"Percentage problems shortcuts" is the much required one to the students who are getting prepared for competitive exams like ACT, SAT.

First let us understand what is percentage.

Percentage can be defined as follows,

Comparing the given number to 100

or

Ratio between the given number and 100

or

Saying per 100

For example,

30 % = 30/100 or 30 : 100

**Shortcut 1 :**

**How to find percentage of a number in another number ?**

**For example,**

**"a" is what percent of "b" ?**

**Shortcut**** 2 :**

**If a number is increased/decreased to another number, how to find the increase in percentage ?**

**For example,**

**"80" is increased to "96"**

**Difference = 96 - 80 = 16**

**Percentage increase = (16 / 80) x 100% = 20% **

**Shortcut 3 :**

**Cost price and profit percentage are given.**

**Cost price = C.P, Profit percentage = P%**

**Then Selling price (S.P) = (100+P)% x C.P**

**Shortcut 4 :**

**Cost price and loss percentage are given.**

**Cost price = C.P, Loss percentage = L%**

**Then Selling price (S.P) = (100 - L)% x C.P**

**Shortcut 5 :**

**Profit = S.P - C.P**

**Loss = C.P - S.P**

**Shortcut 6 :**

**Cost price and Profit are given**

**Cost price = C.P, Profit = "K"**

**Shortcut 7 :**

**Cost price and Loss are given**

**Cost price = C.P, Loss = "K"**

**Shortcut 8 :**

**Selling price and profit percentage are given.**

**How to find cost price ?**

**Use "Hint 1" and solve for C.P**

**Shortcut 9 :**

**Selling price and loss percentage are given.**

**How to find cost price ?**

**Use "Hint 2" and solve for C.P**

**Shortcut 10 :**

**Marked Price : It is the price before discount given**

**Selling price = Marked price - Discount value**

**Shortcut 11 :**

**Marked price = M.P, Discount = D%**

**Then, Discount value = (D% ) x M.P**

** Selling price (S.P) = (100 - D)% x M.P **

**Shortcut 12 :**

**Marked price (M.P) and discount value are given.**

**Then shortcut to find the discount percentage is, **

**Shortcut 13 :**

**RETAILER USING FALSE WEIGHT :**

**A trader cheats his customer to make a profit by stating that he sells at cost price.But he gives his customer less than 1000 grams (false weight) for every kg.**

**Cheated value = Original weight - False weight**

**Then shortcut to find profit percentage is, **

**Shortcut 14 :**

**Shortcut 15 :**

**Problem 1 :**

What is 20% of 50 ?

**Solution :**

20 % of 50 = 0.2 x 50 = 10

**Hence, 20% of 50 is 10**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 2 :**

If A's salary is 20% less than B's salary. By what percent is B's salary more than A's salary ?

**Solution :**

Let us assume B's salary = $ 100 ----------(1)

Then, A's salary = $ 80 --------(2)

Now we have to find the percentage increase from (2) to (1).

Difference between (1) and (2) = $ 20

Percentage increase from (2) to (1) = (20/80) x 100% = 25%

**Hence, B's salary is 25% more than A's salary. **

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 3 :**

In an election, a candidate who gets 84% of votes is elected by majority with 588 votes. What is the total number of votes polled ?

**Solution :**

Let "x" be the total number of votes polled.

Given : A candidate who gets 84% of votes is elected by majority of 476 votes

From the above information, we have

84% of x = 588 ---------> 0.84x = 588

x = 588 / 0.84 = 700

**Hence, the total number of votes polled 700.**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 4 :**

When the price of a product was decreased by 10 % , the number sold increased by 30 %. What was the effect on the total revenue ?

**Solution :**

Before decrease in price and increase in sale,

Let us assume that price per unit = $ 100.

Let us assume that the number of units sold = 100

Then the total revenue = 100 x 100 = 10000 -----------(1)

After decrease 10 % in price and increase 30 % in sale,

Price per unit = $ 90.

Number of units sold = 130

Then the total revenue = 90 x 130 = 11700 -----------(2)

From (1) and (2), it is clear that the revenue is increased.

