Focus on problems involving rates, such as speed or price per item. Learn the method of dimensional analysis to accurately and efficiently convert between different units of measurement, a key skill for science and math problems.
A unit rate is a rate where the second quantity in the comparison is \(1\) unit. Examples include miles per hour (miles per \(1\) hour), dollars per pound (dollars per \(1\) pound), or words per minute (words per \(1\) minute).
If a box of cereal costs \(\$3.60\) for \(18\) ounces, what is the unit price in cents per ounce?
First, convert dollars to cents: \(\$3.60 = 360\) cents. Then divide the total cost by the number of ounces: \(\frac{360 \text{ cents}}{18 \text{ ounces}} = 20\) cents per ounce.
What is dimensional analysis?
Dimensional analysis is a method used to convert units by multiplying by conversion factors. A conversion factor is a fraction where the numerator and denominator are equal quantities but in different units (e.g., \(\frac{12 \text{ inches}}{1 \text{ foot}}\)). This allows units to be canceled out systematically.
Using dimensional analysis, convert \(7\) feet into inches. (Given: \(1\) foot = \(12\) inches)
$$ 7 \text{ feet} \times \frac{12 \text{ inches}}{1 \text{ foot}} = 84 \text{ inches} $$ The 'feet' unit in the numerator and denominator cancel out, leaving inches.
A car travels \(180\) miles in \(3\) hours. What is its average speed in miles per hour (mph)?
Speed is a rate calculated as distance divided by time.
$$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{180 \text{ miles}}{3 \text{ hours}} = 60 \text{ mph} $$
How many seconds are in \(2.5\) hours?
Use dimensional analysis with two conversion factors: \(60\) minutes/hour and \(60\) seconds/minute.
$$ 2.5 \text{ hours} \times \frac{60 \text{ minutes}}{1 \text{ hour}} \times \frac{60 \text{ seconds}}{1 \text{ minute}} = 9000 \text{ seconds} $$
A snail moves at a rate of \(2\) centimeters per minute. Convert this speed to meters per hour.
Use two conversion factors: \(1\) meter / \(100\) cm, and \(60\) minutes / \(1\) hour.
$$ \frac{2 \text{ cm}}{1 \text{ min}} \times \frac{1 \text{ meter}}{100 \text{ cm}} \times \frac{60 \text{ min}}{1 \text{ hour}} = \frac{120}{100} \frac{\text{meters}}{\text{hour}} = 1.2 \frac{\text{meters}}{\text{hour}} $$
Convert a speed of \(60\) miles per hour to feet per second. (Given: \(1\) mile = \(5280\) feet)
If \(1\) U.S. dollar is worth \(0.92\) euros, how many U.S. dollars can you get for \(230\) euros?
Set up the conversion factor to cancel euros: \(\frac{1 \text{ dollar}}{0.92 \text{ euros}}\).
$$ 230 \text{ euros} \times \frac{1 \text{ dollar}}{0.92 \text{ euros}} = \frac{230}{0.92} \text{ dollars} = 250 \text{ dollars} $$
How do you convert an area from square feet (\(\text{ft}^2\)) to square inches (\(\text{in}^2\))?
Since \(1 \text{ ft} = 12 \text{ in}\), you must square the entire conversion factor: \((1 \text{ ft})^2 = (12 \text{ in})^2\), which means \(1 \text{ ft}^2 = 144 \text{ in}^2\). To convert, multiply the area in \(\text{ft}^2\) by \(144\).
A room has an area of \(10\) square yards (\(\text{yd}^2\)). What is its area in square feet (\(\text{ft}^2\))? (Given: \(1\) yard = \(3\) feet)
The conversion factor for area is \(1 \text{ yd}^2 = (3 \text{ ft})^2 = 9 \text{ ft}^2\).
$$ 10 \text{ yd}^2 \times \frac{9 \text{ ft}^2}{1 \text{ yd}^2} = 90 \text{ ft}^2 $$
A tank has a volume of \(2\) cubic meters (\(\text{m}^3\)). What is its volume in cubic centimeters (\(\text{cm}^3\))? (Given: \(1\) m = \(100\) cm)
Density is defined as mass per unit volume. If a substance has a density of \(800\) kg/m³, what is its density in g/cm³? (\(1\) kg = \(1000\) g, \(1\) m = \(100\) cm)
