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Rates | Lesson

What are rates?

A rate is a

with different units. When we write a rate, it has units in the form of a fraction,

\frac{A}{B}

. For example, speed, a ratio of distance and time, has units like

\frac{meters}{second}

\frac{miles}{hour}

. The word per represents the fraction bar: meters per second is equivalent to

\frac{meters}{second}

An unit rate is a rate with a denominator of

1

. When we say "

4

meters per second," we mean "

4

meters per

1

second."

What skills are tested?

Calculating a unit rate
Using a unit rate in calculations
Solving word problems involving rates

How do we calculate unit rates?

To calculate a unit rate with units

\frac{A}{B}

, divide the quantity with unit

A

by the quantity with unit

B

\frac{quantity A}{quantity B} = unit rate \frac{A}{B}

For example, if Balto ran

33

miles in

3

hours, then his speed was:

\begin{aligned} \frac{33 miles}{3 hours} & = \frac{3 \times 11}{1 \times 3} \frac{miles}{hour} \\ = 11 \frac{miles}{hour} \end{aligned}

How do we apply unit rates?

Two common applications of unit rates are speed and unit price. These unit rates can be used to calculate total distance and total cost.

quantity A = unit rate \frac{A}{B} \times quantity B

$total distance = speed \times time$ ‍
$total cost = unit price \times quantity$ ‍

We can calculate any of the three parts in the equations above given the other two. To do so, we:

Write down the appropriate equation.
Plug in the values for the known quantities.
Solve for the unknown quantity.

A killer whale swims at a speed of

20

miles per hour. How many miles does the killer whale swim in

2

hours?

\begin{aligned} distance & = speed \times time \\ = 20 \frac{miles}{hour} \times 2 hours \\ = 40 miles \end{aligned}

Adrienne spent a total of

$ 160

on concert tickets that cost

$ 40

each. How many concert tickets did she buy?

Using

x

to represent the number of tickets Adrienne bought:

\begin{aligned} total cost & = unit price \times quantity \\ 160 dollars & = 40 \frac{dollars}{ticket} \times x tickets \\ \frac{160 dollars}{40 \frac{dollars}{ticket}} & = \frac{40 \frac{dollars}{ticket} \times x tickets}{40 \frac{dollars}{ticket}} \\ 4 tickets & = x tickets \end{aligned}

What's the connection between rates and proportional relationships?

Since rates are ratios, we can apply them to different quantities. To do so, we can either multiply the unit rate by the new quantity or find the desired quantity by solving a proportional relationship.

3

hot dogs cost

$ 10.50

, how much does

8

hot dogs cost?

We can calculate the price of a single hot dog first, then multiply it by

8

\begin{aligned} \frac{10.5 dollars}{3 hot dogs} & = 3.5 \frac{dollars}{hot dog} \\ total cost & = unit price \times quantity \\ = 3.5 \frac{dollars}{hot dog} \times 8 hot dogs \\ = 28 dollars \end{aligned}

We can also set up a proportional relationship to solve for

x

, the cost of

8

hot dogs in dollars:

\begin{aligned} \frac{10.5 dollars}{3 hot dogs} & = \frac{x dollars}{8 hot dogs} \\ \frac{10.5 dollars}{3 hot dogs} \times 8 hot dogs & = \frac{x dollars}{8 hot dogs} \times 8 hot dogs \\ 28 dollars & = x dollars \end{aligned}

Your turn!

TRY: CALCULATING UNIT PRICE

Craig bought

96

rolls or toilet paper for

48

dollars. What is the cost of a single roll of toilet paper?

dollars

TRY: APPLYING A RATE

While eating ants, a giant anteater flicks its tongue

150

times per minute. At this rate, how many times does the giant anteater flick its tongue in

6

minutes?

TRY: USING UNIT RATE OR PROPORTIONAL RELATIONSHIP

A mouse's heart beats

900

times in

1.5

minutes. At this rate, how many times does the mouse's heart beat in

10

minutes?

TRY: USING UNIT PRICES FROM A TABLE

Item	Price
Burger	$$ 7$ ‍
Fries	$$ 3$ ‍

The prices of items at Bob's Burgers are shown in the table above. If Teddy orders

3

burgers and

2

fries, what is the total cost of his order?

Things to remember

A rate is a ratio of two different units. An unit rate is a rate with a denominator of

1

To calculate a unit rate with units

\frac{A}{B}

, divide the quantity with unit

A

by the quantity with unit

B

\frac{quantity A}{quantity B} = unit rate \frac{A}{B}

Two common applications of the unit rate equation are speed and price. The equations for them are written in the form of:

quantity A = unit rate \frac{A}{B} \times quantity B

$distance = speed \times time$ ‍
$total price = unit price \times quantity$ ‍

We can calculate any of the three parts in the equations above given the other two. To do so, we:

Write down the appropriate equation.
Plug in the values for the unit rate and the known quantity.
Solve for the unknown quantity.

All rate questions can also be solved using proportional relationships.

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CadenceC
Posted 3 years ago. Direct link to CadenceC's post “I like how you guys added...”
I like how you guys added Bob's Burgers. Lol its my favorite show.
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Faith'AnneJ
Posted 3 years ago. Direct link to Faith'AnneJ's post “I could use a little help...”
I could use a little help and more practice
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844973
Posted 7 months ago. Direct link to 844973's post “i get it....but i dont ge...”
i get it....but i dont get it at the same time.
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Hamza DAWG
Posted 5 months ago. Direct link to Hamza DAWG's post “($_$) How do you do math”
($_$)
How do you do math
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matthew
Posted 12 days ago. Direct link to matthew's post “I just wanted to let you ...”
I just wanted to let you know that the first example that talks about the toilet rolls, it says 96 rolls or toilet paper. Is it suppost to say that or is it suppost to say 96 rolls of toilet paper?
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MojabE
Posted 3 months ago. Direct link to MojabE's post “this is a good resource f...”
this is a good resource for a test or homework when you need it at school
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Slushe
Posted 2 years ago. Direct link to Slushe's post “I am wondering, where the...”
I am wondering, where the practice for this is at and if there even is one.
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Praxis Core Math

Course: Praxis Core Math > Unit 1