Percent of Change - PSAT Math
Card 0 of 119
The cost of a shirt in January is 
dollars. In February, the cost is decreased by 10%. In March, the cost is decreased by another 10%. By what percentage did the shirt decrease in total between January and March?
The cost of a shirt in January is
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The best way to answer this question is to plug in a number for n. Since you are working with percentages, it may be easiest to use 100 for n.
We know that in the month of February, the cost of this shirt was decreased by 10%. Because 10% of 100 is $10, the new cost of the shirt is $90.
In March, the cost of the shirt decreased another 10%. 10% of 90 is 9, so the cost of the shirt is now $81.
To find the total percentage decrease, you must divide 81 by 100 and subtract it from 1.
1 – (81/100) = 1 – 0.81 = 0.19
The total decrease was 19%.
The best way to answer this question is to plug in a number for n. Since you are working with percentages, it may be easiest to use 100 for n.
We know that in the month of February, the cost of this shirt was decreased by 10%. Because 10% of 100 is $10, the new cost of the shirt is $90.
In March, the cost of the shirt decreased another 10%. 10% of 90 is 9, so the cost of the shirt is now $81.
To find the total percentage decrease, you must divide 81 by 100 and subtract it from 1.
1 – (81/100) = 1 – 0.81 = 0.19
The total decrease was 19%.
At a certain store, prices for all items were assigned in January. Each month after that, the price was 20% less than the price the previous month. If the price of an item was x dollars in January, approximately what was the price in dollars of the item in June?
At a certain store, prices for all items were assigned in January. Each month after that, the price was 20% less than the price the previous month. If the price of an item was x dollars in January, approximately what was the price in dollars of the item in June?
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The question tells us that the price in January was x. To find the price in February, we decrease the price of x by 20%, which is the same as taking 80% of x. (In general, a P% decrease of a number is the same as (100 – P)% times that number). Continue to take 0.8 times the previous month's price to find the next month's price until we have the price for June, as follows:
January Price: x
February Price: 0.8 * January Price = 0.8_x_
March Price: 0.8 * February Price = 0.8 * 0.8_x_ = 0.82 * x
April Price: 0.8 * March Price = 0.8 * 0.8 * 0.8_x_ = 0.83 * x
May Price: 0.8 * April Price = 0.8 * 0.8 * 0.8 * 0.8_x_ = 0.84 * x
June Price: 0.8 * May Price = 0.8 * 0.8 * 0.8 * 0.8 * 0.8_x_ = 0.85 * x; therefore, the price in June was 0.85 ≈ 0.328 ≈ 0.33 times the original price.
The question tells us that the price in January was x. To find the price in February, we decrease the price of x by 20%, which is the same as taking 80% of x. (In general, a P% decrease of a number is the same as (100 – P)% times that number). Continue to take 0.8 times the previous month's price to find the next month's price until we have the price for June, as follows:
January Price: x
February Price: 0.8 * January Price = 0.8_x_
March Price: 0.8 * February Price = 0.8 * 0.8_x_ = 0.82 * x
April Price: 0.8 * March Price = 0.8 * 0.8 * 0.8_x_ = 0.83 * x
May Price: 0.8 * April Price = 0.8 * 0.8 * 0.8 * 0.8_x_ = 0.84 * x
June Price: 0.8 * May Price = 0.8 * 0.8 * 0.8 * 0.8 * 0.8_x_ = 0.85 * x; therefore, the price in June was 0.85 ≈ 0.328 ≈ 0.33 times the original price.
The sale of a tablet decreased from $500 to $450. By what percentage did the cost decrease?
The sale of a tablet decreased from $500 to $450. By what percentage did the cost decrease?
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Set up the following ratio
50/500 = n/100
The cost of the tablet decreased by $50. The original cost is $500; therefore, 50 is the numerator and 500 is the denominator on the left side of the ratio.
Since a percentage is a part of a whole, n symbolizes the the percentage decrease.
