Proportion / Ratio / Rate - ACT Math
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In a school relay race, every 5-man team must sprint 110 yards total. However, Jim’s team is short a person. Thus, Jim must run two sections for his team. How far does Jim have to run?
In a school relay race, every 5-man team must sprint 110 yards total. However, Jim’s team is short a person. Thus, Jim must run two sections for his team. How far does Jim have to run?
We first find out how much each person must run. Thus we take 110/5 to get 22 yards. However, Jim must run twice the amount of a normal competitor, so we multiply by 2 to get 44 yards.
We first find out how much each person must run. Thus we take 110/5 to get 22 yards. However, Jim must run twice the amount of a normal competitor, so we multiply by 2 to get 44 yards.
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A building that is 30 feet tall casts a shadow that is 50 feet long. If another building casts a shadow that is 100 feet long, how tall is the building?
A building that is 30 feet tall casts a shadow that is 50 feet long. If another building casts a shadow that is 100 feet long, how tall is the building?
This problem can be set up as a proportion: 30 feet/x feet = 50 feet/100 feet. To solve, we simply cross multiply: (30 feet * 100 feet) = (50 feet * x feet). Thus, 3000 feet = 50_x_ feet. To solve for x, divide each side by 50. Therefore, x = 60 feet. If you got 167 feet, you may have set up the proportion incorrectly by mixing up the height of the building with the length of the shadow. If you got 6 feet or 600 feet, you may have made a computational error. If you got 1670 feet, you may have set the proportion up incorrectly and made a computational error.
This problem can be set up as a proportion: 30 feet/x feet = 50 feet/100 feet. To solve, we simply cross multiply: (30 feet * 100 feet) = (50 feet * x feet). Thus, 3000 feet = 50_x_ feet. To solve for x, divide each side by 50. Therefore, x = 60 feet. If you got 167 feet, you may have set up the proportion incorrectly by mixing up the height of the building with the length of the shadow. If you got 6 feet or 600 feet, you may have made a computational error. If you got 1670 feet, you may have set the proportion up incorrectly and made a computational error.
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If a bicyclist can bike 24 miles per hour, how far (in miles) can he travel in 2 minutes, assuming he bikes at a constant speed (answer rounded to the nearest tenth)?
If a bicyclist can bike 24 miles per hour, how far (in miles) can he travel in 2 minutes, assuming he bikes at a constant speed (answer rounded to the nearest tenth)?
0.8 mile. Using some conversions: ( (24mi/1hr)*(1hr/60min)*2min = 0.8 mile
0.8 mile. Using some conversions: ( (24mi/1hr)*(1hr/60min)*2min = 0.8 mile
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If 10,000 lbs of cement makes 85,000 lbs of concrete, how many pounds of concrete can be made with 3,000 pounds of cement?
If 10,000 lbs of cement makes 85,000 lbs of concrete, how many pounds of concrete can be made with 3,000 pounds of cement?
25,500 lbs of concrete. Setting up a ratio with x representing the number of pounds concrete the 3,000 of cement produces, we obtain the relation: (10,000/85,000) = (3,000/x), x = 25,500 pounds of concrete.
25,500 lbs of concrete. Setting up a ratio with x representing the number of pounds concrete the 3,000 of cement produces, we obtain the relation: (10,000/85,000) = (3,000/x), x = 25,500 pounds of concrete.
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A group of 15 friends is having lunch together. Each person eats at least 2/3 of a pizza. What is the smallest number of whole pizzas needed for lunch?
A group of 15 friends is having lunch together. Each person eats at least 2/3 of a pizza. What is the smallest number of whole pizzas needed for lunch?
The minimum number of whole pizzas needed is 15(2/3) = 10.
The minimum number of whole pizzas needed is 15(2/3) = 10.
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There are 150 students in a lecture hall class in college. 12% of the students received an A. 20 students received a B. Twice the number of students who earned an A received a C. The remainder of the students received a D. Which grade did the students receive more than any other?
There are 150 students in a lecture hall class in college. 12% of the students received an A. 20 students received a B. Twice the number of students who earned an A received a C. The remainder of the students received a D. Which grade did the students receive more than any other?