Difference between (1) and (2) = 1700

Percent increase in revenue

= (Actual increase / Original revenue) x 100 %

= (1700/10000) x 100 %

= 17 %

**Hence, the net effect in the total revenue is 17 % increase. **

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 5 :**

A student multiplied a number by 3/5 instead of 5/3. What is the percentage error in the calculation ?

**Solution :**

In the given two fractions, the denominators are 5 and 3.

Let assume a number which is divisible by both 5 and 3.

Least common multiple of (5, 3) = 15.

So, let the number be 15.

15 x 3/5 = 9 ----------(1) ---------incorrect

15 x 5/3 = 25 ---------(2) --------correct

Difference between (1) and (2) is 16

Percentage error = (Actual error / Correct answer ) x 100 %

= (16 / 25) x 100 %

= 64 %

**Hence, the percentage error in the calculation is 64 %. **

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 6 :**

A student multiplied a number by 3/5 instead of 5/3. What is the percentage error in the calculation ?

**Solution :**

In the given two fractions, the denominators are 5 and 3.

Let assume a number which is divisible by both 5 and 3.

Least common multiple of (5, 3) = 15.

So, let the number be 15.

15 x 3/5 = 9 ----------(1) ---------incorrect

15 x 5/3 = 25 ---------(2) --------correct

Difference between (1) and (2) is 16

Percentage error = (Actual error / Correct answer ) x 100 %

= (16 / 25) x 100 %

= 64 %

**Hence, the percentage error in the calculation is 64 %. **

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 7 :**

If there are 3 boys and 7 girls in a class then, what percent of the class is made up of boys ?

**Solution :**

Total number of students in the class = 3 + 7 = 10

Percentage of boys = (No. of boys / Total no. of students) x 100 %

= (3 / 10) x 100 %

= 30 %

**Hence, 30 % of the class is made up of boys **

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 8 :**

If there are 3 boys and 7 girls in a class then, what percent of the class is made up of boys ?

**Solution :**

Total number of students in the class = 3 + 7 = 10

Percentage of boys = (No. of boys / Total no. of students) x 100 %

= (3 / 10) x 100 %

= 30 %

**Hence, 30 % of the class is made up of boys.**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 9 :**

Two numbers are respectively 20% and 50% are more than a third number, Find the the ratio of the two numbers.

**Solution :**

Let "x" be the third number.

Then,

the first number = (100+20)% of x = 120% of x = 1.2x

the first number = (100+50)% of x = 150% of x = 1.5x

First no. : second no. = 1.2x = 1.5x

1.2x : 1.5x---------------> 12x : 15x

Dividing by (3x), we get 4 : 5

**Hence, the ratio of two numbers is 4:5.**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 10 :**

In a triangle, the first angle is 20% more than the third angle. Second angle is 20% less than the third angle. Then find the three angles of the triangle.

**Solution :**

Let "x" be the third angle.

Then the first angle = 120% of x = 1.2x

The second angle = 80% of x = 0.8x

Sum of the three angles in any triangle = 180°

Then, we have x + 1.2x + 0.8x = 180°

3x = 180° --------> x = 60°

Then the first angle = 1.2(60°) = 72°

The second angle = 0.8(60°) = 48°

**Hence, the three angles of the triangle are 72°, 60° and 48°. **

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 11 :**

A trader cheats his customer to make a profit by stating that he sells at cost price but gives his customers only 800 grams. for every 1000 grams. What is his profit percentage?

**Solution :**

Cheated Value = 1000 - 800 = 200

False weight = 800

Profit % = (Cheated value/False weight)x100%

Profit % = (200/800)x100%**Profit % = 25%**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 12 :**

A sells to B an item at 15% profit. B sells the same item to C at 20% profit. If C pays $ 1656 for it. What is the price at which A bought the item?

**Solution :**

**Hence, the price at which A bought the item is $1200 **

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 13 :**

Mr. Lenin sold a chair at a loss of 15%. If he had sold at a profit of 10%, he would have got $100 more. What is the cost is the cost price of the chair?

**Solution :**

Let "x" be the cost price of the chair

S.P (-15%) = 85% of x

= 0.85x --------(1)

S.P (+10%) = 110% of x

= 1.1x ---------(2)

In (2), he got $100 more than (1). So we have

(2) - (1) = 100

1.1x - 0.85x = 100

0.25x = 100

25x = 10000

x = 400

**Hence, the cost price of the chair is $400**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 14 :**

If good are purchased for $ 1500 and one fifth of them sold at a loss of 15%. Then at what profit percentage should the rest be sold to obtain a profit of 15%?

