Age ProblemsBasic Tips:

If the condition is like that David is 2 years older than James.Then it means age of David = 2 + age of James

If the condition says David is twice as old as James then age of David = 2 x age of James

After fifteen years Father will be twice as old as his son. But five years ago he was four times as old as his son. Their present ages are:

Solution:Suppose that age of father F and age of son is S.

After fifteen years age of father will be F + 15 and age of son will be S + 15.And the condition will become

F + 15 = 2 x (S + 15)

And five years ago Father’s age was F – 5 and son’s age was S – 5.And the condition will become

F – 5 = 4 x (S – 5)

By solving both equation you can get F = 45 and S = 15.

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AREA AND VOLUME

Basic Tips:

Area of triangle = x base x height

height

Area of rectangle = length x breadth base

Area of Square = (Length) 2

Area of Square =

diagonal

Area of Circle = r2

A

x

Area of the sector= r2 O radius B

Arc length AOB = 2 r

Volume of Cube = (length) 3

Surface Area of the Cube = 6 (length) 2

lenght

Volume of Cuboids = length x width x height

Surface Area of cuboids = 2 x (length x width + width x height + height x length)

radius

Volume of Cylinder = r h Surface Area of Cylinder = 2 r (r + h) height

Example: A picture in a museum is 7 ft wide and 9 ft long. If its frame has a width of

1 ft all around; The area of frame is:

2

Solution: 9 ft

1ft

11 ft 1ft 9ft 1ft

7 ft1 ft

area of picture is 7ft x 9 ft = 63 ft

After the increment of 1 ft all around its width become 9 ft and its length become 11 ft. Therefore the Total area of the figure = Area of picture + Area of Frame = 9 ft x 11 ft = 99 ft

The area of frame =Total Area – Area of Picture= 99 - 63= 36 ft Answer

Right Angle Triangles to remember:

3 5 6 10 5 13 4 8 12

45 60

3

x x x 2x 45 30 x x

Angles 45 : 45 : 90 30 : 60 : 90Side opposite to respective angle x : x : x : :

Perimeter:Perimeter means sum of outlines.Perimeter of square = 4 x length Perimeter of rectangle = 2 x (length + width)Perimeter of Circle = Circumference of Circle = 2 r

Special Case:

In this case the perimeter of shaded region is the sum of mentioned lengths + arc length

Special Case:

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In this special case the area of the rectangle is half of the area of square or rectangle.

Special Case:

If the area of shaded region is 75% of the area of circle;

IxThe value of x = ?

If the shaded region is 75% then the unshaded region is25% of the total circle and the angle of unshaded region is also25 % of 360 = 90 answser.

Averages

Average=

Basic Tips:

5

x

* If you want to find the mean or an average of an arithmetic progression (a series which has same difference), like consecutive numbers, consecutive even numbers or consecutive odd numbers, it has Mean=Median.

(a) Find the mean of first ’11’ integers.

SOLUTION:

Arrange all the 11 integers in ascending or descending order

In ascending order

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

In this case ‘6’ is a middle most number which is a median because it is a arithmetic progression so the mean/average= median therefore mean = ‘6’

(b) ‘X’ has an average of 82 in the first four tests. After taking the next text his average dropped to 80. Find his recent test grade?

Solution:

In this case we use this number line method as given below. We know that in 4 tests he has an average of 82 but in 5 tests his average decreased to 80.

No. of tests Average

5 80

-2

x4 82

To find the score the score of final test just multiply the initial no. of tests to the raise or decrease of the average and add to the new average we will get the answerInitial no. of tests= 4 and the decrease of average= -2 and the new average is 80.So the score of the final test will be 4x(-2)+80=72.

Score of the final test= initial no. of tests x (raise or decrease of average) + new average

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(b) ‘Y’ has an average of 60 in first four innings. After taking the sixth inning his average become 64. What was his last inning’s score?

Solution:

No. of innings Average

6 64

+4x

5 60

Score of the final inning= initial no. of inning x raise or decrease of the average + new average

Initial innings= 5, raise/decrease of average= +4 and new average= 64

Soothe final score = 5 x (+4) + 64 = 84.

(c) James travels a certain distance at 60 mph and returns over the same route at 40 mph. What is the average rate for the entire trip?

Solution:

For this special case in which an object travels a same route for the both given speeds you should apply the formula

Average speed=

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Where x and y are the two speeds.

