Profit and Loss - Practice Question with concepts and Tricks

PROFIT and LOSS

Example 1: By what percent must the cost price be raised in fixing the sale price in order that there may be a profit of 20% after allowing a commission of 10%?

Solution:  Method 1:  Basic - Let C.P = 100
   ⇒ S.P = 100 + 20% = 120

Let marked price M.P = y
Discount or commission = 10%
   ⇒ S.P = (100 – 10)% of y = 9y/10

Both the S.P are equal. Therefore: 9y/10 = 120
   ⇒ y = 1200/9 = 400/3 = 133.33
Cost price must be raised by Rs 33.33 or (33.33/100) x 100 = 33.3%

 Method 2:  By formula of successive percentage change:
Net percentage = x% + y% + xy/100%

Here overall profit = 20% ; Discount = 10%
Let C.P be raised by x%
   ⇒ 20 = x – 10 – 10x/100
   ⇒ x = 100/3 or 33.33%

Example 2: By selling 33 metres of cloth, a person gains the cost of 11 metres. Find his gain%.?

Solution:  Method 1:  Basic: Let the C.P of one metre cloth = Rs 1
   ⇒ C.P. of 33 metres of cloth = Rs 33
Gain is C.P of 11 metres of cloth = Rs 11
   ⇒ Gain% = (11/33) x 100 = 33.3%

 Method 2:  Formula: If C.P of m articles is equal to S.P of n articles:
Profit or loss percentage = [(m - n)/n] x 100
Here gain is given.
   ⇒ S.P of 33 metres of cloth – C.P of 33 metres of cloth = C.P of 11 metres of cloth
   ⇒ S.P of 33 metres of cloth = C.P of 44 metres of cloth
   ⇒ Gain% = [(44 - 33)/33] x 100 = 33.3%

Example 3: A man buys a certain number of oranges at 20 for Rs 60 and an equal number at 30 for Rs 60. He mixes them and sells them at 25 for Rs 60. What is his gain or loss percent?

Solution:  Method 1:  Basic - Let he bought x oranges of each kind.
   ⇒ C.P of x oranges of first type = 60x/20 = Rs 3x
   ⇒ C.P of x oranges of second type = 60x/30 = 2x

Now, S.P of 2x oranges = 60 x 2x/25 = Rs 4.8x   
   ⇒ Loss = S.P – C.P = 5x – 4.8x = 0.2x
   ⇒ Loss% = (0.2x/5x) x 100 = 4%

 Method 2:  Formula: If a man purchases a certain number of articles at 'x' for R rupees and the same number at 'y' for R rupees and mixes them together and sells them at 'z' for R rupees.
Then his gain or loss % =

   ⇒ [{(2 x 20 x 30)/25(20 + 30)} - 1] x 100 = -4%
Here Negative sign means loss

Example 4: A fruit seller buys lemons at 2 for a rupee and sells them at 5 for three rupees. His profit percent is?

Solution:  Method 1:  Basic - C.P of 2 lemons = Rs 1
   ⇒ C.P of 1 lemon = Rs 1/2

Also, S.P of 5 lemons = Rs 3
   ⇒ S.P of 1 lemon = Rs 3/5

Gain = 3/5 – 1/2 = 1/10 = Rs 0.1
   ⇒ Gain% = (0.1/0.5) x 100 = 20%

 Method 2:  By cross multiplication:

Profit or loss % = [(2 x 3 – 5 x 1)/(5 x 1)] x 100 = 20%
Positive sign means gain

Example 5: A person bought 50 pens for Rs 50 each. He sold 40 of them at a loss of 5%. He wants to gain 10% on the whole. Then his gain percent on the remaining pens should be?

Solution: Let profit on remaining 10 pens should be x%.
   ⇒ x% of 10  – 5% of 40 = 10% of 50
   ⇒ x = 70%

Example 6: A table is sold at a profit of 13%. If it is sold for Rs 25 more, profit is 18%. Cost price of table is?

Solution: C.P = [ More gain or loss/Difference in percentage profit or loss ] x 100
   ⇒ 25/5 x 100 = Rs 500

Example 7: Two toys are sold at Rs 504 each. One toy brings the dealer a gain of 12% and the other a loss of 4%. The gain or loss percent by selling both the toys is?

Solution:  Method 1:  Basic: S.P of first toy = Rs 504 and Gain is 12%.
   ⇒ C.P = 504 x 100/112 = Rs 450

S.P of second toy = Rs 504 and Loss is 4%
   ⇒ C.P = 504 x 100/96 = Rs 525

Total S.P = 504 + 504 = Rs 1008
Total C.P = 450 + 525 = Rs 975
   ⇒ Gain% = [(1008 - 975)/975] x 100 = (44/13)%

 Method 2:  By Direct formula: Let Gain = x% and Loss = y%.
Overall Gain or loss % will be:

   ⇒ [{100(12 – 4) + 2 x 12 x (-4)}/{(100 + 12) + (100 – 4)}]
   ⇒ (44/13)%

Example 8: A man sells two pipes at Rs 12 each. He gains 20% on one and loses 20% on the other. Find gain or loss in whole transaction?

