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Chapter 6: Problem 1

Define the key terms. a. Rate b. Unit rate c. Unit cost

Short Answer

Expert verified
Rate is a ratio of two quantities; unit rate is a rate with a denominator of one; unit cost is cost per unit.

Step by step solution

01

Define Rate

A rate is a ratio between two related quantities. It expresses how one quantity changes in relation to another. For example, speed is a rate expressing distance over time, such as miles per hour.
02

Define Unit Rate

A unit rate is a rate in which the second quantity in the comparison is one unit. It simplifies a rate to per single unit of the denominator, such as 60 miles per hour or 5 dollars per pound.
03

Define Unit Cost

A unit cost is a type of unit rate where the total cost is divided by the number of units produced or consumed. This shows the cost per one unit of the good or service, such as 2 dollars per item.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate
When we talk about a rate, we're discussing a special kind of ratio. A rate compares two different types of quantities.
For instance, take speed. Speed measures how much distance (like miles) you can travel over a set amount of time (like hours or minutes).
If you say your car is traveling at 60 miles per hour, that means for every hour you drive, you cover 60 miles.
Rates can apply to many different situations:
  • Heartbeats per minute (bpm)
  • Steps taken per second
  • Calories burned per workout
Remember, what makes a rate special is how it reveals the relationship between two different quantities.
Unit Rate
A unit rate takes the idea of a rate and simplifies it even further.
It describes how much of one quantity corresponds to just a single unit of another quantity.
For example, if you say a car travels at 60 miles per hour, you鈥檙e describing a unit rate. That鈥檚 because you鈥檙e stating the distance you travel (60 miles) in exactly one hour.
Simplifying rates to unit rates can make them easier to understand and compare:
  • If 10 apples cost 5 dollars, the unit rate (cost per apple) would be 5 dollars/10 apples = 0.5 dollars per apple.
  • If a printer prints 300 pages in 5 minutes, the unit rate (pages per minute) would be 300 pages/5 minutes = 60 pages per minute.
By converting rates to unit rates, you can more easily compare different situations and make informed decisions.
Unit Cost
The concept of unit cost is very similar to unit rate, but it specifically focuses on cost.
Unit cost tells you the price for each single unit of a product or service.
For example, if a pack of 5 pens costs \(10, the unit cost is the total cost divided by the number of units:
  • \text{Unit cost} = \frac{\text{Total cost}}{\text{Number of units}} = \frac{10}{5} = 2 \text{ dollars per pen}
Knowing the unit cost is particularly helpful when comparing prices.
Say you鈥檙e in a store and you see that a 12-pack of soda costs \)6, while a 24-pack costs $11. You can find the unit cost for each option and decide which is the better deal:
  • 12-pack unit cost: \frac{6}{12} = 0.50 \text{ dollars per can}
  • 24-pack unit cost: \frac{11}{24} \text{ (approximately)} = 0.46 \text{ dollars per can}
In this case, the 24-pack offers a lower unit cost per can, making it the better buy.

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Most popular questions from this chapter

Erik bought \(\$ 624\) Canadian with \(\$ 600\) American. At this rate, how many Canadian dollars can he buy with \(\$ 250\) American?

Refer to the table showing Alex Rodriguez's salary (rounded to the nearest \(\$ 100,000\) ) for selected years during his career. Write each ratio in lowest terms. \begin{array}{l|l|l|l} \hline \text { Year } & \text { Team } & \text { Salary } & \text { Position } \\ \hline 2007 & \text { New York Yankees } & \$ 22,700,000 & \text { Third baseman } \\ \hline 2004 & \text { New York Yankees } & \$ 22,000,000 & \text { Third baseman } \\ \hline 2000 & \text { Seattle Mariners } & \$ 4,400,000 & \text { Shortstop } \\ \hline 1996 & \text { Seattle Mariners } & \$ 400,000 & \text { Shortstop } \\ \hline \end{array} Write the ratio of Alex's salary for the year 1996 to the year 2000 .

Solve the proportion. Be sure to check your answers. $$\frac{m}{12}=\frac{5}{8}$$

Solve the proportion. Be sure to check your answers. $$\frac{9}{21}=\frac{x}{7}$$

Solve the proportions. $$\frac{x-2}{4 x}=\frac{3}{8}$$
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