If the distance between the initial and the final point is D and the speeds are x and y and assume that ‘t1‘ is the time through which the object is traveling with speed x and ‘t2’ is the time required for the speed ’ is the time required for the speed y.Therefore t1=D/a and t2=D/b

Average Speed= Total distance Total time

= 2D

t1 + t2

= 2D

= 2 a b a + b

In this case a=60 and b=40 therefore

Average Speed= = = 48 mph

Otherwise the formula for average speed is

Average Speed=Total distance Total time

Fractions

Basic Tips:

Mr. X loses of the money in the first game and of the remainder in the next

game. What proportion of the money was he left with?

Solution:Suppose that he had total money = x

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On first game he lost of the total money

x

x

Then 2nd time he lost of the remainder ( )

x

x

Therefore in the last he remains with = Answer

Mixture Problems

Basic Tips:

Average Cost =

% or concentration of solute = weight of solute x 100

Total weight of mixture

9

lost

remainder

lost

remainder lost

remainder

In a mixture of 100 ml of acid and water, there is only 20% acid.

Tips:

In this question you have 20% acid which is 20 ml out of 100 ml of mixture. If you

want to increase the strength or percentage or concentration of an acid then acid

would be added.

Other wise if you want to decrease the concentration of acid then water would be

added and the quantity of acid remains constant.

Ina mixture of 100ml of acid and water is only 20% acidic. If you want to

increase the strength of acid up to 40%. How much acid would be added?

Solution:

Mixture = 100 ml acid = 20% of 100 = 20 ml

Suppose that x ml of acid would be added to make mixture 40% acidic.

Now after addition of x ml of acid

Acid = x + 20 Mixture = x + 100

% of acid = 100 = 40

x = 33.33 ml

In a mixture of 100ml of acid and water is only 20% acidic. If you want to

decrease the concentration of acid up to 10%. How much water would be added?

Solution:

Mixture = 100 ml Acid = 20 ml

Let us suppose that x ml of water to make the mixture 10% acidic.

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Now after addition of x ml of water

Acid remains constant which is 20 ml and Mixture = 100 + x

% of acid = 100 = 10

x = 100 ml

* A certain variety of tea worth Rs.30 per kg mixed with a lower quality of tea priced

at 20 per kg, so that the mixture is worth Rs.26 per kg. The ratio of the two varieties

in the resulting mixture is:

Solution:

In the questions which you have given the prices of two varieties of such commodity

and you have to find out the ratio between the quantity of those two types in the

resulting mixture, and the average cost of the resulting mixture has also been given.

In these type of questions we use ‘X’ method.

Tea A Tea B

30 Rs. /kg 20 Rs. /kg

26 Rs. /kg

26- 20= 6 30 – 26 = 4

Ratio between Tea A and Tea B = 6 : 4 = 3 : 2

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NUMBERS AND INTEGERS

Basic Tips:

In case of arithmetic progression likeConsecutive no: 1, 2, 3, 4, 5, 6, 7, ..….Consecutive even no: 2, 4, 6, 8, 10, 12, ........Consecutive odd no: 1, 3, 5, 7, 9, 11, .....

Median (middle most number of the series) = Mean of the series

the mean of 1, 3, 5, 7, 9, 11 is:

12

Solution:The difference of the above series is constant; it refers that it is an arithmetic progression, identify the middle number of the series which is 5; therefore the mean of the series is also 5.

the mean of 2, 4, 6, 8, 10, 12 is:Solution:The difference of the above series is constant; it refers that it is an arithmetic progression, identify the middle number of the series, there are two middle numbers 6 and 8; take an average of 6 and 8 which is 7, it is the median of the sequence as well as the mean.

Examples: The sum of five consecutive numbers is 220. Find the 1st number of

the sequence.Solution:We know that it is an arithmetic progression; therefore the median or middle number of the sequence is equal to the mean of the sequence.Our first step is to find the average or mean of the sequence.

Average or Mean of the sequence = = 44

44 is the middle most number of the sequence; the sequence has 5 numbers therefore the 3rd number of the sequence is the middle most number. Therefore 44 is the 3rd number of the sequence.So, the 1st number of the sequence is 42.

The sum of seven consecutive odd numbers is 91. Find the 4th number of the sequence.

Solution:It is also an arithmetic progression; therefore the 4th number is the median of the sequence as well as the mean of the sequence.

Mean or average of the sequence = = 13

Therefore 13 is the 4th number of sequence.

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The sum of six consecutive even numbers is 90. Find the 6th number of the sequence.

Solution:It is an arithmetic progression but sequence contains 6 numbers; therefore the average of 3rd and 4th number of sequence is the median of the sequence as well as the mean of the sequence.

Mean or average of the sequence = = 15

15 is the average of 14 and 16; therefore 14 and 16 is the 3rd and 4th number of the sequence respectively.Therefore 6th number of the sequence is 20.