**Solution :**

As per the question, we need 15% profit on $1500.

Selling price for 15% on 1500

S.P =115% x 1500 = 1.15x1500 = **1725**

**When all the good sold, we must have received $1725 for 15% profit.**

When we look at the above picture, in order to reach 15% profit overall, the rest of the goods ($1200) has to be sold for $1470.

That is,

C.P = $1200, S.P = $1470, Profit = $270

Profit percentage = (270/1200) x 100

Profit percentage = 22.5 %

**Hence, the rest of the goods to be sold at 22.5% profit in order to obtain 15% profit overall.**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 15 :**

By selling 20 articles, a trader gained the selling price of 5 articles. Find the profit percent.

**Solution :**

Let "X" be the S.P of 5 articles.

Given : Profit of 20 articles = S.P of 5 articles

So, **profit of 20 articles = X**

S.P of 20 articles = 4 . (S.P of 5 articles) = 4X

C.P of 20 articles = S.P of 20 articles - Profit of 20 articles

C.P of 20 articles = 4X - X

**C.P of 20 articles = 3X**

Profit percentage = (X / 3X).100% = (1/3).100%

Profit percentage = 33.33%

**Hence, the profit percentage is 33.33**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 16 :**

I purchased 120 books at the rate of $3 each and sold 1/3 of them at the rate of $4 each. 1/2 of them at the rate of $ 5 each and rest at the cost price. Find my profit percentage.

**Solution :**

Total money invested = 120x3 = $360 -------(1)

Let us see, how 120 books are sold in different prices.

From the above picture,

Total money received = 160 + 300 +60 = $ 520 --------(2)

Profit = (2) - (1) = 520 - 360 = $160

Profit percentage = (160/360)x100 % = 44.44%

**Hence the profit percentage is 44.44**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 17 :**

A trader marks his goods 20% above the cost price and allows a discount of 10% for cash. Find the profit percentage

**Solution :**

Let the cost price be $100.

Then, marked price (M.P) = $120

Let the selling price be "X"

From the above picture, we get

90% of (M.P) = X

(0.9).120 = X

108 = X --------> S.P = 108

Cost price = $100, Selling Price = $108 ---------> **Profit % = 8%**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 18 :**

A person wants to get 20% profit after selling his object at 20% discount. Find the required percentage increase in marked price.

**Solution :**

Let the cost price be $100.

Then, the selling price = $120

Let the marked price be "X"

From the above picture, we get

80% of (M.P) = S.P

(0.8)X = 120

X = 150 --------> M.P = 150

Cost price = $100, Marked Price = $150

**Hence, the required percentage increase = 50%**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 19 :**

A
person buys 8 articles for $15 and sells them at 10 for $18. Find the profit or loss percentage.

**Solution :**

**Cost price : **

8 articles -------> $15

40 articles = 5 x 8 articles = 5x15 = $75

**C.P of 40 articles = $75 ----------(1)**

**Selling price :**

10 articles -------> $18

40 articles = 4 x 10 articles = 4(18) = $72

**S.P of 40 articles = $72 ----------(2)**

From (1) and (2), we get C.P > S.P.

So there is loss.

And loss = (1) - (2) = 75 - 72 = 3

Loss percentage = (3/75)x100 % = 4%

**Hence, the loss percentage is 4. **

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 20 :**

The selling price of 10 articles is the cost price of 15 articles. Find profit or loss percentage.

**Solution :**

Let the cost price of one article be $1 -------(1)

**Given : **

S.P of 10 articles = C.P of 15 articles

S.P of 10 articles = 15x1 = $15

S.P of one article = 15/10 = $1.5 -------(2)

From (1) and (2), we get S.P > C.P

So, there is profit.

Profit = (2) - (1) = 1.5 - 1 = 0.5

Profit percentage = (0.5/1)x100 = 50%

**Hence, the profit percentage = 50%**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 21 :**

Sum of the cost price of two products is $50. Sum of the selling price of the same two products is $52. If one is sold at 20% profit and other one is sold at 20% loss, find the cost price of each product.

