Converting Units

Put inches on the bottom so they cancel and hands are left. $$36\text{ in}\times \frac{1\text{ hand}}{4\text{ in}}=\frac{36}{4}\text{ hands}$$ The pony is $\boxed{{\displaystyle 9}}$ hands tall. Flip that factor the other way and the units come out as $\frac{{\text{in}}^2}{\text{hand}}$, which is how the units warn you the factor is upside down.

Put fathoms on the bottom so they cancel and feet are left. $$23\text{ fathom}\times \frac{6\text{ ft}}{1\text{ fathom}}=23\times 6\text{ ft}=138\text{ ft}$$ The rope is $\boxed{{\displaystyle 138}}$ feet long. That factor equals $1$ because 6 feet and 1 fathom are the same distance, so it changes the label without changing the rope.

Jars is the unit you want gone, so put it on the bottom. $$270\text{ jars}\times \frac{1\text{ crate}}{18\text{ jars}}=\frac{270}{18}\text{ crates}=15\text{ crates}$$ So 270 jars fill $\boxed{{\displaystyle 15}}$ full crates. The other orientation would have left units of $\frac{{\text{jars}}^2}{\text{crate}}$, which is how the units catch a flipped factor.

Put pounds on the bottom so they cancel and stone is left. $$91\text{ pounds}\times \frac{1\text{ stone}}{14\text{ pounds}}=\frac{91}{14}\text{ stone}$$ Both numbers share a factor of 7, since $91=7\times 13$ and $14=7\times 2$, so the luggage weighs $\boxed{{\displaystyle 13\mathrm{/}2}}$ stone, the same as 6.5 stone. Conversion factors work just as well when the answer lands between two whole numbers.

No single furlong-to-feet factor exists, so chain two through yards, each turned so the unwanted unit is on the bottom. $$4\text{ furlong}\times \frac{220\text{ yd}}{1\text{ furlong}}\times \frac{3\text{ ft}}{1\text{ yd}}=4\times 220\times 3\text{ ft}$$ Furlong cancels against the first denominator and yard against the second, leaving feet, and $4\times 220\times 3=2,640$. So 4 furlongs is $\boxed{{\displaystyle 2640}}$ feet.

There is no yard-to-inch fact, so chain two factors through feet. $$6\text{ yd}\times \frac{3\text{ ft}}{1\text{ yd}}\times \frac{12\text{ in}}{1\text{ ft}}=6\times 3\times 12\text{ in}$$ The yd cancels, then the ft cancels, and only in survives. The cloth is $\boxed{{\displaystyle 216}}$ inches long.

Chain three factors, each turned so the unwanted unit sits on the bottom. $$1\text{ fortnight}\times \frac{14\text{ days}}{1\text{ fortnight}}\times \frac{24\text{ hr}}{1\text{ day}}\times \frac{60\text{ min}}{1\text{ hr}}$$ Fortnight, day, and hr each cancel in turn, leaving minutes. Then $14\times 24=336$ and $336\times 60=20,160$, so 1 fortnight is $\boxed{{\displaystyle 20160}}$ minutes.

Chain the two exchanges through tokens, each factor turned so the unwanted unit is on the bottom. $$9\text{ dollars}\times \frac{4\text{ tokens}}{1\text{ dollar}}\times \frac{3\text{ tickets}}{1\text{ token}}=9\times 4\times 3\text{ tickets}$$ Dollars cancel, then tokens cancel, and only tickets survive, so 9 dollars is worth $\boxed{{\displaystyle 108}}$ prize tickets. A posted exchange rate is a conversion factor like any other.

Write the speed as a fraction and attach two factors, one for the top unit and one for the bottom. $$\frac{45\text{ mi}}{1\text{ hr}}\times \frac{5,280\text{ ft}}{1\text{ mi}}\times \frac{1\text{ hr}}{3,600\text{ s}}=\frac{45\times 5,280}{3,600}\text{ ft/s}$$ The mi cancels and the hr cancels, leaving feet over seconds, and $\frac{237,600}{3,600}=66$. The speed is $\boxed{{\displaystyle 66}}$ ft/s.