PERCENTAGES

Important formulas:

(1) x 100

For e .g: 20 is what percent of 80?

x 100 = 25 %

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(2) Percentage Increase/ Decrease = x 100

For e.g. *If the value of stock increases from $25 to $30. Find the percentage change or increase.

% increase = x 100 = 20% increase

* Salary of ‘X’ decreases from 200 to 150. Find the percentage change or decrease.

% decrease = = 25% decrease.

One special case of Percentage

# If the length of a rectangle is increased by 30% and the width of the rectangle is decreased by 30%, then the area of the rectangle is increased or decreased by what %.

Solution:

If the length is x% increasing and the width is x% decreasing the net result will always

decreasing and it will decreased by the formula is ‘ % decreased’

In this case x= 30

Therefore ‘ % = % = % = 9% decreased

PROFIT AND LOSS

Important Formulas:

If S.P. > C.P. it refers profitProfit = S.P. – C.P.

% Profit =

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if profit is based on S.P. then % Profit =

If C.P. > S.P. it refers lossLoss = C.P. – S.P.

% Loss =

if profit is based on S.P. then % Loss =

If you have 20% Profit then Your S.P. = 120% of C.P.

If you have 60 % Profit then Your S.P. = 160% of C.P.

Find the C.P. of an article, if it made a profit of 40% and its S.P = 42.Solution:

If you have 40% profit then your S.P. = 140% of C.P. therefore

42 = 140% of C.P. => C.P. = 30.

Find the S.P. of an article whose C.P. = 60 and if it is sell on 25% Profit.Solution:

If you have 25% profit then Your S.P. = 125% of C.P.

S.P. = 125% of 60=> S.P. = 75

If you have 20% Loss then Your S.P. = 80% of C.P.

If you have 60 % Loss then Your S.P. = 40% of C.P.

Find the C.P. of an article, if it made a loss of 40% and its S.P = 42.Solution:

If you have 40% loss then your S.P. = 60% of C.P. therefore

42 = 60% of C.P. => C.P. = 70.

Find the S.P. of an article whose C.P. = 60 and if it is sell on 25% loss.Solution:

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If you have 25% loss then Your S.P. = 75% of C.P.

S.P. = 75% of 60=> S.P. = 45

If profit is 100% then S.P = 2 C.P. and similarly If profit is 200% then S.P. = 3 C.P.

Discount = M.P. – S.P.

% Discount = x 100

Ratio & ProportionImportant Formulas :

Direct Proportion If the two quantities are directly related it refers that if one quantity is increased

and other one will also be increased and if one is declining the value of other is also declining.

For e.g.: If 5 monkeys eat 20 bananas then how many monkeys will eat 28 bananas?

Monkeys Bananas

5 20

x 28

We will do cross multiplication method

5 x 28 = x x 20

x = 7 Ans

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Inverse Proportion If the two quantities are inversely related it refers that if one quantity is increased

and other one will be declined and if one is decreased the value of other will increased.

For e.g.:

If 200 people have a provision for 20 days then how many days 150 people will take to finish the same food?

We will do linear multiplication for inverse proportion

People Days

200 20

150 x

200 x 20 = 150 x x

x = 26 days Ans

Simple and Compound Interest

Basic Tips:

Simple Interest is the interest in which Principal amount (amount you invest, lend or borrow) is fixed through out the number of years. In Compound Interest the Principal amount is not remains constant, it increases as annually or semi annually or quarterly.

In banking questions or taking loan from bank you will always apply Compound Interest.

Simple Interest = P = Principal amount, R = Rate % per annum

T = Time period per year

Total Amount = Principal + InterestTotal Amount > Principal

A person invested 1500 for 2 years at 5% simple interest. Find his Simple Interest and Amount he will get back.

Solution:

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P = 1500 T = 2 R = 5

Simple Interest = = = 150

Amount = P + S.I. = 1500 + 150

Amount = 1650

In Compound Interest we don’t have direct formula for C.I. we can evaluate Amount directly then we can evaluate C.I. by

Compound Interest = Amount – Principal

Amount =P (1 + )

P = Principal amount, R = Rate % per annum, T = Time period per year

A person invested 1000 for 2 years at 5% compounded annually. Find the Compound Interest after 2 years.

