Percentage - PSAT Math
Card 1 of 700
Cindy is running for student body president and is making circular pins for her campaign. She enlarges her campaign image to fit the entire surface of a circular pin. After the image is enlarged, its new diameter is 75 percent larger than the original. By approximately what percentage has the area of the image increased?
Cindy is running for student body president and is making circular pins for her campaign. She enlarges her campaign image to fit the entire surface of a circular pin. After the image is enlarged, its new diameter is 75 percent larger than the original. By approximately what percentage has the area of the image increased?
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Pick any number to be the original diameter. 10 is easy to work with. If the diameter is 10, the radius is 5. The area of the original image is A = πr2, so the original area = 25π. Now we increase the diameter by 75%, so the new diameter is 17.5. The radius is then 8.75. The area of the enlarged image is approximately 77π. To find the percentage by which the area has increased, take the difference in areas divided by the original area. (77π - 25π)/25π = 51π/25π = 51/25 = 2.04 or approximately 200%
Pick any number to be the original diameter. 10 is easy to work with. If the diameter is 10, the radius is 5. The area of the original image is A = πr2, so the original area = 25π. Now we increase the diameter by 75%, so the new diameter is 17.5. The radius is then 8.75. The area of the enlarged image is approximately 77π. To find the percentage by which the area has increased, take the difference in areas divided by the original area. (77π - 25π)/25π = 51π/25π = 51/25 = 2.04 or approximately 200%
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The radius of a given circle is increased by 20%. What is the percent increase of the area of the circle.
The radius of a given circle is increased by 20%. What is the percent increase of the area of the circle.
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If we plug-in a radius of 5, then a 20% increase would give us a new radius of 6 (which is 1.2 x 5). The area of the new circle is π(6)2 = 36π, and the area of the original circle was π(5)2 = 25π . The numerical increase (or difference) is 36π - 25π = 11π. Next we have to divide this difference by the original area: 11π/25π = .44, which multiplied by 100 gives us a percent increase of 44%. The percent increase = (the numerical increase between the new and original values)/(original value) x 100. The algebraic solution gives us the same answer. If radius r of a certain circle is increased by 20%, then the new radius would be (1.2)r. The area of the new circle would be 1.44 π r2 and the area of the original circle πr2. The difference between the areas is .44 π r2, which divided by the original area, π r2, would give us a percent increase of .44 x 100 = 44%.
If we plug-in a radius of 5, then a 20% increase would give us a new radius of 6 (which is 1.2 x 5). The area of the new circle is π(6)2 = 36π, and the area of the original circle was π(5)2 = 25π . The numerical increase (or difference) is 36π - 25π = 11π. Next we have to divide this difference by the original area: 11π/25π = .44, which multiplied by 100 gives us a percent increase of 44%. The percent increase = (the numerical increase between the new and original values)/(original value) x 100. The algebraic solution gives us the same answer. If radius r of a certain circle is increased by 20%, then the new radius would be (1.2)r. The area of the new circle would be 1.44 π r2 and the area of the original circle πr2. The difference between the areas is .44 π r2, which divided by the original area, π r2, would give us a percent increase of .44 x 100 = 44%.
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Phoenicia is a grocery store that is expanding quickly.
In 2011 Phoenicia's total sales were $1,800,800.
In 2012 their sales rose to $2,130,346.
By what percentage did the store increase its income from 2011 to 2012.
(Round answer to the nearest tenth.)
Phoenicia is a grocery store that is expanding quickly.
In 2011 Phoenicia's total sales were $1,800,800.
In 2012 their sales rose to $2,130,346.
By what percentage did the store increase its income from 2011 to 2012.
(Round answer to the nearest tenth.)
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$1,800,800 divided by 100 equals 18,008 and $2,130,346 divided by 18,008 is 118.3
So we know that $2,130,346 is 118.3% of the sales in the previous year. Hence sales increased by 18.3%.
$1,800,800 divided by 100 equals 18,008 and $2,130,346 divided by 18,008 is 118.3
So we know that $2,130,346 is 118.3% of the sales in the previous year. Hence sales increased by 18.3%.
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If the side of a square is doubled in length, what is the percentage increase in area?
If the side of a square is doubled in length, what is the percentage increase in area?
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The area of a square is given by 
, and if the side is doubled, the new area becomes 
. The increase is a factor of 4, which is 400%.
The area of a square is given by
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Let x and y be numbers such that x and y are both nonzero, and x > y. If half of x is equal to thirty percent of the positive difference between x and y, then what is the ratio of x to y?