To solve for n, you can cross multiply. So, 50(100) = n(500).
n = 10%
Set up the following ratio
50/500 = n/100
The cost of the tablet decreased by $50. The original cost is $500; therefore, 50 is the numerator and 500 is the denominator on the left side of the ratio.
Since a percentage is a part of a whole, n symbolizes the the percentage decrease.
To solve for n, you can cross multiply. So, 50(100) = n(500).
n = 10%
On Monday, the price of a shirt costs x dollars. On Tuesday, the manager puts the shirt on sale for 10% off Monday's price. On Wednesday, the manager increases the price of the shirt by 10% of Tuesday's price. Describe the change in price from Monday to Wednesday.
On Monday, the price of a shirt costs x dollars. On Tuesday, the manager puts the shirt on sale for 10% off Monday's price. On Wednesday, the manager increases the price of the shirt by 10% of Tuesday's price. Describe the change in price from Monday to Wednesday.
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To find the cost on Tuesday, take 10% off Monday's price. In other words, find 90% of Monday's price. This is simply 0.9_x_. If we are to now add 10% of this value back onto itself to find Wednesday's price, we want 100% + 10%, or 110% of 0.9_x_.
1.1(0.9_x_) = 0.99_x_
This value is 1% smaller than x.
To find the cost on Tuesday, take 10% off Monday's price. In other words, find 90% of Monday's price. This is simply 0.9_x_. If we are to now add 10% of this value back onto itself to find Wednesday's price, we want 100% + 10%, or 110% of 0.9_x_.
1.1(0.9_x_) = 0.99_x_
This value is 1% smaller than x.
The cost of a hat increases by 15% and then decreases by 35%. After the two price changes, the new price of the hat is what percent of the original?
The cost of a hat increases by 15% and then decreases by 35%. After the two price changes, the new price of the hat is what percent of the original?
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The easiest way to do percentage changes is to keep them all in one equation. Therefore, we would say that an increase of 15% is the same as multiplying the original value by 1.15. Likewise, we would say that a discount by 35% is the same as multiplying the original by .65.
For our problem, let the hat cost X dollars originally. Therefore, after its increase, it costs 1.15_X_ dollars. Now, we can consider this new price as the whole to which the discount will be applied. Therefore, a 35% reduction is (1.15_X_) * 0.65.
Simplifying, we get 0.7475, or 74.75%.
The easiest way to do percentage changes is to keep them all in one equation. Therefore, we would say that an increase of 15% is the same as multiplying the original value by 1.15. Likewise, we would say that a discount by 35% is the same as multiplying the original by .65.
For our problem, let the hat cost X dollars originally. Therefore, after its increase, it costs 1.15_X_ dollars. Now, we can consider this new price as the whole to which the discount will be applied. Therefore, a 35% reduction is (1.15_X_) * 0.65.
Simplifying, we get 0.7475, or 74.75%.
If the length of a rectangle is increased by thirty percent, which of the following most closely approximates the percent by which the rectangle's width must decrease, so that the area of the rectangle remains unchanged?
If the length of a rectangle is increased by thirty percent, which of the following most closely approximates the percent by which the rectangle's width must decrease, so that the area of the rectangle remains unchanged?
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If the price of a TV was decreased from $3,000 to $1,800, by what percent was the price decreased?
If the price of a TV was decreased from $3,000 to $1,800, by what percent was the price decreased?
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The price was lowered by $1,200 which is 40% of $3,000.
The price was lowered by $1,200 which is 40% of $3,000.
If a rectangle's length decreases by fifteen percent, and its width decreases by twenty percent, then by what percent does the rectangle's area decrease?
If a rectangle's length decreases by fifteen percent, and its width decreases by twenty percent, then by what percent does the rectangle's area decrease?
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Let's call the original length and width of the rectangle $l_{1}$ and $w_{1}$ , respectively.
The initial area, $A_{1}$, of the rectangle is equal to the product of the length and the width. We can represent this with the following equation:
$A_{1}$$=l_{1}$cdot $w_{1}$
Next, let $l_{2}$ and $w_{2}$ represent the length and width, respectively, after they have been decreased. The final area will be equal to $A_{2}$, which will be equal to the product of the final length and width.
