attached: the answer is about $3.17$ dollars per meter, not a bare $3.17$.

Take the exact route first. Twenty-seven meters is four and a half reel lengths, since $6 \times 4.5 = 27$, so scale both quantities by $4.5$, which is the equivalent-ratio move that leaves a rate unchanged:

$\frac{19 \text{ dollars}}{6 \text{ meters}} = \frac{19 \times 4.5}{6 \times 4.5} = \frac{85.50 \text{ dollars}}{27 \text{ meters}}.$

So $27$ meters costs $85.50$ dollars, and no rounding entered that calculation anywhere.

Now the till's route. It holds a price to the nearest cent and multiplies:

$3.17 \times 27 = 85.59.$

The two totals are not the same:

$85.59 - 85.50 = 0.09.$

Neither multiplication is wrong. The gap comes from the rounding in part A: raising $3.1666\ldots$ to $3.17$ added a third of a cent to the price of every meter, and twenty-seven of those thirds of a cent come to a whole nine cents. The rounding itself is tiny, a third of a cent on a single meter. What makes it visible is that multiplying a rounded figure repeats that same small error once per unit.

Start with the exact total. Forty-eight meters is eight reel lengths, since $48 \div 6 = 8$, so scale both quantities by $8$:

$8 \times 19 = 152.$

The order costs exactly $152$ dollars. Now the rounded per-meter price:

$3.17 \times 48 = 152.16.$

The clerk's premise is perfectly true. The rounded price $3.17$ really is within half a cent of $3.1666\ldots$, and it is off by only a third of a cent. What is false is the conclusion drawn from it, because that half-cent is a limit on the error in the price of ONE meter, and an order of $48$ meters multiplies the error by $48$ at the same moment that it multiplies the price by $48$:

$0.00\overline{3} \times 48 = 0.16.$

Sixteen cents, which is exactly the gap between the two totals. So the error in the total is not bounded by half a cent at all; it grows with the size of the order, while the error per meter stays fixed. On an order ten times larger it would be ten times larger again.

The habit that avoids all of this is to scale from the exact rate and round the total at the end, rather than round a unit rate first and then multiply the rounded figure. Rounding is a last step, not a first one.

Dollars per ounce puts ounces underneath, so each bag's price is divided by that bag's own weight. The small bag:

$2.99 \div 13 = 0.23.$

The large bag:

$5.04 \div 21 = 0.24.$

Both figures now answer the same question, what one single ounce of seed costs, so they are finally on the same footing and can be set side by side. For a price, the lower figure is the better buy, because it is less money for the same amount of seed, and

$0.23 < 0.24,$

so the small bag is cheaper per ounce, by a cent an ounce.

That is worth pausing on, because the sticker prices give no hint of it. The large bag costs more in total and holds more seed, and neither of those facts settles anything on its own. Only the division does.

The bin's price is already a unit rate: it tells you what one single ounce costs. To predict the cost of many ounces, scale that unit rate up by multiplying, which is the same move as scaling a ratio up from its simplest form:

$0.22 \times 21 = 4.62.$

The ounces underneath the rate pair off with the $21$ ounces being bought, leaving dollars behind, which is why the answer is money and not a rate. So the bin charges $4.62$ dollars for the same weight the large bag holds.

Set that against the large bag's sticker price:

$5.04 - 4.62 = 0.42.$

The bin is $0.42$ dollars cheaper for the same $21$ ounces of seed. Notice that this comparison is fair precisely because both sides now describe the same weight; comparing the bin's rate directly with the bag's sticker price would compare a price for one ounce with a price for twenty-one.

A sticker price answers exactly one question: how much money leaves your pocket. It says nothing at all about how much seed arrives, and here the two bags hold different amounts, so the cheaper sticker is cheaper partly because there is less in the bag. The shopper's comparison silently assumes the two bags hold the same weight, and that assumption is simply false.

It is worth being honest about what happened here, because it is what makes the habit so hard to shake: the shopper's verdict lands on the same bag the division picks out. The reasoning was wrong and the answer still came out right. That is the most dangerous shape a rule can have, because the only way to discover that it agreed with the division this time is to carry out the division, and a rule you can only check by doing the work it was meant to save is not a rule at all. Change the two prices a little and it fails, with nothing on the shelf to warn you.

What does settle the question is a rate. Divide each bag's price by that bag's own weight, so that both figures answer the identical question, what one ounce costs:

$\frac{2.99 \text{ dollars}}{13 \text{ ounces}} \quad \text{against} \quad \frac{5.04 \text{ dollars}}{21 \text{ ounces}}.$

The two weights disappear into the denominators, and what is left is two numbers measuring the same thing, which is the whole reason unit rates are worth computing.

Two conditions have to hold before any two rates may be set against each other. First, the same units: dollars per ounce cannot be compared with dollars per pound until one of them is converted, because a pound is many ounces and the figures are not measuring the same thing. Second, the same order: dollars per ounce and ounces per dollar are different rates from the same data, and mixing them reverses which one looks better. Once both conditions hold, the direction of the reading is decided by what the rate counts. For a price you want the lower figure per unit; for a speed you want the higher.

Liters per minute names minutes as the denominator, so the juice is what gets shared out: the $29$ liters are split equally across the $5$ minutes, one share per minute.

