Determine whether each pair of ratios are equivalent ratios. Write

Transcription

2-6 Ratios and Proportions
Determine whether each pair of ratios are equivalent ratios. Write yes or no.
No, the ratios are not equivalent.
Yes, the ratios are equivalent.
Solve each proportion. If necessary, round to the nearest hundredth.
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RACE Jennie ran the first 6 miles of a marathon in 58 minutes. If she is able to maintain the same pace, how long
will it take her to finish the 26.2 miles?
Let t represent the time it will take her to finish 26.2 miles. Write a proportion.
It will take Jennie 253.3 minutes or 4 hours and 13.3 minutes.
MAPS On a map of North Carolina, Raleigh and Asheville are about 8 inches apart. If the scale is 1 inch = 12 miles,
how far apart are the cities?
Let d represent the distance between the two cities. Write a proportion.
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It will take Jennie 253.3 minutes or 4 hours and 13.3 minutes.
MAPS On a map of North Carolina, Raleigh and Asheville are about 8 inches apart. If the scale is 1 inch = 12 miles,
how far apart are the cities?
Let d represent the distance between the two cities. Write a proportion.
Raleigh and Asheville are about 96 miles apart.
Determine whether each pair of ratios are equivalent ratios. Write yes or no.
Use cross products to check if the ratios are equivalent.
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CAR WASH The B Clean Car Wash washed 128 cars in 3 hours. At that rate, how many cars can they wash in 8
hours?
Let c represent the number of cars they wash in 8 hours. Write a proportion.
They will wash about 341 cars.
GEOGRAPHY On a map of Florida, the distance between Jacksonville and Tallahassee is 2.6 centimeters. If 2
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centimeters = 120 miles, what is the distance between the two cities?
They will wash about 341 cars.
GEOGRAPHY On a map of Florida, the distance between Jacksonville and Tallahassee is 2.6 centimeters. If 2
centimeters = 120 miles, what is the distance between the two cities?
Use the given scale to convert the map distance to the real world distance.
CCSS PRECISION An artist used interlocking building blocks to build a scale model of Kennedy Space Center,
Florida. In the model, 1 inch equals 1.67 feet of an actual space shuttle. The model is 110.3 inches tall. How tall is
the actual space shuttle? Round to the nearest tenth.
MENU On Monday, a restaurant made $545 from selling 110 hamburgers. If they sold 53 hamburgers on Tuesday,
how much did they make?
Let p represent their profit from lunch. Write a proportion.
Their profit from lunch is about $262.59.
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Their profit from lunch is about $262.59.
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ATHLETES At Piedmont High School, 3 out of every 8 students are athletes. If there are 1280 students at the
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school, how many are not athletes?
ATHLETES At Piedmont High School, 3 out of every 8 students are athletes. If there are 1280 students at the
school, how many are not athletes?
Let s represent the number of students who are athletes. If
of the students are athletes, then
of the students
are not athletes. Write a proportion.
So, 800 students are not athletes.
BRACES Two out of five students in the ninth grade have braces. If there are 325 students in the ninth grade, how
many have braces?
Let s represent the number of students that have braces. Write a proportion.
So, 130 students in the ninth grade have braces.
PAINT Joel used a half gallon of paint to cover 84 square feet of wall. He has 932 square feet of wall to paint. How
many gallons of paint should he purchase?
Let p represent the number of gallons of paint needed. Write a proportion.
It will take about 5.55 gallons of paint, so Paul should purchase 6 gallons of paint.
MOVIE THEATERS
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So, 130 students in the ninth grade have braces.
PAINT Joel used a half gallon of paint to cover 84 square feet of wall. He has 932 square feet of wall to paint. How
many gallons of paint should he purchase?
Let p represent the number of gallons of paint needed. Write a proportion.
MOVIE THEATERS
a. Write a ratio of the number of indoor theaters to the total number of theaters for each year.
b. Do any two of the ratios you wrote for part a form a proportion? If so, explain the real-world meaning of the
proportion.
a.
2003:
, 2004:
, 2005:
, 2006:
, 2007:
, 2008:
, 2009:
b.
Year
Indoor
2003 35,361
2004 36,012
2005 37,092
2006 37,776
2007 38,159
2008 38,201
2009 38,605
Total
35,995
36,652
37,740
38,425
38,794
38,834
39,233
Ratio
0.9823
0.9825
0.9828
0.9832
0.9836
0.9836
0.9839
No, none of the ratios from part a form a proportion. If two of these ratios formed a proportion, the indoor theaters
compared to the total number of theaters would produce equivalent fractions.
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DIARIES In a survey, 36% of the students said that they kept an electronic diary. There were 900 students who
kept an electronic diary. How many students were in the survey?