**Solution :**

Let "x" and "y" be the cost prices of two products.

Then, x + y = 50 --------(1)

Let us assume thatr "x" is sold at 20% profit

Then, the selling price of "x" = 120% of "x"

selling price of "x" = 1.2x

Let us assume thatr "y" is sold at 20% loss

Then, the selling price of "y" = 80% of "y"

selling price of "x" = 0.8y

Given : Selling price of "x" + Selling price of "y" = 52

1.2x + 0.8y = 52 -------> 12x + 8y = 520

3x + 2y = 130 --------(2)

Solving (1) and (2), we get x = 30 and y = 20

**Hence the cost prices of two products are $30 and $20.**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 22 : **

A computer store used a markup rate of 40%. Find the selling price of a computer game that cost the retailer $25.

**Solution : **

Selling price (S.P) = (100+M)% x C.P

Here, M = 40, C.P = $25

Then, S.P = (100 + 40)% x 25

S.P = 140% x 25

S.P = 1.4 x 25 = $35

**Hence, the selling price is $35**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 23 : **

A golf store pays its wholesaler $40 for a certain club, and then sells it to a golfer for $75. What is the markup rate?

**Solution : **

Cost price (C.P) = $ 40

Selling price (S.P) = $ 75

Mark up value = 75 - 40 = $ 35

Mark up rate = (35/40)x100 % = 87.5 %

**Hence, the mark up rate is 87.5 %**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 24 : **

A store uses a 40% markup on cost. Find the cost of a pair of shoes that sells for $63.

**Solution : **

Selling price (S.P) = (100+M)% x C.P ---------(1)

Here, S.P = $ 63, M = 40

Plugging the above values in (1)

(1)-----------> 63 = (100+40)% x C.P

63 = 140% x C.P ---------> 63 = 1.4 x C.P

63/1.4 = C.P ---------> 45 = C.P

**Hence, the cost of a pair of shoes is $ 45**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 25 : **

A product is originally priced at $55 is marked 25% off. What is the sale price?

**Solution : **

Selling price (S.P) = (100 - M)% x L.P ---------(1)

Here, L.P = $ 55, M = 25

Plugging the above values in (1)

(1)-----------> S.P = (100 - 25)% x 55

S.P = 75% x 55 ---------> S.P = 0.75 x 55

S.P = 41.25

**Hence, the selling price is $ 41.25**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 26 : **

A product that regularly sells for $425 is marked down to $318.75. What is the discount rate?

**Solution : **

Regular price = $ 425

Marked down price = $ 318.75

Marked down value = 425 - 318.75 = 106.25

Marked down rate = (106.25 / 425) x 100 %

Marked down rate = 25 %

**Hence, the marked down rate is 25 %**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 27 : **

A product is marked down 15%; the sale price is $127.46. What was the original price?

**Solution : **

Selling price (S.P) = (100 - M)% x Original price ---------(1)

Here, S.P = 127.46, M = 15

Plugging the above values in (1), we get

127.46 = (100 - 15) x Original price

127.46 = 85% x Original price

127.46 = 0.85 x Original price

127.46 / 0.85 = Original price

149.95 = Original price

**Hence, the original price is $ 149.95**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 28 :**

On selling 20 units of an item, the profit is equal to cost price of 5 units. Find the mark mark up rate.

**Solution :**

Let "m" be the cost price of one unit.

Then, the cost price of 20 units = 20m

Profit on selling 20 units = C.P of 5 units = 5m

Mark up rate = ( profit / cost ) x 100 %

= (5m / 20m) x 100 %

= 25%

**Hence, the mark up rate is 25%**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 29 :**

On selling an item, a trader gets a profit of $20. If the selling price is five times the profit, find the mark up rate.

**Solution :**

Profit = $20

Selling price = 5 x profit = 5 x 20 = $100

Cost price = Selling price - Profit

Cost price = 100 - 20 = $80

Mark up rate = ( profit / cost ) x 100 %

= (20 / 80) x 100 %

= 25%

**Hence, the mark up rate is 25%**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 30 :**

Difference between the cost price of two products is $10. Difference between the selling price is $20. If one is sold at 20% profit and other one is sold at 20% loss, find the cost price of each product.

**Solution :**

Let "x" and "y" be the cost prices of two products.