$A_{2}$$=l_{2}$cdot $w_{2}$
We are asked to find the change in the area, which essentially means we want to compare $A_{1}$ and $A_{2}$. In order to do this, we will need to find an expression for $A_{2}$ in terms of $l_{1}$ and $w_{1}$ . We can rewrite $l_{2}$ and $w_{2}$ in terms of $l_{1}$ and $w_{1}$.
First, we are told that the length is decreased by fifteen percent. We can think of the full length as 100% of the length. If we take away fifteen percent, we are left with 100 – 15, or 85% of the length. In other words, the final length is 85% of the original length. We can represent 85% as a decimal by moving the decimal two places to the left.
$l_{2}$ = 85% of $l_{1}$ = $0.85l_{1}$
Similarly, if we decrease the width by 20%, we are only left with 80% of the width.
$w_{2}$ = 80% of $w_{1}$ = $0.80w_{1}$
We can now express the final area in terms of $l_{1}$ and $w_{1}$ by substituting the expressions we just found for the final length and width.
$A_{2}$$=l_{2}$cdot $w_{2}$ = $(0.85l_{1}$$)(0.80w_{1}$) = $0.68l_{1}$$w_{1}$
Lastly, let's apply the formula for percent of change, which will equal the change in the area divided by the original area. The change in the area is equal to the final area minus the original area.
percent change = $$\frac{(A_{2}$$$-A_{1}$$)}{A_{1}$}(100%)
$=\frac{(0.68l_{1}$$$w_{1}$$-l_{1}$$w_{1}$$)}{l_{1}$$w_{1}$}(100%)
$=\frac{-0.32l_{1}$$$w_{1}$$}{l_{1}$$w_{1}$}(100%) = –0.32(100%) = –32%
The negative sign indicates that the rectangle's area decreased. The change in the area was a decrease of 32%.
The answer is 32.
Let's call the original length and width of the rectangle $l_{1}$ and $w_{1}$ , respectively.
The initial area, $A_{1}$, of the rectangle is equal to the product of the length and the width. We can represent this with the following equation:
$A_{1}$$=l_{1}$cdot $w_{1}$
Next, let $l_{2}$ and $w_{2}$ represent the length and width, respectively, after they have been decreased. The final area will be equal to $A_{2}$, which will be equal to the product of the final length and width.
$A_{2}$$=l_{2}$cdot $w_{2}$
We are asked to find the change in the area, which essentially means we want to compare $A_{1}$ and $A_{2}$. In order to do this, we will need to find an expression for $A_{2}$ in terms of $l_{1}$ and $w_{1}$ . We can rewrite $l_{2}$ and $w_{2}$ in terms of $l_{1}$ and $w_{1}$.
First, we are told that the length is decreased by fifteen percent. We can think of the full length as 100% of the length. If we take away fifteen percent, we are left with 100 – 15, or 85% of the length. In other words, the final length is 85% of the original length. We can represent 85% as a decimal by moving the decimal two places to the left.
$l_{2}$ = 85% of $l_{1}$ = $0.85l_{1}$
Similarly, if we decrease the width by 20%, we are only left with 80% of the width.
$w_{2}$ = 80% of $w_{1}$ = $0.80w_{1}$
We can now express the final area in terms of $l_{1}$ and $w_{1}$ by substituting the expressions we just found for the final length and width.
$A_{2}$$=l_{2}$cdot $w_{2}$ = $(0.85l_{1}$$)(0.80w_{1}$) = $0.68l_{1}$$w_{1}$
Lastly, let's apply the formula for percent of change, which will equal the change in the area divided by the original area. The change in the area is equal to the final area minus the original area.
percent change = $$\frac{(A_{2}$$$-A_{1}$$)}{A_{1}$}(100%)
$=\frac{(0.68l_{1}$$$w_{1}$$-l_{1}$$w_{1}$$)}{l_{1}$$w_{1}$}(100%)
$=\frac{-0.32l_{1}$$$w_{1}$$}{l_{1}$$w_{1}$}(100%) = –0.32(100%) = –32%
The negative sign indicates that the rectangle's area decreased. The change in the area was a decrease of 32%.