$29 \div 5 = 5.8.$

So the press runs at $5.8$ liters per minute. The quotient is a decimal, which is no obstacle at all: a unit rate is whatever one unit accounts for, whole or not, and half-liters are perfectly real juice.

Multiplying back is a quick check that the share size is right:

$5.8 \times 5 = 29,$

five minutes of $5.8$ liters each rebuilds the $29$ liters the press started from.

An hour is $60$ minutes, and $60$ is $12$ lots of $5$, so the time has to be scaled up by a factor of $12$. Scaling a rate is the equivalent-ratio move from the last lesson: multiply both quantities by the same number and the rate itself is unchanged, only the units it is reported in have moved.

$\frac{29 \text{ liters}}{5 \text{ minutes}} = \frac{29 \times 12}{5 \times 12} = \frac{348 \text{ liters}}{60 \text{ minutes}}.$

Since $60$ minutes is exactly one hour, the press produces $348$ liters per hour.

The per-minute figure agrees, as it must, since nothing about the press has changed. Sixty minutes at $5.8$ liters each gives

$5.8 \times 60 = 348,$

the same total. Two routes, one rate.

Read each statement back as a sentence about the press. The first trainee's rate is

$\frac{29 \text{ liters}}{60 \text{ minutes}},$

which claims that an hour of pressing yields $29$ liters. The press actually yields that in five minutes, so this statement stretches the same juice over twelve times as long and describes a press running twelve times too slowly.

The second trainee's rate is

$\frac{348 \text{ liters}}{5 \text{ minutes}},$

which claims a full hour's juice arrives in five minutes. That is the opposite error, a press running twelve times too fast. Each trainee changed one quantity by a factor of $12$ and left the other where it was, so in each case the pair no longer describes the same machine.

What scaling both quantities preserves is the amount per single unit, which is what the rate actually says. Divide either correct version and the same number comes back:

$29 \div 5 = 5.8, \qquad 348 \div 60 = 5.8.$

Both report $5.8$ liters in one minute, and that is precisely why they are the same rate written in different units. Change one quantity on its own and that quotient changes with it, so the result is a different rate wearing the old rate's label.

Miles per gallon asks how far the van travels on one single gallon, so the gallons are the quantity being divided by, and the miles are the quantity being shared out. The word standing after per always names the denominator, and that is the whole of the rule:

$\frac{279 \text{ miles}}{9 \text{ gallons}} = (279 \div 9) \text{ miles per gallon} = 31 \text{ miles per gallon}.$

The van does $31$ miles on a gallon. The trainee divided the other way round, forming the reciprocal, and then attached the label belonging to the division that was not performed. Nothing was wrong with the arithmetic itself; what was wrong was the order, and then the units written beside it.

A sense check catches this kind of slip without any calculation. A figure of about $0.032$ miles per gallon would mean a whole gallon of fuel carried the van roughly a thirtieth of a mile, a few dozen paces. Whenever a rate comes out absurdly small or absurdly large, suspect the order of the division first.

Dividing the gallons by the miles shares the fuel out among the miles, so what comes back is the fuel that one single mile consumes:

$9 \div 279 = 0.032258\ldots \approx 0.032.$

The units follow the order of the division exactly as they always do: gallons on top, miles underneath, so the reading is gallons per mile, not miles per gallon. Three decimal places is a sensible stopping point here, because two places would flatten the figure to $0.03$, too coarse to separate one van's fuel use from another's.

Both figures describe the same journey, from opposite ends. Thirty-one miles per gallon answers how far the van gets on one gallon; about $0.032$ gallons per mile answers how much fuel one mile costs. Each is exactly the other turned upside down: since $279 \div 9 = 31$, the reversed division $9 \div 279$ is precisely one thirty-first, and $0.032$ is that fraction rounded.

So the trainee's only errors were the order of the division and the label. The number itself is a perfectly good rate, and it answers a question a fleet manager might genuinely ask when budgeting fuel for a known distance.

Work the second van exactly as the first, in both orders:

$208 \div 6.5 = 32, \qquad 6.5 \div 208 = 0.03125.$

So the second van covers $32$ miles on a gallon and burns $0.03125$ gallons in each mile. That second quotient terminates, so unlike the first van's it needs no rounding at all and can be written out exactly.

Now test the clerk's rule. A gallons-per-mile figure counts fuel consumed over a fixed distance, so a larger figure means more fuel burned to cover the very same mile. That is worse, not better, and the rule is upside down. The mistake is treating a bigger number as a better result without asking what the number counts.

The two orderings agree with each other, as they must, since they are two readings of the same logs. In miles per gallon the second van is ahead:

$32 > 31.$

And in gallons per mile the second van is below the first:

$0.03125 < 0.032258\ldots$

More ground on a gallon, less fuel in a mile: the second van is the more efficient, whichever way the rates are written.

What fixes the direction of the comparison is not the size of the number but what the quantity on top is counting. When the top quantity is something you want more of, such as miles travelled, a bigger unit rate is better. When the top quantity is something you want less of, such as fuel burned or dollars spent, a smaller unit rate is better. That single idea is why the lower figure wins on a price per ounce while the higher one wins on a speed, and it is why two rates must be written in the same units in the same order before either reading is attempted.