Then, x - y = 10 --------(1)

Let us assume thatr "x" is sold at 20% profit

Then, the selling price of "x" = 120% of "x"

selling price of "x" = 1.2x

Let us assume thatr "y" is sold at 20% loss

Then, the selling price of "y" = 80% of "y"

selling price of "x" = 0.8y

Given : Selling price of "x" - Selling price of "y" = 12

1.2x - 0.8y = 20 -------> 12x - 8y = 200

3x - 2y = 50 --------(2)

Solving (1) and (2), we get x = 30 and y = 20

**Hence, the cost prices of two products are $30 and $20.**

**Let us look at the next problem on "**Percentage problems shortcuts"

**Problem 31 :**

The production of rice increased by 50% from 1995 to 1996.By what percentage should the production of rice be increased from 1996 to 1997, so that the production of rice in 1997 becomes six times that of 1995?

**Solution :**

Let 100 tons be the production of rice in 1995

1995 ===> 100 tons

1995-1996 ===> 150 toms (because production has been increased by 50%)

1996-1997 ===> 600 tons (six times production in 1995)

When we look in to the above calculations, it is very clear that the production of rice has been increased 450 tons in 1996 - 97 from 150 tons in 1996.

Percentage of increase in 1996-1997

= (450/150)X100%

= 3X100%=300%

**Hence percentage of rice production increased from 1996 to 1997 is 300%**

Let us look at the next problem on "Percentage problems shortcuts"

**Problem 32 :**

15% of income of A is equal to 25% of income of B and 10% of income of B is equal to 30% of income of C. If income of C is $ 1600, then find the total income of A, B and C.

**Solution :**

Let A, B and C be the incomes of A, B and C respectively

From the given information, we have C = $1600

10% of B = 30% of C

(10/100)B = (30/100) x 1600

B = $4800

15% of A = 25% of B

(15/100)A = (25/100) x 4800

A = $8000

A + B + C = 8000 + 4800 + 1600 = 14400

**Hence, the total income of A, B and C is $14400**

Let us look at the next problem on "Percentage problems shortcuts"

**Problem 33 :**

The length of a rectangle is increased by 50%. By what percent would the width have to be decreased to maintain the same area?

**Solution :**

Let the length and width of the rectangle be 100 cm each.

Area of the rectangle = l x W = 100 x 100 = 10000 cm²

The length of the rectangle is increased by 50%

Let the width of the rectangle be decreased by P% to maintain the same area

After changes, length = 150, width = (100-P)% of 100 = 100-P

Even after the above two changes, area will be same

Therefore 150X(100-P) = 10000

15000 - 150P = 10000 ===> 150P = 5000 ===> P = 33.33% **Hence, the width has to be decreased by 33.33% to maintain the same area**

Let us look at the next problem on "Percentage problems shortcuts"

**Problem 34 :**

The production of wheat was increased by 20% from the year 1994 to 1995. It was further increased by 25% from 1995 to 1996. The percentage change in the production of wheat from 1994 to 1996 was

**Solution :**

Let 100 tons be the production of wheat in 1994

In 1994 ===> 100 tons

In 1994-1995 ===> 120 tons (20% increment)

In 1995-1996 ===> 150 tons (further 25% increment)

When we look in to the above calculations, it is very clear that the production of wheat has been increased by 50 tons in 1996 from 100 tons in 1994

Percentage change = (50/100) x 100 % = 50% **Hence, the percentage change in the production of wheat from 1994 to 1996 was 50%**

Let us look at the next problem on "Percentage problems shortcuts"

**Problem 35 :**

The price of a table is $ 400 more than that of a chair. If 4 tables and 6 chairs together cost $3600, by what percentage is the price of the chair less than that of the table ?

**Solution :**

Let "x" be the price of a chair. Then the price of a table = x+400

4 tables + 6 chairs = 3600

4(x+400) + 6x = 3600

4x+1600+6x = 3600

10x = 2000 ===>x = 200

So, price of a chair = $400 and price of a table = $800

Price of a chair is $400 less than that of the table

Percentage = (400/800) x 100% = 50% **Hence, the price of a chair is 50% less than that of the table.**

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**25. Word Problems on Quadratic equations**

**26. Pipes and Cisterns Shortcuts**

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