The answer is 32.
The cost of a load of laundry is reduced by 35%. The cost is then reduced 2 weeks later by another 25%. What is the overall reduction?
The cost of a load of laundry is reduced by 35%. The cost is then reduced 2 weeks later by another 25%. What is the overall reduction?
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The original reduction brings the total to 65% of the original value. Taking a 25% discount off that price gives 48.75% of the original value. This means the reduction had been 51%.
The original reduction brings the total to 65% of the original value. Taking a 25% discount off that price gives 48.75% of the original value. This means the reduction had been 51%.
A dress is reduced in price by 35%, but it still doesn't sell, so the manager discounts it by another 10%. What is the total percentage discount?
A dress is reduced in price by 35%, but it still doesn't sell, so the manager discounts it by another 10%. What is the total percentage discount?
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For these type of questions, it is always best to pretend that we are beginning with a $100 item and to calculate from there.
100times(1-0.35)=100times0.65=65
65times(1-0.1)=65times 0.9=58.5
If an item that is $100 is discounted by 35%, and then another 10%, the new price is 58.5%.
100-58.5=41.5
The price difference (discount) is $41.5 for every $100, or:
frac{41.5}{100}=0.415=41.5%
The total discount is 41.5%.
For these type of questions, it is always best to pretend that we are beginning with a $100 item and to calculate from there.
100times(1-0.35)=100times0.65=65
65times(1-0.1)=65times 0.9=58.5
If an item that is $100 is discounted by 35%, and then another 10%, the new price is 58.5%.
100-58.5=41.5
The price difference (discount) is $41.5 for every $100, or:
frac{41.5}{100}=0.415=41.5%
The total discount is 41.5%.
The population of Town A is 12,979 people in 1995. The population, when measured again in 2005, is 22,752. What was the change in population to the nearest whole percentage point?
The population of Town A is 12,979 people in 1995. The population, when measured again in 2005, is 22,752. What was the change in population to the nearest whole percentage point?
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Since we are looking for the change, we must take the
(Ending Point – Starting Point)/Starting Point * 100%
(22752 – 12979)/12979 * 100%
9773/12979 * 100%
0.753 * 100%
75%
Since we are looking for the change, we must take the
(Ending Point – Starting Point)/Starting Point * 100%
(22752 – 12979)/12979 * 100%
9773/12979 * 100%
0.753 * 100%
75%
A factory produced 2500 units during the month of September. In order to increase production by 12% in the month of October, the factory hired more workers. How many units were produced in October?
A factory produced 2500 units during the month of September. In order to increase production by 12% in the month of October, the factory hired more workers. How many units were produced in October?
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This is a percentage increase problem.
Easiest approach : 2500 x 1.12 = 2800
In this way you are adding 12% to the original.
Using the formula, find 12% of 2500
12/100 = x/2500,
30000 = 100x
300 = x
Now add that to the original to find the new production:
2500 + 300 = 2800
This is a percentage increase problem.
Easiest approach : 2500 x 1.12 = 2800
In this way you are adding 12% to the original.
Using the formula, find 12% of 2500
12/100 = x/2500,
30000 = 100x
300 = x
Now add that to the original to find the new production:
2500 + 300 = 2800
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%
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%.
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%.
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 A = $s^{2}$, and if the side is doubled, the new area becomes A = $(2s)^{2}$ = $4s^{^{2}$}. The increase is a factor of 4, which is 400%.
The area of a square is given by A = $s^{2}$, and if the side is doubled, the new area becomes A = $(2s)^{2}$ = $4s^{^{2}$}. The increase is a factor of 4, which is 400%.
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
The cost of a shirt in January is dollars. In February, the cost is decreased by 10%. In March, the cost is decreased by another 10%. By what percentage did the shirt decrease in total between January and March?
The cost of a shirt in January is
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The best way to answer this question is to plug in a number for n. Since you are working with percentages, it may be easiest to use 100 for n.