Send liters to milliliters first, then let the density turn milliliters into grams, each factor turned so the unwanted unit is on the bottom. $$4\text{ liters}\times \frac{1000\text{ mL}}{1\text{ liter}}\times \frac{1.5\text{ g}}{1\text{ mL}}$$ Liter cancels, then mL cancels, leaving grams. Since $4\times 1000=4000$ and $4000\times 1.5=6000$, the tank holds $\boxed{{\displaystyle 6000}}$ grams. A density is a conversion factor too, one that trades volume for mass.

A square inch is $1\text{ in}\times 1\text{ in}$, so the factor $\frac{2.5\text{ cm}}{1\text{ in}}$ gets used twice, once for each side. $$12{\text{ in}}^2\times \frac{2.5\text{ cm}}{1\text{ in}}\times \frac{2.5\text{ cm}}{1\text{ in}}$$ Both inch units cancel and two centimeter units are left, which make ${\text{cm}}^2$. Then $2.5\times 2.5=6.25$ and $12\times 6.25=75$, so the sticker is $\boxed{{\displaystyle 75}}$ square centimeters. Using the linear factor twice is the same as squaring it.

Lay the three facts down as factors, each turned so the unwanted unit is on the bottom, and walk brindle to floon to tick to mote. $$3\text{ brindle}\times \frac{5\text{ floons}}{1\text{ brindle}}\times \frac{4\text{ ticks}}{1\text{ floon}}\times \frac{6\text{ motes}}{1\text{ tick}}$$ Brindle, floon, and tick each cancel, leaving motes, and $3\times 5\times 4\times 6=360$. The path is $\boxed{{\displaystyle 360}}$ motes long. You never had to picture any of these units, only the relationships between them.

Put leagues on the bottom so they cancel and miles are left. $$8\text{ leagues}\times \frac{3\text{ miles}}{1\text{ league}}=8\times 3\text{ miles}=24\text{ miles}$$ The trail is $\boxed{{\displaystyle 24}}$ miles long.

Put pencils on the bottom so they cancel and gross is left. $$360\text{ pencils}\times \frac{1\text{ gross}}{144\text{ pencils}}=\frac{360}{144}\text{ gross}$$ Both numbers share the factor $72$, and $360÷72=5$ while $144÷72=2$, so the warehouse has $\boxed{{\displaystyle 5\mathrm{/}2}}$ gross, which is two and a half gross.

Start with the amount you have and multiply by the conversion factor oriented so sheets cancels. $$3,500\text{ sheets}\times \frac{1\text{ ream}}{500\text{ sheets}}=\frac{3,500}{500}\text{ reams}$$ The unit sheets appears on top and bottom, so it cancels and reams remains. Now divide. $$\frac{3,500}{500}=7$$ So $3,500$ sheets is $\boxed{{\displaystyle 7}}$ reams.

Start with the wall length and attach the conversion factor with inches on the bottom, so the inches cancel and bricks are left. $$162\text{ in}\times \frac{1\text{ brick}}{9\text{ in}}$$ The inch units cancel, top and bottom. $$162\cancel{\text{ in}}\times \frac{1\text{ brick}}{9\cancel{\text{ in}}}=\frac{162}{9}\text{ bricks}$$ Now divide. $$\frac{162}{9}=18\text{ bricks}$$ So the wall is $\boxed{{\displaystyle 18}}$ bricks long.

Chain two factors through furlongs, each turned so the unwanted unit is on the bottom. $$2\text{ mile}\times \frac{8\text{ furlong}}{1\text{ mile}}\times \frac{40\text{ rod}}{1\text{ furlong}}=2\times 8\times 40\text{ rod}$$ Mile cancels, then furlong cancels, leaving rods. The boundary is $\boxed{{\displaystyle 640}}$ rods long.

Start with $5$ gallons and chain two conversion factors so the unwanted units cancel. Orient each factor with the unit you want to remove on the bottom. $$5\text{ gallon}\times \frac{4\text{ quart}}{1\text{ gallon}}\times \frac{2\text{ pint}}{1\text{ quart}}$$ The gallons cancel against the gallon on the bottom of the first factor, and the quarts cancel against the quart on the bottom of the second factor, leaving only pints. $$5\times 4\times 2=40\text{ pint}$$ So $5$ gallons is $\boxed{{\displaystyle 40}}$ pints.

