Unit Conversion
A cake recipe says "350 g of flour," but the scale at your house reads in kilograms. On the Mexico City–Puebla highway a sign says 110 km/h, but in your Physics problem the speed has to be in m/s. Changing units without changing the real quantity is a tool you use in science and in everyday life.
The Conversion Factor
These are the most common factors with SI prefixes: $1\ \text{km} = 1000\ \text{m}$, $1\ \text{m} = 100\ \text{cm}$, $1\ \text{cm} = 10\ \text{mm}$, $1\ \text{kg} = 1000\ \text{g}$, $1\ \text{t} = 1000\ \text{kg}$.
U.S. customary system
In Mexico you also see U.S. customary units: inches on screens and pipes, pounds per square inch (psi) in tires, pounds and ounces in U.S. recipes, gallons of paint. They are converted the same way, with their factor:
| Quantity | Equivalence |
|---|---|
| Length | $1\ \text{inch} = 2.54\ \text{cm}$; $1\ \text{yard} = 0.9144\ \text{m}$ |
| Mass | $1\ \text{pound} \approx 0.454\ \text{kg}$; $1\ \text{ounce} \approx 28.35\ \text{g}$ |
| Capacity | $1\ \text{gallon} \approx 3.785\ \text{L}$ |
The wardrobe in Bruno's room is $2\ \text{m}$ high. Express this height in centimeters.
The wardrobe in Bruno's room is $2\ \text{m}$ high. Express this height in centimeters.
A fun run has a route of $5\ \text{km}$. How many meters is that?
A fun run has a route of $5\ \text{km}$. How many meters is that?
A packet of rice weighs $2\ \text{kg}$. How many grams is that?
A packet of rice weighs $2\ \text{kg}$. How many grams is that?
Leo's ruler measures $30\ \text{cm}$, like the one in the figure. How many millimeters is that? And how many meters?
Leo's ruler measures $30\ \text{cm}$, like the one in the figure.
How many millimeters is that? And how many meters?
A sugar cube weighs $6\ \text{g}$. Express this mass in kilograms.
A sugar cube weighs $6\ \text{g}$. Express this mass in kilograms.
A basketball player is $1.98\ \text{m}$ tall. Express his height in centimeters and in millimeters.
A basketball player is $1.98\ \text{m}$ tall. Express his height in centimeters and in millimeters.
An old bridge restricts crossing to $3\ \text{t}$. A loaded van weighs $2800\ \text{kg}$, as shown in the figure. Can it cross? Justify your answer by converting both…
An old bridge restricts crossing to $3\ \text{t}$. A loaded van weighs $2800\ \text{kg}$, as shown in the figure.
Can it cross? Justify your answer by converting both masses to the same unit.
An athletics track measures $400\ \text{m}$ per lap, like the one in the figure. How many laps must be run to complete $2\ \text{km}$?
An athletics track measures $400\ \text{m}$ per lap, like the one in the figure.
How many laps must be run to complete $2\ \text{km}$?
Time and Capacity
Time does not use base 10 but base 60, and that is why it can be tricky. Capacity, on the other hand, is decimal.
The film Nerea watched yesterday lasted $1.5\ \text{h}$. How many minutes is that?
The film Nerea watched yesterday lasted $1.5\ \text{h}$. How many minutes is that?
A bottle of water contains $1.5\ \text{L}$. How many milliliters is that?
A bottle of water contains $1.5\ \text{L}$. How many milliliters is that?
Break time lasts $30\ \text{min}$. How many seconds is that?
Break time lasts $30\ \text{min}$. How many seconds is that?
A drink can contains $330\ \text{cm}^3$. a) How many milliliters is that? b) And how many liters?
A drink can contains $330\ \text{cm}^3$. a) How many milliliters is that? b) And how many liters?
A bottle of perfume contains $75\ \text{mL}$. a) How many liters is that? b) How many such bottles can be filled with a $3\ \text{L}$ container?
A bottle of perfume contains $75\ \text{mL}$. a) How many liters is that? b) How many such bottles can be filled with a $3\ \text{L}$ container?
A glass of juice has $250\ \text{mL}$ and a jug contains $1.8\ \text{L}$, as shown in the figure. a) How many full glasses can be served from a jug? b) How many…
A glass of juice has $250\ \text{mL}$ and a jug contains $1.8\ \text{L}$, as shown in the figure.
- a) How many full glasses can be served from a jug? b) How many milliliters are left over? c) Express the leftover amount in liters.
Area and Volume: Watch Out for Squares and Cubes!
Volume works the same way, but with three sides: $1\ \text{m}^3 = 100 \times 100 \times 100 = 1{,}000{,}000\ \text{cm}^3$. A very useful case is $1\ \text{L} = 1000\ \text{cm}^3$ (that is, $1\ \text{mL} = 1\ \text{cm}^3$) and $1\ \text{m}^3 = 1000\ \text{L}$.