Let x and y be numbers such that x and y are both nonzero, and x > y. If half of x is equal to thirty percent of the positive difference between x and y, then what is the ratio of x to y?
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We need to find expressions for fifty percent of x and for thirty percent of the positive difference between x and y. Then, we can set these two expressions equal to each other and determine the ratio of x to y.
Fifty percent of x is equal to one-half of x, which is the same as multiplying x by 0.50.
50% of x = 0.5_x_
Thirty percent of the positive difference between x and y means that we need to multiply the positive difference between x and y by thirty percent. Because x > y, the positive difference between x and y is equal to x – y. We then need to take thirty percent of the quantity x – y. Remember that to convert from a percent to a decimal, we move the decimal two spaces to the left. Therefore, 30% = 0.30. We can now multiply this by (x – y).
30% of x – y = 0.30(x – y)
Now, we set the two expressions equal to one another.
0.5_x_ = 0.30(x – y)
Distribute the right side.
0.5_x_ = 0.3_x_ – 0.3_y_
The ratio of x to y is represent by x/y. Thus, we want to group the x and y terms on opposite sides of the equations, and then divide both sides by y.
0.5_x_ = 0.3_x_ – 0.3_y_
Subtract 0.3_x_ from both sides.
0.2_x_ = –0.3_y_
Divide both sides by 0.2
x = (–0.3/0.2)y
Divide both sides by y to find x/y.
x/y = (–0.3/0.2) = –1.5.
Because the answers are in fractions, we want to rewrite –1.5 as a fraction. We can write –1.5 as –1.5/1 and then mutiply the top and bottom by 2.
(–1.5/1)(2/2) = –3/2
The answer is –3/2
We need to find expressions for fifty percent of x and for thirty percent of the positive difference between x and y. Then, we can set these two expressions equal to each other and determine the ratio of x to y.
Fifty percent of x is equal to one-half of x, which is the same as multiplying x by 0.50.
50% of x = 0.5_x_
Thirty percent of the positive difference between x and y means that we need to multiply the positive difference between x and y by thirty percent. Because x > y, the positive difference between x and y is equal to x – y. We then need to take thirty percent of the quantity x – y. Remember that to convert from a percent to a decimal, we move the decimal two spaces to the left. Therefore, 30% = 0.30. We can now multiply this by (x – y).
30% of x – y = 0.30(x – y)
Now, we set the two expressions equal to one another.
0.5_x_ = 0.30(x – y)
Distribute the right side.
0.5_x_ = 0.3_x_ – 0.3_y_
The ratio of x to y is represent by x/y. Thus, we want to group the x and y terms on opposite sides of the equations, and then divide both sides by y.
0.5_x_ = 0.3_x_ – 0.3_y_
Subtract 0.3_x_ from both sides.
0.2_x_ = –0.3_y_
Divide both sides by 0.2
x = (–0.3/0.2)y
Divide both sides by y to find x/y.
x/y = (–0.3/0.2) = –1.5.
Because the answers are in fractions, we want to rewrite –1.5 as a fraction. We can write –1.5 as –1.5/1 and then mutiply the top and bottom by 2.
(–1.5/1)(2/2) = –3/2
The answer is –3/2
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If 
of 
is equal to 
of 
, and 
of 
is equal to 
of 
, then what percent of 
is 
?
If
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We are told that 50% of x is equal to 25% of y. We need to represent these two pieces of information as algebraic expressions. We can convert 50% and 25% to decimals by moving the decimals two places to the left. Thus, 50% = 0.50, and 25% = 0.25. To find 50% of x, we multiply x by 0.50. In other words, 50% of x = 0.50_x_. Likewise, 25% of y = 0.25_y_. We now set 0.50_x_ and 0.25_y_ equal to one another.
0.50_x_ = 0.25_y_
Let's divide both sides by 0.25 to get rid of decimals.
2_x_ = y
Next, we are told that 40% of y is equal to 60% of z. We will represent 40% and 60% as 0.40 and 0.60, respectively. Thus, we can write the following equation:
0.40_y_ = 0.60_z_
Ultimately, we are asked to find x as a percentage of z. This means we want to find an equation with x and z, but not y. If we solve for y in the second equation, and then substitute this value into the first, we can eliminate y.
Let's take the equation 0.40_y_ = 0.60_z_ and divide both sides by 0.40.
y = 1.5_z_
Now, we can take 1.5_z_ and substitute this for y in the first equation.
