SHORTCUT METHODS IN MATHS

 Here is some very very important shortcuts to my shishyas , learn them practice them and  apply your weapon at the time of exams .you have to  know on which type of questions you can apply and get  answers quickly .so you must practice on this.

 

1.  When any value increases by

1.     10%, it becomes 1.1 times of itself. (since 100+10 = 110% = 1.1)

2.     20%, it becomes 1.2 times of itself.

3.     36%, it becomes 1.36 times of itself.

4.     4%, it becomes 1.04 times of itself.

Thus we can see the effects on the values due to various percentage increases.

2.  When any value decreases by

1.     10%, it becomes 0.9 times of itself. (Since 100-10 = 90% = 0.9)

2.     20%, it becomes 0.8 times of itself

3.     36%, it becomes 0.64 times of itself

4.     4%, it becomes 0.96 times of itself.

Thus we can see the effects on a value due to various percentage decreases.

Note:

1. When a value is multiplied by a decimal more than 1 it will be increased and when multiplied by less than 1 it will be decreased.

2. The percentage increase or decrease depends on the decimal multiplied.

Eg: 0.7 => 30% decrease, 0.67 => 33% decrease, 0. 956 => 4.4% decrease and so on.

Eg: When the actual value is x, find the value when it is 30% decreased.

Soln: 30% decrease => 0.7 x.

Eg: A value after an increase of 20% became 600. What is the value?

Soln: 1.2x = 600 (since 20% increase)

ð     x = 500.

Eg: If 600 is decrease by 20%, what is the new value?

Soln: new value = 0.8 X 600 = 480. (Since 20% decrease)

Thus depending on the decimal we can decide the % change and vice versa.

Eg: When a value is increased by 20%, by what percent should it be reduced to get the actual value?

Soln: (It is equivalent to 1.2 reduced to 1 and we can use % decrease formula)

% decrease = (1.2 – 1)/1.2 X 100 = 16.66%.

3.             When a value is subjected multiple changes, the overall effect of all the changes can be obtained by multiplying all the individual factors of the changes.

Eg: The population of a town increased by 10%, 20% and then decreased by 30%. The new population is what % of the original?

Soln: The overall effect = 1.1 X 1.2 X 0.7   (Since 10%, 20% increase and 30% decrease)

= 0.924 = 92.4%.

Eg: Two successive discounts of 10% and 20% are equal to a single discount of ___

Soln: Discount is same as decrease of price.

So, decrease = 0.9 X 0.8 = 0.72 => 28% decrease (Since only 72% is remaining).

practice problems:

1.     If 20% of 40% of a = 25% of a% of b, then what is b?

a. 8/5               b. 16/25                       c. 8/25             d. None

2. By what % is 200 more than 50?

a. 100              b. 200                          c. 300              d. None

3. A value changes from 30 to 80. What is the percentage change?

a. 125              b. 166.66                     c. 156              d. None

4. The population of a city is increased by 30% and thus became 78000. What is the original population?

a. 76000          b. 64200                      c. 60000          d. None

5. In a theatre, the number of seats is increased by 20% and the price per ticket is increased by 10% but the public response decreased by 30%. What is the net effect on the economy of the theatre?

a.10% rise        b. 7% fall                      c. 7% rise         d. None

6. A saves 20% of his income. His income is increased by 20% and so he increased his expenditure by 30%. What is the percentage change in his savings?

a. 20% fall        b. 4% fall                      c. 20% rise       d. 4% rise

7. The price of petrol is increased by 25%. By what percent the consumption be reduced to make the expenditure remain the same?

a. 25%             b. 33.33%                    c. 20%             d. None

8. The side of a square is increased by 20%. The percentage change in its area is ___

a. 20%             b. 44%                         c. 36%             d. None

9. If the length of a rectangle is increased by 33.33%, by what percentage should the breadth be reduced to make the area same?

a. 20%             b. 33.33%                    c. 25%             d. None

10. In an election between two candidates, A and B, A secured 56% of the votes and won by 48000 votes. Find the total number of votes polled if 20% of the votes were declared invalid.

a. 500000        b. 400000                    c. 600000        d. None

clear explanation for  above problems:

1.     1/5 X 2/5 X a = ¼ X a X b  =>  b = 8/25

2.     % difference = (200-50)/50 X 100 = 300 %

3.     % increase = (80-30)/30 X 100 = 166.66 %

4.     1.3 x = 78000  =>  x = 60000.

