Percent of Change - PSAT Math
Card 1 of 119
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?
Tap to reveal answer
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
← Didn't Know|Knew It →
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
Tap to reveal answer
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%.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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%
← Didn't Know|Knew It →
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.
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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%.
← Didn't Know|Knew It →
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?
Tap to reveal answer
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
Let's call the original length and width of the rectangle 
and 
, respectively.
The initial area, 
, of the rectangle is equal to the product of the length and the width. We can represent this with the following equation:
Next, let 
and 
represent the length and width, respectively, after they have been decreased. The final area will be equal to 
, which will be equal to the product of the final length and width.
We are asked to find the change in the area, which essentially means we want to compare 
and 
. In order to do this, we will need to find an expression for 
in terms of 
and 
. We can rewrite 
and 
in terms of 
and 
.
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.
= 85% of 
= 
Similarly, if we decrease the width by 20%, we are only left with 80% of the width.
= 80% of 
= 
We can now express the final area in terms of 
and 
by substituting the expressions we just found for the final length and width.
= (
)(
) = 
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 = 
(100%)
=
(100%)
=
(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
The initial area,
Next, let
We are asked to find the change in the area, which essentially means we want to compare
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.
Similarly, if we decrease the width by 20%, we are only left with 80% of the width.
We can now express the final area in terms of
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 =
=
=
The negative sign indicates that the rectangle's area decreased. The change in the area was a decrease of 32%.
The answer is 32.
← Didn't Know|Knew It →
The cost of a load of laundry is reduced by 
. The cost is then reduced 2 weeks later by another 
. What is the overall reduction?
The cost of a load of laundry is reduced by
Tap to reveal answer
The original reduction brings the total to 
of the original value. Taking a 
discount off that price gives 
of the original value. This means the reduction had been 
.
The original reduction brings the total to
← Didn't Know|Knew It →
A dress is reduced in price by 
, but it still doesn't sell, so the manager discounts it by another 
. What is the total percentage discount?
A dress is reduced in price by
Tap to reveal answer
For these type of questions, it is always best to pretend that we are beginning with a $100 item and to calculate from there.
If an item that is $100 is discounted by 35%, and then another 10%, the new price is 58.5%.
The price difference (discount) is $41.5 for every $100, or:
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.
If an item that is $100 is discounted by 35%, and then another 10%, the new price is 58.5%.
The price difference (discount) is $41.5 for every $100, or:
The total discount is 41.5%.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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%
← Didn't Know|Knew It →
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?
Tap to reveal answer
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
← Didn't Know|Knew It →
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
Tap to reveal answer
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%.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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%
← Didn't Know|Knew It →
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.
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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%.
← Didn't Know|Knew It →
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?
Tap to reveal answer
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →