Rates

What are Rates?

A rate is a special type of ratio that compares quantities with different units. Rates are used to show how one quantity changes in relation to another.

Examples of Rates

Speed

60 miles per hour

Distance per unit of time

Price

$3.50 per gallon

Cost per unit of volume

Wages

$15 per hour

Money per unit of time

Density

8.9 g/cm³

Mass per unit of volume

Unit Rates

A unit rate is a rate where the second quantity (the denominator) is 1. Unit rates make it easier to compare different rates.

Converting to Unit Rates

To convert a rate to a unit rate, divide both quantities by the second quantity.

If you travel 240 miles in 4 hours, what is your unit rate (speed)?

Rate = 240 miles / 4 hours

Unit rate = 240 ÷ 4 = 60 miles per hour

Comparing Unit Rates

Unit rates make it easy to compare different options and find the best value.

Option A: 24 oz for $3.99

Unit rate = $3.99 ÷ 24 = $0.166 per oz

Option B: 32 oz for $4.99

Unit rate = $4.99 ÷ 32 = $0.156 per oz

Option B is the better value because it has a lower cost per ounce.

Solving Rate Problems

Distance, Rate, and Time

The formula d = rt (distance = rate × time) is used to solve many rate problems.

Example 1: If you drive at 65 miles per hour for 3.5 hours, how far will you travel?

d = rt

d = 65 × 3.5

d = 227.5 miles

Finding the Rate

If you know the distance and time, you can find the rate by dividing: r = d/t.

Example 2: If you travel 210 miles in 3 hours, what is your average speed?

r = d/t

r = 210 ÷ 3

r = 70 miles per hour

Finding the Time

If you know the distance and rate, you can find the time by dividing: t = d/r.

Example 3: If you need to travel 300 miles at 60 miles per hour, how long will it take?

t = d/r

t = 300 ÷ 60

t = 5 hours

Complex Rate Problems

Work Rate Problems

Work rate problems involve calculating how long it takes to complete a task based on individual rates.

Example: If Alice can paint a room in 3 hours and Bob can paint the same room in 4 hours, how long would it take them working together?

Alice's rate: 1/3 of the room per hour

Bob's rate: 1/4 of the room per hour

Combined rate: 1/3 + 1/4 = 4/12 + 3/12 = 7/12 of the room per hour

Time to complete: 1 ÷ (7/12) = 12/7 ≈ 1.71 hours

Mixture Problems

Mixture problems involve combining substances with different rates or concentrations.

Example: How many liters of 20% acid solution should be mixed with 5 liters of 60% acid solution to get a 30% acid solution?

Let x = liters of 20% solution

0.20x + 0.60(5) = 0.30(x + 5)

0.20x + 3 = 0.30x + 1.5

0.20x - 0.30x = 1.5 - 3

-0.10x = -1.5

x = 15 liters

Practice Rates

Test your knowledge with our interactive rate practice exercises.

Start Practice

Unit Converter

Use our calculator to convert between different units and solve rate problems.

Open Converter

Real-World Applications

1
Calculating travel time based on speed and distance
2
Comparing prices to find the best value (price per unit)
3
Calculating fuel efficiency (miles per gallon)
4
Determining cooking times based on weight (minutes per pound)
5
Calculating wages based on hourly rates and time worked

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