2_x_ = 1.5_z_
In order to find x as a percent of z, we must solve for x in terms of z. This means we must divide both sides of the equation by 2.
x = 0.75_z_
x is 0.75 times z. We can represent 0.75 as 75%, because in order to convert from a decimal to a percent, we need to move the decimal two spaces to the right. Therefore, if x = 0.75_z_, then x = 75% of z.
The answer is 75.
We are told that 50% of x is equal to 25% of y. We need to represent these two pieces of information as algebraic expressions. We can convert 50% and 25% to decimals by moving the decimals two places to the left. Thus, 50% = 0.50, and 25% = 0.25. To find 50% of x, we multiply x by 0.50. In other words, 50% of x = 0.50_x_. Likewise, 25% of y = 0.25_y_. We now set 0.50_x_ and 0.25_y_ equal to one another.
0.50_x_ = 0.25_y_
Let's divide both sides by 0.25 to get rid of decimals.
2_x_ = y
Next, we are told that 40% of y is equal to 60% of z. We will represent 40% and 60% as 0.40 and 0.60, respectively. Thus, we can write the following equation:
0.40_y_ = 0.60_z_
Ultimately, we are asked to find x as a percentage of z. This means we want to find an equation with x and z, but not y. If we solve for y in the second equation, and then substitute this value into the first, we can eliminate y.
Let's take the equation 0.40_y_ = 0.60_z_ and divide both sides by 0.40.
y = 1.5_z_
Now, we can take 1.5_z_ and substitute this for y in the first equation.
2_x_ = 1.5_z_
In order to find x as a percent of z, we must solve for x in terms of z. This means we must divide both sides of the equation by 2.
x = 0.75_z_
x is 0.75 times z. We can represent 0.75 as 75%, because in order to convert from a decimal to a percent, we need to move the decimal two spaces to the right. Therefore, if x = 0.75_z_, then x = 75% of z.
The answer is 75.
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Find the simplified fraction for 
Find the simplified fraction for
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For any percent 
we can write it as a fraction in the form 
So we can write 
as 
Then we simplify
So our simplified answer is 
For any percent
So we can write
Then we simplify
So our simplified answer is
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You have 2 fair dice that each have 6 faces. On one roll, what's the probability that both dice land on an even number?
You have 2 fair dice that each have 6 faces. On one roll, what's the probability that both dice land on an even number?
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In order to find the probability of rolling two even numbers on 2 dice we need to find the probability of each dice having an even number on the roll and then multiply them together.
The probability of rolling an even number is
because 2, 4, and 6 are even numbers which goes in the numerator and there are 6 total numbers which goes in the denominator. After this we reduce the fraction and get one half. Then we need to muliply this by probability of the second dice having an even number.
In order to find the probability of rolling two even numbers on 2 dice we need to find the probability of each dice having an even number on the roll and then multiply them together.
The probability of rolling an even number is
because 2, 4, and 6 are even numbers which goes in the numerator and there are 6 total numbers which goes in the denominator. After this we reduce the fraction and get one half. Then we need to muliply this by probability of the second dice having an even number.
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What is 15% of a $16.73 bill?
What is 15% of a $16.73 bill?
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We can re write 15% as a fraction and then use proportions to solve.
From here we cross multiply and divide to solve for 
.
We can re write 15% as a fraction and then use proportions to solve.
From here we cross multiply and divide to solve for
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If a test has a total of 
questions and 
of the questions are multiple choice, how many questions are multiple choice?
If a test has a total of
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To solve this problem we set up a ratio. We want to find 
of 
. Therefore, we set up the following ratio:
In the case 
represents the number of questions that are multiple choice. From here we cross multiply and divide.
To solve this problem we set up a ratio. We want to find
In the case
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If there are 3 boys in a class and 7 girls. What percent of the class is made up of boys?
If there are 3 boys in a class and 7 girls. What percent of the class is made up of boys?
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To solve this problem we set up a ratio of part of total. The part is the number of boys in the class and the total is the number of boys and girls in the class.
now to find the percent we can multiply this fraction by 10/10
From here we can see that it is 30%
To solve this problem we set up a ratio of part of total. The part is the number of boys in the class and the total is the number of boys and girls in the class.
now to find the percent we can multiply this fraction by 10/10
From here we can see that it is 30%
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Marker Colors Students Blue 13 Pink 10 Orange 5 Brown 5 Green 7
The above chart shows the number of students in a class who chose each of the five marker colors available.
What percentage of the class chose a green marker?
| Marker Colors | Students |
|---|---|
| Blue | 13 |
| Pink | 10 |
| Orange | 5 |
| Brown | 5 |
| Green | 7 |
The above chart shows the number of students in a class who chose each of the five marker colors available.