First find 12% of 150, so 0.12 * 150 = 18 students received an A.
20 students received a B, and 36 students received a C (double the A's).
To find the number of D-grades, all we have to do is subtract these from the total (since there were no grades of F),
Thus: 150 – 18 – 20 – 36 = 76 students who received a D in the course, which is the most common grade.
First find 12% of 150, so 0.12 * 150 = 18 students received an A.
20 students received a B, and 36 students received a C (double the A's).
To find the number of D-grades, all we have to do is subtract these from the total (since there were no grades of F),
Thus: 150 – 18 – 20 – 36 = 76 students who received a D in the course, which is the most common grade.
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A brownie recipes calls for a 1:5 ratio of water to brownie mix. If you need 90 cups of brownie mix, how much water do you need?
A brownie recipes calls for a 1:5 ratio of water to brownie mix. If you need 90 cups of brownie mix, how much water do you need?
First set up a proportion, 1/5 = x/90, then solve for x: 5x = 90 → x = 18 cups.
First set up a proportion, 1/5 = x/90, then solve for x: 5x = 90 → x = 18 cups.
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If a 12 oz can of lemonade has 75 calories; how many calories are in an 8 oz can of lemonade?
If a 12 oz can of lemonade has 75 calories; how many calories are in an 8 oz can of lemonade?
A proportion is a statement of equality between two fractions or two ratios. Set up a proportion between the size of the drink and the calories. To solve a proportion cross multiply and solve the resulting equation.
75/12 = x/8 → 150/24 = 3x/24 → 50/8 = x/8 → x = 50
A proportion is a statement of equality between two fractions or two ratios. Set up a proportion between the size of the drink and the calories. To solve a proportion cross multiply and solve the resulting equation.
75/12 = x/8 → 150/24 = 3x/24 → 50/8 = x/8 → x = 50
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A farmer has a piece of property that is 10,000 feet by 40,000 feet. His annual property taxes are paid at a rate of \$3.50 per acre. If one acre = 43,560 ft2, how much will the farmer pay in taxes this year? Round to the nearest dollar.
A farmer has a piece of property that is 10,000 feet by 40,000 feet. His annual property taxes are paid at a rate of \$3.50 per acre. If one acre = 43,560 ft2, how much will the farmer pay in taxes this year? Round to the nearest dollar.
Property area = 10,000 ft x 40,000 ft = 400,000,000 ft2
Acreage = 400,000,000 ft2 / 43,560 ft2 per acre = 9,183 acres
Taxes = \$3.50 per acre x 9,183 acres = \$32,140
Property area = 10,000 ft x 40,000 ft = 400,000,000 ft2
Acreage = 400,000,000 ft2 / 43,560 ft2 per acre = 9,183 acres
Taxes = \$3.50 per acre x 9,183 acres = \$32,140
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When television remotes are shipped from a certain factory, 1 out of every 200 is defective. What is the ratio of defective to nondefective remotes?
When television remotes are shipped from a certain factory, 1 out of every 200 is defective. What is the ratio of defective to nondefective remotes?
One remote is defective for every 199 non-defective remotes.
One remote is defective for every 199 non-defective remotes.
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On a desk, there are 
papers for every 
paper clips and 
papers for every 
greeting card. What is the ratio of paper clips to total items on the desk?
On a desk, there are
Begin by making your life easier: presume that there are 
papers on the desk. Immediately, we know that there are 
paper clips. Now, if there are 
papers, you know that there also must be 
greeting cards. Technically you figure this out by using the ratio:
By cross-multiplying you get:
Solving for 
, you clearly get 
.
(Many students will likely see this fact without doing the algebra, however. The numbers are rather simple.)
Now, this means that our desk has on it:
papers
paper clips
greeting cards
Therefore, you have 
total items. Based on this, your ratio of paper clips to total items is:
, which is the same as 
.
Begin by making your life easier: presume that there are
By cross-multiplying you get:
Solving for
(Many students will likely see this fact without doing the algebra, however. The numbers are rather simple.)
