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Problem 1 :

The price of a product is increased from $25 to $30. Find the percent increase. 

Problem 2 :

Find the percent decrease from 90 inches to 63 inches.

Problem 3 :

The price of a product is $30. If the price is increased by 15%, then find the new price. 

Problem 4 :

The cost price of a product is $150 and it is sold at a loss of 30%. Find the selling price of the product.   

Problem 5 : 

John got 50% of the questions correct on a 40-question test and 80% on a 80-question test. What percent of all questions did John get correct?   

Problem 6 : 

The price of a dress is increased by 20%, then decreased by 40%, then increased by 25%. the final price is what percent of the original price.?

Problem 7 : 

If x is 50% larger than z, and y is 20% larger than z, then x is what percent larger than y?

Problem 8 : 

The number of students at a school decreased 20% from 2015 to 2016. If the number of students enrolled in 2016 was k, find the number of students enrolled in 2015 in terms of k. 

Problem 9 : 

Townsend Realty purchased a property for $140,000 after having received a 40% discount off the original price along with an additional 20% off the discounted price for purchasing the property in cash. Find the original price of the property (Round the answer to the nearest dollar).

Problem 10 : 

Among 9th graders at a school, 30% of the students are Red Sox fans. Among those Red Sox fans, 40% are also Celtics fans. What percent of the 9th graders at the school are both Red Sox fans and Celtics fans? 

Detailed Answer Key

Problem 1 :

The price of a product is increased from $25 to $30. Find the percent increase. 

Solution :

Find the amount of change. 

Amount of change  =  Greater value - Lesser value 

=  30 - 25

=  5

Percentage change is 

=  (Amount of change / Original amount) x 100 %

=  (5 / 25)  100%

=  0.2  100 %

=  20 %

So, the percent increase from $25 to $30 is 20%.

Problem 2 :

Find the percent decrease from 90 inches to 63 inches.

Solution :

Find the amount of change. 

Amount of change  =  Greater value - Lesser value 

=  90 - 63

=  27

Percentage change is  

=  (Amount of change / Original amount) x 100 %

=  (27 / 90)  100%

=  0.3  100 %

=  30 %

So, the percent decrease from 90 inches to 63 inches is 30%. 

Problem 3 :

The price of a product is $30. If the price is increased by 15%, then find the new price. 

Solution :

The new price is 

=  (100 + 15)% of 30

=  115%  30

=  (115/100)  30

=  1.15  30

=  34.5

So, the new price after 15% increase is $34.50

Problem 4 :

The cost price of a product is $150 and it is sold at a loss of 30%. Find the selling price of the product.     

Solution :

Because the product is sold in loss, the cost price has to be decreased by 30% to find the selling price.

The selling price of the product is  

=  (100 - 30)% of 150

=  70%  150

=  (70/100)  150

=  0.7  150

=  105

So, the selling price of the product is $105.

Problem 5 : 

John got 50% of the questions correct on a 40-question test and 80% on a 80-question test. What percent of all questions did John get correct?   

Solution :

Number of questions he got correct in 40-question test is

=  50% ⋅ 40

=  0.5 ⋅ 40

=  20

Number of questions he got correct in 80-question test is

=  80% ⋅ 80

=  0.8 ⋅ 80

=  64

Number of questions he got correct in both the tests is 

=  20 + 64

=  84

Total number of questions in both the tests is 

=  40 + 80

=  120

Percent of questions John got correct is 

=  (84/120) ⋅ 100%

=  70%

So, John got 70% of all questions correct. 

Problem 6 : 

The price of a dress is increased by 20%, then decreased by 40%, then increased by 25%. the final price is what percent of the original price.?

Solution :

Let $100 be the original price. 

Price after 20% increase : 

=  (100 + 20)% ⋅ 100

=  120⋅ 100

=  1.2 ⋅ 100

=  120

Price after 40% decrease : 

=  (100 - 40)% ⋅ 120

=  60⋅ 120

=  0.6 ⋅ 120

=  72

Price after 25% increase : 

=  (100 + 25)% ⋅ 72

=  125⋅ 72

=  1.25 ⋅ 72

=  90

Original price is $100 and final price is $90. 

So, the final price is 90% of the original price.  

Problem 7 : 

If x is 50% larger than z, and y is 20% larger than z, then x is what percent larger than y?

Solution :

Let z  =  100

Finding the value of x :

x  =  (100 + 50)%  100

x  =  150%  100

x  =  1.5  100

x  =  150

Finding the value of y :

y  =  (100 + 20)%  100

y  =  120%  100

y  =  1.2  100

y  =  120

Difference between the value of x and y is 

=  150 - 120

=  30

Percent change :

=  (30 / 120)  100 %

=  25 %

So, x is 25% larger than y.

Problem 8 : 

The number of students at a school decreased 20% from 2015 to 2016. If the number of students enrolled in 2016 was k, find the number of students enrolled in 2015 in terms of k. 

Solution :

Number of students enrolled in 2016  =  k 

Let x be the number of students enrolled in 2015. 

Number of students enrolled in 2016 was decreased by 20% compared to 2015. 

So, we have

Number of students enrolled in 2016  =  (100 - 20)%  x

k  =  80%  x

k  =  (80/100)  x

k  =  (4/5)  x

Multiply each side by 5/4. 

(5/4)  k  =  x

1.25k  =  x

So, the number of students enrolled in 2015 is 1.25k.

Problem 9 : 

Townsend Realty purchased a property for $140,000 after having received a 40% discount off the original price along with an additional 20% off the discounted price for purchasing the property in cash. Find the original price of the property (Round the answer to the nearest dollar). 

Solution :

Let x be the original price of the property. 

The price of the property after 40% discount :

=  (100 - 40)% of x

=  60%  x

=  0.6x

The price of the property after additional 20% off :

=  (100 - 20)% of 0.6x

=  80%  0.6x

=  0.8  0.6x

=  0.48x

Therefore, the purchased price is 0.48x. 

But, the purchased price given in the question is $140,000.

Then, we can have

0.48x  =  140000

Divide each side by 0.32

x    291,667

So, original price of the property is about $291,667.

Problem 10 : 

Among 9th graders at a school, 30% of the students are Red Sox fans. Among those Red Sox fans, 40% are also Celtics fans. What percent of the 9th graders at the school are both Red Sox fans and Celtics fans?  

Solution :

Let 100 be the strength of 9th graders. 

Number of Red Sox fans :

=  30%  100

=  0.3  100

=  30

Number of Celtics and Red Sox fans :

=  40%  30

=  0.4  30

=  12

Out of 100 students in 9th grade, 12 students are both Red Sox and Celtics fans. 

So, 12% of the 9th graders are both Red Sox and Celtics fans.

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