A fish tank contains $12\ \text{L}$ of water. How many cubic centimeters is that? Remember: $1\ \text{L} = 1000\ \text{cm}^3$.
A fish tank contains $12\ \text{L}$ of water. How many cubic centimeters is that? Remember: $1\ \text{L} = 1000\ \text{cm}^3$.
The floor of Julia's room has an area of $12\ \text{m}^2$. a) How many square centimeters is that? Note: $1\ \text{m}^2$ is not $100\ \text{cm}^2$. b) Explain in words…
The floor of Julia's room has an area of $12\ \text{m}^2$. a) How many square centimeters is that? Note: $1\ \text{m}^2$ is not $100\ \text{cm}^2$. b) Explain in words (or with a drawing of a square with a side of $1\ \text{m}$) why $1\ \text{m}^2 = 10\,000\ \text{cm}^2$. The square in the figure may help you:
A square tile has sides of $50\ \text{cm}$, as shown in the figure. a) What is its area in square centimeters? b) And in square meters? c) How many tiles are needed to…
A square tile has sides of $50\ \text{cm}$, as shown in the figure.
- a) What is its area in square centimeters? b) And in square meters? c) How many tiles are needed to cover $1\ \text{m}^2$?
A tanker truck carries $20\,000\ \text{L}$ of milk, like the one in the figure. a) How many cubic meters is that? Hint: $1\ \text{m}^3 = 1000\ \text{L}$. b) If the milk…
A tanker truck carries $20\,000\ \text{L}$ of milk, like the one in the figure.
- a) How many cubic meters is that? Hint: $1\ \text{m}^3 = 1000\ \text{L}$. b) If the milk is packaged into bottles of $2\ \text{L}$, how many bottles are filled?
An inflatable pool is box-shaped, $2\ \text{m}$ long, $1\ \text{m}$ wide, and $0.5\ \text{m}$ high, like the one in the figure. a) Calculate its volume in cubic meters.…
An inflatable pool is box-shaped, $2\ \text{m}$ long, $1\ \text{m}$ wide, and $0.5\ \text{m}$ high, like the one in the figure.
- a) Calculate its volume in cubic meters. b) Convert it into liters knowing that $1\ \text{m}^3 = 1000\ \text{L}$. c) If it is filled using a hose that pumps $20\ \text{L}$ per minute, how many minutes does it take to fill?
Speed: km/h and m/s
A cheetah runs at $108\ \text{km/h}$ and a gazelle moves $28\ \text{m}$ every second. Since they are in different units, you can't tell which one is faster. How many m/s does the cheetah run at? To compare them, you have to convert.
A car travels along the highway at $90\ \text{km/h}$, as shown in the figure. Convert its speed into meters per second by dividing by $3.6$.
A car travels along the highway at $90\ \text{km/h}$, as shown in the figure.
Convert its speed into meters per second by dividing by $3.6$.
A sprinter runs at $10\ \text{m/s}$. Express his speed in $\text{km/h}$ by multiplying by $3.6$.
A sprinter runs at $10\ \text{m/s}$. Express his speed in $\text{km/h}$ by multiplying by $3.6$.
Who travels faster: a cyclist riding at $8\ \text{m/s}$ or a scooter rolling at $25\ \text{km/h}$? Express both speeds in the same unit before comparing them.
Who travels faster: a cyclist riding at $8\ \text{m/s}$ or a scooter rolling at $25\ \text{km/h}$? Express both speeds in the same unit before comparing them.
An airplane flies at $900\ \text{km/h}$. a) Convert its speed into meters per second. b) How many meters does it travel in $10\ \text{s}$? c) And in one minute? Express…
An airplane flies at $900\ \text{km/h}$. a) Convert its speed into meters per second. b) How many meters does it travel in $10\ \text{s}$? c) And in one minute? Express this last result in kilometers as well.
In a city, the speed limit is $50\ \text{km/h}$ and a car passes a speed camera at $15\ \text{m/s}$. a) Is it exceeding the limit? Convert the units before answering. b)…
In a city, the speed limit is $50\ \text{km/h}$ and a car passes a speed camera at $15\ \text{m/s}$. a) Is it exceeding the limit? Convert the units before answering. b) What distance does it travel in $10\ \text{s}$ at that speed? c) The number $15$ seems small compared to $50$: explain why, despite this, the car is driving faster than allowed.
Back to the beginning: the $350\ \text{g}$ of flour is $350 \div 1000 = 0.35\ \text{kg}$; and the $110\ \text{km/h}$ on the sign, in your Physics problem, is $110 \div 3.6 \approx 30.6\ \text{m/s}$.