Unit Conversion
One Sunday, Yaritza sees "350 g of meat" in a sancocho recipe, but the scale at the colmado reads in pounds. And on the Duarte Highway the speedometer shows km/h, while the Physical Science problem asks for m/s. Changing units without changing the actual quantity is one of the most widely used tools in science and in everyday life.
The Conversion Factor
Converting means expressing the same quantity with another unit. To do this we use a conversion factor: the number that tells how many small units fit in a large one (for example, $1\ \text{m} = 100\ \text{cm}$). It is also called a unit factor, because multiplying by it does not change the quantity, only the way it is written.
Common local units
At the colmado you buy by the pound, gasoline is sold by the gallon, and land is measured in tareas. These units are not part of the SI, but they are converted the same way, using their factor:
| Common unit | Equals |
|---|---|
| $1\ \text{lb}$ (pound) | $\approx 0.454\ \text{kg}$ |
| $1\ \text{gal}$ (gallon) | $\approx 3.785\ \text{L}$ |
| $1\ \text{foot}$ | $\approx 30.48\ \text{cm}$ |
| $1\ \text{inch}$ | $\approx 2.54\ \text{cm}$ |
| $1\ \text{tarea}$ | $\approx 629\ \text{m}^2$ |
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.
Time and Capacity
Time is not decimal: an hour has $60$ minutes and a minute has $60$ seconds. Capacity is decimal: a liter has $1000$ milliliters.
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?
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 $2\ \text{km}$?
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!
A square with a side of $1\ \text{m}$ has a side of $100\ \text{cm}$. Its area is $100 \times 100 = 10{,}000\ \text{cm}^2$. The factor is applied twice, once for each dimension. For a volume it is applied three times: $1\ \text{m}^3 = 100 \times 100 \times 100 = 1{,}000{,}000\ \text{cm}^3$.
A fish tank contains $12\ \text{L}$ of water. How many cubic centimeters is that? Remember: $1\ \text{L} = 1000\ \text{cm}^3$.
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
Guaguas and cars use km/h, but in Physical Science we work with m/s. A car goes $90\ \text{km/h}$ on the Duarte Highway: how many meters per second is it moving? Watch how an object moving at $8\ \text{m/s}$ advances: every second it covers $8\ \text{m}$.
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 meat is converted to kilograms by dividing by $1000$ ($0.35\ \text{kg}$), and to pounds by dividing by $454$ ($\approx 0.77\ \text{lb}$). And the $90\ \text{km/h}$ on the speedometer is $25\ \text{m/s}$ in the Physical Science problem.