5.     Net effect = 1.2 X 1.1 X 0.7

= 0.924  =>  7.6% decrease.

6.             Let I be the income.

Expenditure = 0.8I                 Savings = 0.2I => 20%

New income = 1.2I  (since 20% rise)

New expenditure = (0.8I) X 1.3  (Since 30% rise)

= 1.04I

So, new savings = 1.2I – 1.04I = 0.16I => 16%

(So income decreased form 20% to 16%)

% decrease = (20-16)/20 X 100 = 20%.

7.             It is equivalent to 1.25 decreased to 1.

% decrease = (1.25-1)/1.25 X 100 = 20%

8. % change in area = 1.2 X 1.2 (since area = side X side)

= 1.44 => 44%.

9.             It is equivalent to 1.25 decreased to 1. So 20% decrease.

10.         Valid Votes:

A got 56%  =>  B got 44%

Difference = 12% = 48000

So, 100% = 400000. These are valid votes.

But valid votes are only 80% of total votes.

So, 80% of total votes  = 400000  =>  total votes = 500000
**********************************************************************************************************************************

 1.     A trader uses a 800gm weight instead of 1 kg. Find his profit %.

Soln: (He is buying 800 gm but selling 1000 gm.

So, CP is for 800 gm and SP is for 1000 gm.)

SP/CP = 1000/800 = 1.25 => 25% profit.

1.     A trader uses 1 kg weight for 800 gm and increases the price by 20%. Find his profit/loss %.

Soln: 1 kg weight for 800 gm => loss (decrease) => 800/1000 = 0.8

20% increase in price => profit (increase) => 1.2

So, net effect = (0.8) X (1.2) = 0.96 => 4% loss.

1.     A milk vendor mixes water to milk such that he gains 25%. Find the percentage of water in the mixture.

Soln: To gain 25%, the volume has to be increased by 25%.

So, for 1 lt of milk, 0.25 lt of water is added => total volume = 1.25 lt

% of water = 0.25 / 1.25 X 100 = 20%.

1.     A trader bought an item for Rs. 200. If he wants a profit of 22%, at what price must he sell it?

Soln: CP=200, Profit = 22%.

So, SP = 1.22CP = 1.22 X 200 = 244/-.

1.     A person buys an item at Rs. 120 and sells to another at a profit of 25%. If the second person sells the item to another at Rs. 180, what is the profit % of the second person?

Soln: SP of 1^st person = CP of 2^nd person = 1.25 X 120 = 150.

SP of 2^nd person = 180.

Profit % = SP/CP = 180/150 = 1.2 => 20%.

1.     A milk vendor mixes water to 20 lt of milk such that the ratio of milk and water is 4:3. He sold the mixture at Rs. 12 per liter but bought the milk at Rs. 10 per liter. Find the profit % of the vendor.

Soln: milk : water = 4:3 => he bought 4 parts (milk) but sold 7 parts (mixture)

CP = 10 and SP = 12.

So, profit % = (SP/CP) X (SP/CP) = (7/4) X (12/10) = 2.1 => 110% gain.

1.     A trader buys some apples at a price of 10 apples for Rs. 8 and sold them at a price of 8 apples for Rs. 10. Find his profit or loss %.

Soln: He bought 10 apples for Rs. 8 and sold 8 apples for Rs. 10 => clearly got profit

ð     SP > CP => (SP/CP) X (SP/CP) = (10/8) X (10/8) = 100/64 = 1.5625 => 56.25 % gain.

1.     A trader allows a discount of 25% on his articles but wants to gain 50% gain. How many times the CP should be marked on the items?

Soln: CP applied with profit = MP applied with discount = SP

ð     1.5CP = 0.75MP (since 50% gain and 25% discount) => MP = 2CP.

1.     By selling an item at a price a trader gains 40%. What is the profit / loss % if the item is sold at half the price?

Soln: SP =1.4CP => (SP/2) = 0.7CP => 30% loss.