Now, this means that our desk has on it:
Therefore, you have
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In a garden, there are 
pansies, 
lilies, 
roses, and 
petunias. What is the ratio of petunias to the total number of flowers in the garden?
In a garden, there are
To begin, you need to do a simple addition to find the total number of flowers in the garden:
Now, the ratio of petunias to the total number of flowers in the garden can be represented by a simple division of the number of petunias by 
. This is:
Next, reduce the fraction by dividing out the common 
from the numerator and the denominator:
This is the same as 
.
To begin, you need to do a simple addition to find the total number of flowers in the garden:
Now, the ratio of petunias to the total number of flowers in the garden can be represented by a simple division of the number of petunias by
Next, reduce the fraction by dividing out the common
This is the same as
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In a classroom of 
students, each student takes a language class (and only one—nobody studies two languages). 
take Latin, 
take Greek, 
take Anglo-Saxon, and the rest take Old Norse. What is the ratio of students taking Old Norse to students taking Greek?
In a classroom of
To begin, you need to calculate how many students are taking Old Norse. This is:
Now, the ratio of students taking Old Norse to students taking Greek is the same thing as the fraction of students taking Old Norse to students taking Greek, or:
Next, just reduce this fraction to its lowest terms by dividing the numerator and denominator by their common factor of 
:
This is the same as 
.
To begin, you need to calculate how many students are taking Old Norse. This is:
Now, the ratio of students taking Old Norse to students taking Greek is the same thing as the fraction of students taking Old Norse to students taking Greek, or:
Next, just reduce this fraction to its lowest terms by dividing the numerator and denominator by their common factor of
This is the same as
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Jeff went to a bookstore where science books cost \$10.00 each and comic books cost \$5.50 each. If Jeff bought twice as many comic books as science books, and spent a total of \$42.00, how many comic books did he buy?
Jeff went to a bookstore where science books cost \$10.00 each and comic books cost \$5.50 each. If Jeff bought twice as many comic books as science books, and spent a total of \$42.00, how many comic books did he buy?
Assign a variable to science books since everything in the question can be written in terms of science books.
Write an expression for the phrase "twice as many comic books as science books."
To create an equation for the cost of the books, we can write the following:
Substitute in the known values and variables.
Jeff purchased 2 science books and 4 comic books.
Assign a variable to science books since everything in the question can be written in terms of science books.
Write an expression for the phrase "twice as many comic books as science books."
To create an equation for the cost of the books, we can write the following:
Substitute in the known values and variables.
Jeff purchased 2 science books and 4 comic books.
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The ratio of 
to 
is 4 to 9, and the ratio of 
to 
is 5 to 6. What is the ratio of 
to 
?
The ratio of 
Using the given information we can generate the following two proportions:
and 
Next, cross-multiply each proportion to come up with the following two equations:
and 
Each equation shares a term with the 
variable. We need to make this variable equal in both equations to continue. Multiply the first equation by a factor of 3 and the second by a factor of 2, so that the 
terms are equivalent. Let's start with the first equation.
Now, we will perform a similar operation on the second equation.
Now, we can set these equations equal to one another.
Drop the equivalent 
terms.
The proportion then becomes the following:
or 
Using the given information we can generate the following two proportions:
Next, cross-multiply each proportion to come up with the following two equations:
Each equation shares a term with the 
Now, we will perform a similar operation on the second equation.
Now, we can set these equations equal to one another.
Drop the equivalent 
The proportion then becomes the following:
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On her birthday in 2013, Molly was three times older than Steve. On her birthday in 2016, Molly was 2 times older than Steve. How old was Steve on Molly's birthday in 2013?
On her birthday in 2013, Molly was three times older than Steve. On her birthday in 2016, Molly was 2 times older than Steve. How old was Steve on Molly's birthday in 2013?
First, let's assign variables to the names of the individuals to represent their age in 2013.
In 2013, Molly was three times older than Steve; therefore, we can write the following expression:
We are also told that in 2016, Molly will be two times older than Steve; thus, we can write another expression:
.
We can then substitute 
in for 
in the second equation to arrive at the following:
First, let's assign variables to the names of the individuals to represent their age in 2013.