MOVIE THEATERS
a. Write a ratio of the number of indoor theaters to the total number of theaters for each year.
b. Do any two of the ratios you wrote for part a form a proportion? If so, explain the real-world meaning of the
proportion.
a.
2003:
, 2004:
, 2005:
, 2006:
, 2007:
, 2008:
, 2009:
b.
Year
Indoor
2003 35,361
2004 36,012
2005 37,092
2006 37,776
2007 38,159
2008 38,201
2009 38,605
Total
35,995
36,652
37,740
38,425
38,794
38,834
39,233
Ratio
0.9823
0.9825
0.9828
0.9832
0.9836
0.9836
0.9839
No, none of the ratios from part a form a proportion. If two of these ratios formed a proportion, the indoor theaters
Let s represents the number of students in the survey. 36% can be written as
. Now, write a proportion.
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2008
2009
38,201
38,605
38,834
39,233
0.9836
0.9839
No, none
ofProportions
the ratios from part a form a proportion. If two of these ratios formed a proportion, the indoor theaters
2-6 Ratios
and
Let s represents the number of students in the survey. 36% can be written as
. Now, write a proportion.
There were 2500 students in the survey.
MULTIPLE REPRESENTATIONS In this problem, you will explore how changing the lengths of the sides of a
shape by a factor changes the perimeter of that shape.
a. GEOMETRIC Draw a square ABCD. Measure and label the sides. Draw a second square MNPQ with sides
twice as long as ABCD. Draw a third square FGHJ with sides half as long as ABCD.
b. TABULAR Complete the table below using the appropriate measures.
c. VERBAL Make a conjecture about the change in the perimeter of a square if the side length is increased or
decreased by a factor.
a.
b.
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c. If the length of a side is increased by a factor, the perimeter is also increased by that factor. If the length of the
There were 2500 students in the survey.
MULTIPLE REPRESENTATIONS In this problem, you will explore how changing the lengths of the sides of a
shape by a factor changes the perimeter of that shape.
a. GEOMETRIC Draw a square ABCD. Measure and label the sides. Draw a second square MNPQ with sides
twice as long as ABCD. Draw a third square FGHJ with sides half as long as ABCD.
b. TABULAR Complete the table below using the appropriate measures.
c. VERBAL Make a conjecture about the change in the perimeter of a square if the side length is increased or
decreased by a factor.
a.
b.
c. If the length of a side is increased by a factor, the perimeter is also increased by that factor. If the length of the
sides are decreased by a factor, the perimeter is also decreased by the same factor.
CCSS STRUCTURE In 2007, organic farms occupied 2.6 million acres in the United States and produced goods
worth about $1.7 billion. Divide one of these numbers by the other and explain the meaning of the result.
0.0015 acres is the average land area used to produce a dollars worth of goods.
REASONING Compare and contrast ratios and rates.
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c. If theand
length
of a side is increased by a factor, the perimeter is also increased by that factor. If the length of the
2-6 Ratios
Proportions
sides are decreased by a factor, the perimeter is also decreased by the same factor.
CCSS STRUCTURE In 2007, organic farms occupied 2.6 million acres in the United States and produced goods
worth about $1.7 billion. Divide one of these numbers by the other and explain the meaning of the result.
0.0015 acres is the average land area used to produce a dollars worth of goods.
Ratios and rates each compare two numbers by using division. However, rates compare two measurements that
involve different units of measure. An example of a ratio is
CHALLENGE
and
. An example of a rate is
.
, find the value of . (Hint: Choose different values of a and b for
which the proportions are true and evaluate the expression
.)
The hint suggests that we choose different values of a and b for which the proportions are true and evaluate the
expression
.
How do we go about choosing these numbers? Let s look at the first equation.
What values of a and b make this equation true? The simplest values to use are a = 4 and b = 2, because 4 + 1 = 5
and 2 1 = 1, making the numerators and denominators both equal.
Now that we have two values that work for the first equation, we can test them in the second equation. As it
happens, these values work in the second equation as well.
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Ratios and rates each compare two numbers by using division. However, rates compare two measurements that
involve different units of measure. An example of a ratio is
. An example of a rate is
.
and
CHALLENGE
, find the value of . (Hint: Choose different values of a and b for
which the proportions are true and evaluate the expression
.)
The hint suggests that we choose different values of a and b for which the proportions are true and evaluate the
expression
.
How do we go about choosing these numbers? Let s look at the first equation.
What values of a and b make this equation true? The simplest values to use are a = 4 and b = 2, because 4 + 1 = 5
and 2 1 = 1, making the numerators and denominators both equal.
Now that we have two values that work for the first equation, we can test them in the second equation. As it
happens, these values work in the second equation as well.
and
.
If the values for a and b did not work for the second equation, then we would have had to find two different values
that work in the first equation and try them in the second equation until we found two that worked for both.