Chain three factors, each turned so the unwanted unit is on the bottom. $$1\text{ day}\times \frac{24\text{ hours}}{1\text{ day}}\times \frac{60\text{ minutes}}{1\text{ hour}}\times \frac{60\text{ seconds}}{1\text{ minute}}=24\times 60\times 60\text{ seconds}$$ Day, hour, and minute each cancel, leaving seconds. One full day holds $\boxed{{\displaystyle 86400}}$ seconds.

Chain one factor per step, each turned so the old unit cancels. $$3\text{ stacks}\times \frac{5\text{ bundles}}{1\text{ stack}}\times \frac{4\text{ packs}}{1\text{ bundle}}\times \frac{12\text{ cards}}{1\text{ pack}}$$ Stacks, bundles, and packs all cancel, leaving cards, and $3\times 5\times 4\times 12=720$. The collection holds $\boxed{{\displaystyle 720}}$ cards.

Put euros on the bottom of the factor so they cancel and tokens are left. $$15\text{ euros}\times \frac{12\text{ tokens}}{1\text{ euro}}=15\times 12\text{ tokens}=180\text{ tokens}$$ The visitor gets $\boxed{{\displaystyle 180}}$ ride tokens.

Chain the two exchanges through chips, each factor turned so the unwanted unit is on the bottom. $$7\text{ dollars}\times \frac{20\text{ chips}}{1\text{ dollar}}\times \frac{3\text{ points}}{1\text{ chip}}=7\times 20\times 3\text{ points}$$ Dollars cancel, then chips cancel, leaving points. So 7 dollars is worth $\boxed{{\displaystyle 420}}$ prize points.

Attach one factor for the top unit and one for the bottom, each turned so the unwanted unit cancels. $$\frac{72\text{ km}}{1\text{ hr}}\times \frac{1,000\text{ m}}{1\text{ km}}\times \frac{1\text{ hr}}{3,600\text{ s}}=\frac{72\times 1,000}{3,600}\text{ m/s}$$ The km cancels and the hr cancels, leaving meters over seconds, and $\frac{72,000}{3,600}=20$. The train moves at $\boxed{{\displaystyle 20}}$ meters per second.

Use $\frac{5,280\text{ ft}}{1\text{ mi}}$ to clear the miles on top and $\frac{1\text{ hr}}{60\text{ min}}$ to clear the hours on the bottom. $$\frac{15\text{ mi}}{1\text{ hr}}\times \frac{5,280\text{ ft}}{1\text{ mi}}\times \frac{1\text{ hr}}{60\text{ min}}=\frac{15\times 5,280}{60}\text{ ft/min}$$ Feet over minutes is what survives, and $\frac{79,200}{60}=1,320$. The runner is moving at $\boxed{{\displaystyle 1320}}$ feet per minute.

Start with the amount you know and attach the density as a conversion factor, oriented so tablespoons cancels and grams stays. $$8\text{ tablespoons}\times \frac{21\text{ g}}{1\text{ tablespoon}}$$ The $\text{tablespoon}$ on top cancels with the $\text{tablespoon}$ on the bottom. $$8\cancel{\text{tablespoons}}\times \frac{21\text{ g}}{1\cancel{\text{tablespoon}}}=(8\times 21)\text{ g}$$ Now finish the multiplication, $8\times 21=168$, and the only surviving unit is grams. So the jar holds $\boxed{{\displaystyle 168}}$ grams of syrup.

Use the factor $\frac{12\text{ in}}{1\text{ ft}}$ twice, since an area carries two lengths. $$6{\text{ ft}}^2\times \frac{12\text{ in}}{1\text{ ft}}\times \frac{12\text{ in}}{1\text{ ft}}=6\times 12\times 12{\text{ in}}^2$$ The ${\text{ft}}^2$ cancels against the two $\text{ft}$ on the bottom, leaving ${\text{in}}^2$, and $6\times 144=864$. The floor is $\boxed{{\displaystyle 864}}$ square inches.