In 2013, Molly was three times older than Steve; therefore, we can write the following expression:
We are also told that in 2016, Molly will be two times older than Steve; thus, we can write another expression:
We can then substitute 
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The ratio of a to b is 9:2, and the ratio of c to b is 5:3. What is the ratio of a to c?
The ratio of a to b is 9:2, and the ratio of c to b is 5:3. What is the ratio of a to c?
Set up the proportions a/b = 9/2 and c/b = 5/3 and cross multiply.
2a = 9b and 3c = 5b.
Next, substitute the b’s in order to express a and c in terms of each other.
10a = 45b and 27c = 45b --> 10a = 27c
Lastly, reverse cross multiply to get a and c back into a proportion.
a/c = 27/10
Set up the proportions a/b = 9/2 and c/b = 5/3 and cross multiply.
2a = 9b and 3c = 5b.
Next, substitute the b’s in order to express a and c in terms of each other.
10a = 45b and 27c = 45b --> 10a = 27c
Lastly, reverse cross multiply to get a and c back into a proportion.
a/c = 27/10
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Joe needs to repair the roof of his house. He finds two companies that can complete the job. Company A charges an initial cost of \$120, plus \$15 per hour of labor, while Company B charges an initial cost of \$95, plus \$20 per hour of labor. After how many hours of labor does Company A cost less than Company B to repair the roof?
Joe needs to repair the roof of his house. He finds two companies that can complete the job. Company A charges an initial cost of \$120, plus \$15 per hour of labor, while Company B charges an initial cost of \$95, plus \$20 per hour of labor. After how many hours of labor does Company A cost less than Company B to repair the roof?
In order to solve this problem, create an equation that summarizes the roof repair cost for each company. Begin by composing a formula for Company A, which charges 120 dollars upfront and 15 dollars per hour of labor.
Now, Company B charges 95 dollars upfront and 20 dollars per hour of labor. We can write the following equation:
The question asks us to find how many hours of labor that a repair must take in order for Company A to be cheaper than Company B. In other words, we need to compose an inequality in which the cost of Company A is less than the cost of Company B. We will substitute the variable 
for hours and solve.
Subtract 
from each side of the inequality.
Subtract 95 from both sides of the inequality.
Divide both sides of the inequality by 5.
If the repair will take more than 5 hours, Company A will be cheaper.
In order to solve this problem, create an equation that summarizes the roof repair cost for each company. Begin by composing a formula for Company A, which charges 120 dollars upfront and 15 dollars per hour of labor.
Now, Company B charges 95 dollars upfront and 20 dollars per hour of labor. We can write the following equation:
The question asks us to find how many hours of labor that a repair must take in order for Company A to be cheaper than Company B. In other words, we need to compose an inequality in which the cost of Company A is less than the cost of Company B. We will substitute the variable 
Subtract 
Subtract 95 from both sides of the inequality.
Divide both sides of the inequality by 5.
If the repair will take more than 5 hours, Company A will be cheaper.
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There is a shipment of 50 radios; 5 of them are defective; what is the ratio of non-defective to defective?
There is a shipment of 50 radios; 5 of them are defective; what is the ratio of non-defective to defective?
Since there are 5 defective radios, there are 45 nondefective radios; therefore, the ratio of non-defective to defective is 45 : 5, or 9 : 1.
Since there are 5 defective radios, there are 45 nondefective radios; therefore, the ratio of non-defective to defective is 45 : 5, or 9 : 1.
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The ratio of 
to 
is 
to 
, while the ratio of 
to 
is 
to 
.
What is the ratio of 
to 
?
The ratio of
What is the ratio of
Since the ratios are fixed, regardless of the actual values of 
, 
, or 
, we can let 
and 
In order to convert to a form where we can relate 
to 
, we must set the coefficient of 
of each ratio equal such that the ratio can be transferred. This is done most easily by finding a common multiple of 
and 
(the ratio of 
to 
and 
, respectively) which is 
Thus, we now have 
and 
.
Setting the 
values equal, we get 
, or a ratio of 
Since the ratios are fixed, regardless of the actual values of
In order to convert to a form where we can relate
Thus, we now have
Setting the
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