Transcript Chapter 4 – Rates, Ratios, and Proportions

Math Skills – Week 5
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Class Project due Mar 16th
 Guidelines on class website
 Examples next week
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Remember you can not miss more than 3
class periods
Equivalent fractions and multiplication /
division
Product = multiply
Quotient = divide
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Ratio – 4.1
Rates – 4.2
Proportions – 4.3
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Units. When we put a unit after a number, we
give that number some physical context
◦ 8 Feet
◦ 8 Cars
◦ 8 Tadpoles
 Feet, Cars, and Tadpoles are all examples of units

A Ratio is a comparison of two quantities that
have the same units.
◦ We write this comparison as
1. As a fraction

$6/$8
2. As two numbers separated by a colon

$6:$8
3. As two numbers separated by the word to

$6 to $8
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To write a ratio in Simplest Form, we write the
two numbers so that they have no other
common factors other than 1.
◦ Review simplest form.
 Steps
1. Find the prime factorization of each number
2. Cancel out all of the like quantities
3. The resulting numbers is the ratio
Example

Write the Ratio 8/10 in simplest form
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
4/5
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Examples
1. Write the comparison $6 to $8 as a ratio in simplest
form using a fraction, a colon, and the word to.
1. $6/$8 = 6/8 = ¾
2. $6:$8 = 6:8 = 3:4
3. $6 to $8 = 6 to 8 = 3 to 4
2. Write the comparison 18 quarts to 6 quarts as a ratio
in simplest form using a fraction, a colon, and the
word to.
1. 18 quarts/6 quarts = 18/6 = 3/1
2. 18 quarts:6 quarts = 18:6 = 3:1
3. 18 quarts to 6 quarts = 18 to 6 = 3 to 1
3. Example 4 pg.176
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Class Examples
1. Write the comparison of 20 pounds to 24 pounds
as a ratio in simplest form using a fraction, a
colon, and the word to.
1. 20 pounds/24 pounds = 20/24 = 5/6
2. 20 pounds:24 pounds = 20:24 = 5:6
3. 20 pounds to 24 pounds = 20 to 24 = 5 to 6
2. Write the comparison of 64 miles to 8 miles as a
ratio in simplest form using a fraction, a colon,
and the word to.
1. 64 miles/8 miles= 64/8 = 8/1
2. 64 miles:8 miles = 64:8 = 8:1
3. 64 miles to 8 miles= 64 to 8 = 8 to 1
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A rate is a comparison of two quantities that
have different units.
◦ Note: Rates are always written as fractions
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Example
1. A distance runner ran 26 miles in 4 hours. The
distance to time rate is:
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26 miles / 4 hours = 13 miles / 2 hours
2. Write 6 roof supports for every 9 feet as a rate in
simplest form.
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6 roof supports / 9 feet = 2 roof supports / 3 feet
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Class Example
1. Write “15 pounds of fertilizer for 12 trees”
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15 pounds / 12 trees = 5 pounds / 4 trees
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A unit rate, is a rate that has 1 in the
denominator
◦ $3.25 / 1 pound or $3.25/pound is read as “$3.25
per pound”
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To write a unit rate:
◦ Steps
1. Divide the number in the numerator by the number
in the denominator of the rate.
Ex: On a trip, I traveled 344 miles before my car ran out
of gas. My tank holds 16 gallons of gas. What is the
unit rate that I traveled?
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344 miles/16 gallons = 21.5 miles/gallon
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Example
1. Write “300 feet in 8 seconds” as a unit rate

300 feet/8 seconds = 3.75 feet/second
2. Pg. 180 You Try It #3
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Class Example
1. Write “260 miles in 8 hours” as a unit rate
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260 miles/8 hours = 32.5 miles/hour
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A proportion is an expression of the equality
of two ratios or rates.
◦ If I say 50 Miles/4 gallons = 25 miles/2 gallons is
this a true statement?
 In order to say Yes or No, Things to check:
1. The units in the numerator and denominator must be
the same for both rates/ratios.
2. Check that one ratio is the same as the other multiplied
by 1 (written in fractional form)
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25/2 x 2/2 = 50/4, thus this is a true proportion
To determine if a proportion is true.
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Steps
◦
Method 1
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1. Write each fraction in simplest form.
2. If the fractions are equal, we prove that the proportion is
true
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Example
1. Is the proportion 3/6 = ½ a true proportion?
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3/6 in simplest form is 1/2 , ½ = ½ thus this is a true
proportion
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The cross product of a proportion is defined
as shown below.
2 8
=
3 12
x
x
2 x 12 = 24
3 x 8 = 24
For a true proportion, the cross products of
the proportion are always equal
To determine if a proportion is true:
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Method 2 (Preferred)
◦
Steps
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1. Write the cross products of the proportion.
2. If the cross products are equal, we say the proportion is
true.
Example:
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Is the proportion 3/6 = ½ a true proportion?


Cross products are 3 x 2 = 6 and 6 x 1. Thus the
proportion is true.
In Summary:
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◦
A proportion is true if:
1. The fractions are equal when reduced to simplest
form OR
2. The cross products are equal
◦
A proportion is not true if:
1. The fractions are not equal when reduced to simplest
form OR
2. The cross products are not equal
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Examples
1. Is 5/8 = 10/16 a true proportion?
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Yes. 5 x 16 = 80 and 8 x 10 = 80, cross products are
equal so this is a true proportion.
2. Is 62 miles/4 gallons = 33 miles/2 gallons a true
proportion?
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Nope. 62 x 2 = 132 and 33 x 4 = 124, cross products are
not equal so this is not a true proportion.
Class examples
1. Is 6/10 = 9/15 a true proportion?
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True. 6 x 15 = 90, and 9 x 10 = 90.
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Not true. 32 x 8 = 256 and 90 x 6 = 540
2. Is $32/6 hours = $90/8 hours a true proportion?
To solve a proportion, we need to find the
missing number.
Think…
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◦
“What does n have to be in order for this
proportion to be true?”
2 n
=
3 12
x
x
2 x 12 = 24
3 x n = 24
n=8
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Examples
1. Solve n/12 = 25/60

60 x n = 12 x 25  60 x n = 300, 300/60 n=5.

9 x n = 4 x 16  9 x n = 64  n = 64/9  n ≈ 7.1
2. Solve 4/9 = n/16
3. Pg.186 YouTryIt 8,9
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Class Examples
1. Solve 15/20 = 12/n
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15 x n = 20 x 12  240 = n x 15  n = 240/15  n
= 16
2. Solve n/12 = 4/1

1 x n = 12 x 4  1 x n = 48  n = 48/1  n = 48