We know that in the month of February, the cost of this shirt was decreased by 10%. Because 10% of 100 is $10, the new cost of the shirt is $90.
In March, the cost of the shirt decreased another 10%. 10% of 90 is 9, so the cost of the shirt is now $81.
To find the total percentage decrease, you must divide 81 by 100 and subtract it from 1.
1 – (81/100) = 1 – 0.81 = 0.19
The total decrease was 19%.
The best way to answer this question is to plug in a number for n. Since you are working with percentages, it may be easiest to use 100 for n.
We know that in the month of February, the cost of this shirt was decreased by 10%. Because 10% of 100 is $10, the new cost of the shirt is $90.
In March, the cost of the shirt decreased another 10%. 10% of 90 is 9, so the cost of the shirt is now $81.
To find the total percentage decrease, you must divide 81 by 100 and subtract it from 1.
1 – (81/100) = 1 – 0.81 = 0.19
The total decrease was 19%.
At a certain store, prices for all items were assigned in January. Each month after that, the price was 20% less than the price the previous month. If the price of an item was x dollars in January, approximately what was the price in dollars of the item in June?
At a certain store, prices for all items were assigned in January. Each month after that, the price was 20% less than the price the previous month. If the price of an item was x dollars in January, approximately what was the price in dollars of the item in June?
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The question tells us that the price in January was x. To find the price in February, we decrease the price of x by 20%, which is the same as taking 80% of x. (In general, a P% decrease of a number is the same as (100 – P)% times that number). Continue to take 0.8 times the previous month's price to find the next month's price until we have the price for June, as follows:
January Price: x
February Price: 0.8 * January Price = 0.8_x_
March Price: 0.8 * February Price = 0.8 * 0.8_x_ = 0.82 * x
April Price: 0.8 * March Price = 0.8 * 0.8 * 0.8_x_ = 0.83 * x
May Price: 0.8 * April Price = 0.8 * 0.8 * 0.8 * 0.8_x_ = 0.84 * x
June Price: 0.8 * May Price = 0.8 * 0.8 * 0.8 * 0.8 * 0.8_x_ = 0.85 * x; therefore, the price in June was 0.85 ≈ 0.328 ≈ 0.33 times the original price.
The question tells us that the price in January was x. To find the price in February, we decrease the price of x by 20%, which is the same as taking 80% of x. (In general, a P% decrease of a number is the same as (100 – P)% times that number). Continue to take 0.8 times the previous month's price to find the next month's price until we have the price for June, as follows:
January Price: x
February Price: 0.8 * January Price = 0.8_x_
March Price: 0.8 * February Price = 0.8 * 0.8_x_ = 0.82 * x
April Price: 0.8 * March Price = 0.8 * 0.8 * 0.8_x_ = 0.83 * x
May Price: 0.8 * April Price = 0.8 * 0.8 * 0.8 * 0.8_x_ = 0.84 * x
June Price: 0.8 * May Price = 0.8 * 0.8 * 0.8 * 0.8 * 0.8_x_ = 0.85 * x; therefore, the price in June was 0.85 ≈ 0.328 ≈ 0.33 times the original price.
The sale of a tablet decreased from $500 to $450. By what percentage did the cost decrease?
The sale of a tablet decreased from $500 to $450. By what percentage did the cost decrease?
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Set up the following ratio
50/500 = n/100
The cost of the tablet decreased by $50. The original cost is $500; therefore, 50 is the numerator and 500 is the denominator on the left side of the ratio.
Since a percentage is a part of a whole, n symbolizes the the percentage decrease.
To solve for n, you can cross multiply. So, 50(100) = n(500).
n = 10%
Set up the following ratio
50/500 = n/100
The cost of the tablet decreased by $50. The original cost is $500; therefore, 50 is the numerator and 500 is the denominator on the left side of the ratio.
Since a percentage is a part of a whole, n symbolizes the the percentage decrease.
To solve for n, you can cross multiply. So, 50(100) = n(500).
n = 10%