What percentage of the class chose a green marker?
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To figure out what percentage of the class chose green markers, you must first figure out what fraction of the class chose green markers. Then, you must convert that fraction into a percentage.
Figure out the fraction:
7 students chose green markers
40 students total
Fraction of students who chose green: 
To convert this fraction to a percentage, you must multiply the fraction times 100, then divide the numerator by the denominator. You multiply the fraction times 100 because in order to figure out the percent, you must figure out what the fraction means "for (per) every hundred (cent)".
Multiply times 100
Therefore, the answer is 
.
To figure out what percentage of the class chose green markers, you must first figure out what fraction of the class chose green markers. Then, you must convert that fraction into a percentage.
Figure out the fraction:
7 students chose green markers
40 students total
Fraction of students who chose green:
To convert this fraction to a percentage, you must multiply the fraction times 100, then divide the numerator by the denominator. You multiply the fraction times 100 because in order to figure out the percent, you must figure out what the fraction means "for (per) every hundred (cent)".
Multiply times 100
Therefore, the answer is
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A rectangle has a width of 6 and a length of 10. If both the width and length are increased by 2, what is the percent increase of the area of the rectangle?
A rectangle has a width of 6 and a length of 10. If both the width and length are increased by 2, what is the percent increase of the area of the rectangle?
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The equation to use for percent increase or decrease is 
to find the change in area, let's first calculate the area of the original. The formula for area of a rectangle is area=length x width. The original area, therefore, is 
.
The new rectangle has side lengths which are increased by 2, so the new lengths are 8 and 12. The area of the new rectangle, then is 
.
The change in area is found by subtracting the the old area from the new area, so 
.
When you input these numbers into the percent increase equation, you get
The equation to use for percent increase or decrease is
to find the change in area, let's first calculate the area of the original. The formula for area of a rectangle is area=length x width. The original area, therefore, is
The new rectangle has side lengths which are increased by 2, so the new lengths are 8 and 12. The area of the new rectangle, then is
The change in area is found by subtracting the the old area from the new area, so
When you input these numbers into the percent increase equation, you get
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55 and 1/2% of 23 is about what?
55 and 1/2% of 23 is about what?
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55 and 1/2% can be written as a decimal: 0.555. To see what number is about 55.5% of 23, multiply 0.555 by 23. Answer: 12.765 or about 13.
Another route is to say that 55.5% is about half of 23. Half of 23 is 11.5. Since 55.5% is greater than 50%, 13 is the logical choice instead of 11.
55 and 1/2% can be written as a decimal: 0.555. To see what number is about 55.5% of 23, multiply 0.555 by 23. Answer: 12.765 or about 13.
Another route is to say that 55.5% is about half of 23. Half of 23 is 11.5. Since 55.5% is greater than 50%, 13 is the logical choice instead of 11.
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Each wooden chair that a carpenter makes requires $20 worth of supplies. He then sells the chairs for $50 each. The carpenter recently discovered a new supplier that would allow him to spend 25% less on supplies. If he doesn't change his selling price, by what percent could the carpenter increase his profit by using the new supplier?
Each wooden chair that a carpenter makes requires $20 worth of supplies. He then sells the chairs for $50 each. The carpenter recently discovered a new supplier that would allow him to spend 25% less on supplies. If he doesn't change his selling price, by what percent could the carpenter increase his profit by using the new supplier?
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Using $20 worth of supplies and selling the chairs for $50 each, the carpenter is originally making a profit of $30 per chair.
The new supplier would reduce costs by 25% or 1/4. One-fourth of $20 is $5, so the new supplier would be $5 less, or $15.
If the selling price is the same ($50), then the carpenter would now make a profit of $35 per chair, a change of $5.
To calculate percent increase, divide the actual change in profit by the original profit amount, and multiply the result by 100%:
(Actual Change ÷ Original Amount) * 100% = 5/30 * 100% = 500%/30 = 16.7%
Using $20 worth of supplies and selling the chairs for $50 each, the carpenter is originally making a profit of $30 per chair.
The new supplier would reduce costs by 25% or 1/4. One-fourth of $20 is $5, so the new supplier would be $5 less, or $15.
If the selling price is the same ($50), then the carpenter would now make a profit of $35 per chair, a change of $5.
To calculate percent increase, divide the actual change in profit by the original profit amount, and multiply the result by 100%:
(Actual Change ÷ Original Amount) * 100% = 5/30 * 100% = 500%/30 = 16.7%
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A sunglasses kiosk at the mall makes a $50 profit for every 6 pairs of sunglasses it sells. How many pairs of sunglasses must it sell to earn $1000 profit?