Solution:

P = 1000 R = 5% T = 2

Amount = P (1 + )

A= 1000 (1 + ) = 1000 ( ) = 1102.5

C.I. = A – P = 1102.5 – 1000 = 102.5

OR

Amount after 1 year = 1000 + 5% of 1000 = 1050

Amount after 2 years = 1050 + 5% of 1050 = 1102.5

C.I. = A – P = 1102.5 – 1000 = 102.5

A person invested 1000 at 20% for a year compounded semi – annually. Find the amount he will receive at the end of 1 year.

Solution:

For this case in which you have compound interest is compounded semi annually,Interest will be applied for every 6 months or half yearly with half rate.

P = 1000 T = 1 year R = 10% (per 6 months)

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Amount after 6 months = 1000 + 10% of 1000 = 1100

Amount after 1 year = 1100 + 10% of 1100 = 1210

SPEED PROBLEMS

*Distance= Speed x time

There are some cases in we have two objects.

Case1:When both objects ‘A’ and ‘B’ are moving in the same direction.

Speed of ‘A’= SA

Speed of ‘B’=SB

So, the Net Speed of the Objects = SA – SB.

Case2:When the both objects ‘A’ and ‘B’ are moving in the opposite direction or moving towards each other from opposite direction.

Speed of ‘A’=SA

Speed of ‘B’= SB

Net Speed= SA + SB

Speed of Stream = Speed of downstream - Speed of the upstream

2

Speed of Boat / man is still water = Speed of downstream + Speed of the upstream

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2

In some special cases in which we have two objects and one of them is natural like ‘Air or Stream’ and the other object is ‘airplane or boat’ is traveling on a certain route and takes his return journey a same route.

We should apply the formula

S’= Speed of the natural objects like Air or Stream, S=Speed of boat or airplane

(a) A man rows upstream for 3 hours. His speed, in still water is 5 km/hr. he rows the same distance downstream in 2 hours. What is the stream in km/hr?

Solution:In this case S= 5 km/hr and S’=unknown t1=3 and t2=2

So put it in the formula S’ = 3 – 2

5 3 + 2

S’= 1 km/hr

(b) A plane flying at a constant air speed of 600 mph takes 5 hours to fly from Karachi to Lahore against the wind and 4 hours to fly back from Lahore to Karachi with the wind. What is the wind speed?

Solution:In this case S=600 mph and S’ = unknown t1=5 and t2=4

Put it in the formula S’ = 5 – 4

600 5 + 4

S’ = 66.67 mph

*A train leaves Lahore to Karachi at 9:30 PM with a speed of 50 km/hr. Another train also leaves Lahore to Karachi at 11:30 PM with a speed of 60 km/hr. How many km from Lahore they will be together?

Solution:

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In this special case you have to find the distance from the initial point where the two objects are traveling in a same direction and the second one is about to catch the first one and they started traveling after a certain gap of time. You should take out the LCM of the two speeds, which are 50 and 60; first train had traveled already 100km before the second train starts its journey therefore they have a gap of 100 km, because they are moving in same direction, the net speed will be the difference of their speed which is 60 – 50 = 10 km/hr it means that the second one is traveling 10 more km than the 1st train. The difference between them is 100 km and the second train will capture the 1st one in 10 hours. Therefore the total distance covered by the second train in 10 hours will be Distance = Speed x Time

= 60 x 10 = 600 km

OR

You should take out the LCM of the two speeds, which are 50 and 60therefore the LCM is ‘300’and the difference of the time leave is 2 hrs. Distance (where the 2nd train will catch the 1st one) =LCM of Speeds x Difference

of the time when they leave

= 300 x 2 = 600 Km.

Work Problem

Basic Tips:

If a man can do a piece of work in 5 days while his wife is able to do the same work in 10 days. If they work together how much time they will take for doing the same work?

Solution:

Man is more efficient than his wife because he is taking less time than his wife for the same work. If they work together, they are able to finish the whole work in less than 5 days because the man is more efficient and he will do the most part of the work as

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compare to his wife. First of all we have to evaluate their individual rates or work of one day.

Man is able to do the work in 5 days => Rate of man =

Wife is able to finish in 10 days => Rate of Wife =

Man + Wife together can finish the work in x days => Rate of Man + Wife =

Rate of Man + Wife = Rate of Man + Rate of Wife

= +

= =

x = 3 days

If 3 men or 6 boys can do a piece of work in 20 days. How many days will 6 men and 8 boys take to do the same work?

Solution:

Men Boys Days

3 or 6 20

6 and 8 x

For this ‘or’ and ‘and’ condition. If you have two categories workers like men and boys make two fractions given as below

Men Boys

‘And’ condition 6 + 8‘Or’ condition 3 x 20 6 x 20

=

Make it reciprocal

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Which is = 6 therefore your answer is ‘6’ days.

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