WRITING IN MATH On a road trip, Marcus reads a highway sign and then looks at his gas gauge.
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.
If the values
for a and b did not work for the second equation, then we would have had to find two different values
2-6 Ratios
and Proportions
that work in the first equation and try them in the second equation until we found two that worked for both.
WRITING IN MATH On a road trip, Marcus reads a highway sign and then looks at his gas gauge.
Marcus s gas tank holds 10 gallons and his car gets 32 miles per gallon at his current speed of 65 miles per hour. If
he maintains this speed, will he make it to Atlanta without having to stop and get gas? Explain your reasoning.
First, determine how much gas is left in the tank. If the tank is about
Then determine the number of miles he can drive on the
full, he has about
gal of gas
gal of gas. At 32 miles per gallon, he will be able to
Since Atlanta is 200 miles away, he will run out of gas about 20 miles before reaching the city if he doesn t stop to
get gas.
WRITING IN MATH Describe how businesses can use ratios. Write about a real-world situation in which a
business would use a ratio.
For example, a business can use ratios to compare how many of the potential customers in an area use their
business.
Also, using ratios, a pizza business can find the number of potential customers in their area and compare them to how
many they have. They can do the same with their competitors.
In the figure, x : y = 2 : 3 and y : z = 3 : 5. If x = 10, find the value of z.
A 15
B 20
C 25
D 30
First, find y.
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business.
Also, using ratios, a pizza business can find the number of potential customers in their area and compare them to how
many they have. They can do the same with their competitors.
In the figure, x : y = 2 : 3 and y : z = 3 : 5. If x = 10, find the value of z.
A 15
B 20
C 25
D 30
First, find y.
Now, find z.
Choice C is correct.
GRIDDED RESPONSE A race car driver records the finishing times for recent practice trials. What is the mean
time, in seconds, for the trials?
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The mean time is 5.02 seconds.
GEOMETRY If
is equal to
, what is z?
Choice C is correct.
GRIDDED RESPONSE A race car driver records the finishing times for recent practice trials. What is the mean
time, in seconds, for the trials?
GEOMETRY If
is equal to
, what is z?
F 240
G 140
H 120
J 70
is equal to
is equal to
. Then since all three angles in
are equal to all three angles in
. Similar triangles have proportional sides. So, set up a proportion of the sides.
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GEOMETRY If
is equal to
, what is z?
F 240
G 140
H 120
J 70
is equal to
is equal to
. Then since all three angles in
are equal to all three angles in
. Similar triangles have proportional sides. So, set up a proportion of the sides.
Choice G is correct.
Which equation below illustrates the Commutative Property?
A (3x + 4y) +2z = 3x + (4y + 2z)
B 7(x + y) = 7x + 7y
C xyz = yxz
Dx+0=x
Choice B represents the Distributive Property, Choice A shows the Associative Property, and Choice D represents
the Additive Identity. The Commutative Property switches the order of multiplying or adding. Therefore, choice C is
correct.
Solve each equation.
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means the distance between x and 5 is 8. Since distance cannot be negative, the solution is the empty
set
.
Choice B represents the Distributive Property, Choice A shows the Associative Property, and Choice D represents
the Additive
Identity. The Commutative Property switches the order of multiplying or adding. Therefore, choice C is
2-6 Ratios
and Proportions
correct.
means the distance between x and 5 is 8. Since distance cannot be negative, the solution is the empty
set
.
Case 1:
Case 2:
The solution set is ( 7, 11).
Case 1:
Case 2:
The solution set is {10, 7}.
Case 1:
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The solution set is {10, 7}.
Case 1:
Case 2:
The solution set is
.
HEALTH When exercising, a person s pulse rate should not exceed a certain limit. This maximum rate is
represented by the expression 0.8(220 a), where a is age in years. Find the age of a person whose maximum pulse
rate is 152.
A person whose maximum pulse rate is 152 is 30 years old.
Solve each equation. Check your solution.
15 = 4a 5
Check:
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A person whose maximum pulse rate is 152 is 30 years old.
Solve each equation. Check your solution.
15 = 4a 5
Check:
7g
14 = 63
Check:
Check:
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Check:
Check:
GEOMETRY Find the area of
ABC if each small triangle has a base of 5.2 inches and a height of 4.5 inches.
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GEOMETRY Find the area of
ABC if each small triangle has a base of 5.2 inches and a height of 4.5 inches.
The area of one small triangle is:
There are 4 small triangles. Multiply the area of the small triangle by 4 to determine the area of the large triangle. So,
the area of
11.7 or 46.8 square inches.
Evaluate each expression.
4p = 22
5h = 33
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5h = 33
1.25y = 4.375
9.8m = 30.87
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Determine whether each pair of ratios are equivalent ratios. Write

Transcription

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