Solution:  Method 1:  Basic: C.P of first pipe = 12 x 100/120  = Rs 10
C.P of second pipe = 12 x 100/80 = Rs 15
   ⇒ Total S.P = 12 + 12 = Rs 24
   ⇒ Total C.P = 10 + 15 = Rs 25
   ⇒ Loss = 25 – 24 = Rs 1

 Method 2:  There is always loss in such cases:
   ⇒ Loss% = 202/100 = 4%
Total S.P = 12 + 12 = Rs 24

Loss = (S.P x loss%)/(100 – Loss%)
   ⇒ 24 x 4/96 = Rs 1

Example 9: A milkman makes 20% profit by selling milk mixed with water at Rs 9 per litre. If the cost price of 1 litre pure milk is Rs 10, then the ratio of milk and water in the said mixture is?

Solution: C.P of milk mixed with water = 9 x 100/120 = Rs 7.5

 Ratio of milk and water = 7.5 : 2.5 = 3:1

Example 10: A radio dealer sold a radio at a loss of 2.5%. Had he sold it for Rs 100 more, he would have gained 7.5%. In order to gain 12.5% he should sell it for?

Solution:  Method 1:  Basic - Let C.P of radio = Rs x.
S.P at 2.5% loss = 97.5x/100
S.P at 7.5% profit = 107.5x/100
Difference in S.P = Rs 100
   ⇒ 107.5x/100 – 97.5x/100 = Rs 100
   ⇒ x = Rs 1000
Therefore to gain 12.5%, S.P = 1000 x 112.5/100 = Rs 1125

 Method 2:  By Direct Formula:

   ⇒ He should sell at = 100 x 112.5/10 = Rs 1125

Example 11: A man bought a horse and a carriage for Rs 40000. He sold the horse at a gain of 10% and the carriage at a loss of 5%. He gained 1% on his whole transaction. The cost price of the horse was?

Solution:  Method 1:  Basic - Let the C.P of horse be x. Then C.P of carriage = Rs (40000 - x)
   ⇒ S.P of horse = 110x/100
And S.P of carriage = 95 x (40000 - x)/100

Total gain is 1%, it means S.P of whole transaction = 101 x 40000/100
   ⇒ 110x/100 + 95 x (40000 - x)/100 = 101 x 40000/100
   ⇒ x = Rs 16000

 Method 2:  By Alligation:

Ratio of price of carriage : Price of horse = 9 : 6 = 3:2
   ⇒ Cost price of horse = 2 x 40000/5 = Rs 16000

Example 12: A man sells an article at a profit of 15%. If he had bought it at 10% less and sold it for Rs 4 less, he would have gained 25%. The cost price of the article is?

Solution:  Method 1:  Basic - Let C.P1 = Rs 100  then S.P1 = Rs 115
If he bought at 10% less, New C.P2 = Rs 90
Now gain is 25%, New S.P2 = 90 x 125/100 = Rs 112.5
Difference in S.P = Rs 115 – Rs 112.5 = Rs 2.5

If difference is Rs 2.5 then C.P = Rs 100
If difference is Rs 4 then C.P = 100 x 4/2.5 = Rs 160

 Method 2:  By Rule of Fraction: Let C.P = x
   ⇒ 115x/100 – 4 = x(90/100)(125/100)
   ⇒ x = Rs 160

Example 13: A man uses wrong measures and gains 25% on C.P. Find how much weight he used to measure instead of 1 kg?

Solution:  Method 2:  Direct formula: (100 + 25)/100 = 1000/less weight used
   ⇒ Weight used = 800 gm

 Method 2:  Weight hidden in profit/loss%:
Here Profit = 25% = 25/100 = 1/4
   ⇒ Weight Saved = 1 and Weight used = 4
   ⇒ Total weight = 1 + 4 = 5
When total weight was 5, he used = 4
When total weight was 1, he used = 4/5
   ⇒ When total weight was 1000, he used = 4 x 1000/5 = 800 gm

Example 14: A man because of illiteracy used more weights instead of 1 kg, and suffer a loss of (50/3)%. Find how much weight he used instead of 1 kg?

Solution:  Method 1:  Direct formula: (100 – 50/3)/100 = 1000/more weight used
   ⇒ Weight used = 1200 gm

 Method 2:  Weight hidden in profit/loss%:
Here Loss = (50/3)% = 50/300 = 1/6
   ⇒ Extra Weight = 1 and Weight used = 6
   ⇒ Total weight = 6 - 1 = 5
When total weight was 5, he used = 6
When total weight was 1, he used = 6/5
   ⇒ When total weight was 1000, he used = 6 x 1000/5 = 1200 gm