Download Using Proportions

7-1 Using Proportions
•Recognize and use ratios and proportions.
•Apply the properties of proportions.
Ratio
• A ratio is a comparison of two
quantities using division.
• Example: The number of boys to girls
in a class.
Ways to express a ratio
• a/b
• a to b
• a:b
Example 1
• A baseball player’s batting average is
the ratio of the number of base hits
to the number of at-bats, not
including walks. Minnesota Twins’ Joe
Mauer had the highest batting
average in Major League Baseball in
2006. If he had 521 official at-bats
and 181 hits, find his batting average.
• Number of hits
= 181 = 0.347
• Number of at-bats 521
1
• Joe Mauer’s batting average was
0.347
Extended ratios
• An extended ratio can be used to
compare three or more quantities.
• The expression a:b:c means that the
ratio of the first two quantities is
a:b, the ratio of the last two
quantities is b:c, and the ratio of the
first and last quantities is a:c.
Ex. 2 The ratio of the measures of the angles
in a triangle is 3:4:5. Find the measures of the
angles.
• Just as ratio ¾ can be written as 3x/4x or
3x:4x, the extended ratio 3:4:5 can be
written as 3x:4x:5x.
• 3x + 4x + 5x = 180
• 12x = 180
• X = 15
• So the measures of the angles are 3(15) or
45, 4(15) or 60, and 5(15) or 75.
•
45 + 60 + 75 = 180
Proportion
• An equation stating that two ratios are
equal is a proportion.
• Extreme
Means
•
extreme a = c
mean
•
mean
b
d extreme
• The product of the extremes ad and
the product of the means bc are called
cross products.
The cross products of a proportion
are equal.
Equality of Cross Product
• For any numbers a and c and any
nonzero numbers b and d,
•
a = c
•
b
d
• If and only if ad = bc.
Example 2
• Solve
•
•
3t – 1 = 7
4
8
• Nikki can word process 7 words in 6
seconds. At that rate, how many
words can she word process in 3
minutes?
• Words
7 words = x words
• Time
6 seconds
180 seconds
Example 3
• In a triangle, the ratio of the
measures of three sides is 8:7:5. and
its perimeter is 240 centimeters.
Find the measure of each side of the
triangle.
• 8x + 7x + 5x = 240
•
20x = 240
•
x = 12
• Side 1 = 8x = 8(12) = 96cm
• Side 2 = 7x = 7(12) = 84cm
• Side 3 = 5x = 5(12) = 60cm
4. Determine which proportions are
equivalent. Explain your reasoning.
• 7 =x
• 8
y
y =8
x 7
y =x
7
8
7 =8
x y
6. 2 inches on a map represent 150
miles. Find a ratio involving 1 inch.
• 2 inches
• 150 miles
=
1 inch
75 miles
7. The perimeter of a rectangle is 84 feet.
The ratio of the width to the length is 2:5,
Find the length and width.
•
•
•
•
•
•
P = 2l + 2w
84= 2(5x) + 2(2x)
84 = 10x + 4x
84 = 14x
6=x
Length 5(6) is 30, width 2(6) is 12
Ex. 8 The area of a rectangle is 108cm2. The
ratio of the width to the length is 3:4. Find
the length and width.
•
•
•
•
•
•
•
A = lw
108 = (3x)(4x)
108 = 12x2
9 = x2
3=x
Length 4(3) is 12cm,
width 3(3) is 9cm
Solve each proportion by using cross
products.
•
•
•
•
x = 11
5
35
35x = 55
x = 1.57
•
•
•
•
13
=
49
91x =
x =
26
7x
1274
14
• X–2 = 3
• x
8
• 8x – 16 = 3x
•
- 16 = -5x
•
3.2 = x
• If a 6-foot post casts a shadow that is 8
feet long, how tall is an antenna that casts
a 60-foot shadow at the same time?
Class work on page 464, problems 1-16
Homework on page 465, problems 1736 even numbers.