A sunglasses kiosk at the mall makes a $50 profit for every 6 pairs of sunglasses it sells. How many pairs of sunglasses must it sell to earn $1000 profit?
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Divide the profit per 6 pairs into the total desired profit $1000/$50 = 20.
Multiply 20 by 6 sunglasses = 120 sunglasses. Or use 6/50 = x/1000 and solve for x.
Divide the profit per 6 pairs into the total desired profit $1000/$50 = 20.
Multiply 20 by 6 sunglasses = 120 sunglasses. Or use 6/50 = x/1000 and solve for x.
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Ricky works at a shoe shop, and earns $40 in commission for each pair of shoes he sells plus a $100 weekly salary. If Ricky receives no other money, which of the following expressions represents the total dollar amount Ricky receives for a week in which he sells n shoes?
Ricky works at a shoe shop, and earns $40 in commission for each pair of shoes he sells plus a $100 weekly salary. If Ricky receives no other money, which of the following expressions represents the total dollar amount Ricky receives for a week in which he sells n shoes?
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If Ricky sells n shoes in a week, he earns $40_n_ in commission. His salary is a constant $100 per week, so his total payout is $100 + $40_n._
If Ricky sells n shoes in a week, he earns $40_n_ in commission. His salary is a constant $100 per week, so his total payout is $100 + $40_n._
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An entrepreneur started a company making floggles. The factory requires $1000 worth of fixed expenses to keep it running every month. She is able to produce one floggle at the cost of $4 and sell one floggle at the cost of $6. If she produces and sells 500 floggles in one month, what is her profit?
An entrepreneur started a company making floggles. The factory requires $1000 worth of fixed expenses to keep it running every month. She is able to produce one floggle at the cost of $4 and sell one floggle at the cost of $6. If she produces and sells 500 floggles in one month, what is her profit?
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Profit = Income - Expenditures
Income = $6/floggle times 500 floggles = $3000
Expenditures = $1000 + $4/floggle times 500 floggles = $1000 + $2000 = $3000
Profit = 3000 - 3000 = 0
Profit = Income - Expenditures
Income = $6/floggle times 500 floggles = $3000
Expenditures = $1000 + $4/floggle times 500 floggles = $1000 + $2000 = $3000
Profit = 3000 - 3000 = 0
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You are planning a New Year’s Eve bash. For each person attending, the caterer will charge you $15 for food, $10 for beverages, $5 for service. The band charges $2000 for the entire evening. You also have to pay the venue $2500 to rent the location for the night and $3 for parking for each attendee. If you expect 500 people to attend and you would like to make a $10000 profit for planning the event, how much must each ticket cost?
You are planning a New Year’s Eve bash. For each person attending, the caterer will charge you $15 for food, $10 for beverages, $5 for service. The band charges $2000 for the entire evening. You also have to pay the venue $2500 to rent the location for the night and $3 for parking for each attendee. If you expect 500 people to attend and you would like to make a $10000 profit for planning the event, how much must each ticket cost?
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First determine total cost.
Caterer: Per person = $15 + $10 + $5 = $30 per person
Parking: Per person = $3 per person
Total per person = $33
$33 * 500 people = $16,500
Plus cost of renting venue + band = $2500 + $2000 = $4500
Total (net) cost = $16,500 + $4500 = $21,000
Total (gross) cost = net cost + profit = $21,000 + $10,000 = $31,000
Cost per ticket = Gross cost / # of attendee = $31,000 / 500 = $62
First determine total cost.
Caterer: Per person = $15 + $10 + $5 = $30 per person
Parking: Per person = $3 per person
Total per person = $33
$33 * 500 people = $16,500
Plus cost of renting venue + band = $2500 + $2000 = $4500
Total (net) cost = $16,500 + $4500 = $21,000
Total (gross) cost = net cost + profit = $21,000 + $10,000 = $31,000
Cost per ticket = Gross cost / # of attendee = $31,000 / 500 = $62
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The Widget Company has annual revenues of $150,000. Their expenses over the same time frame was $75,000. What was the percent profit?
The Widget Company has annual revenues of $150,000. Their expenses over the same time frame was $75,000. What was the percent profit?
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Profit = Revenue – Expense
% Profit = $ Profit ÷ $ Total Revenue
% Profit = ($150,000 – $75,000) ÷ $150,000 = 50%
Profit = Revenue – Expense
% Profit = $ Profit ÷ $ Total Revenue
% Profit = ($150,000 – $75,000) ÷ $150,000 = 50%
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