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Aptitude Round Question-Ratio & Proportion-Free Download

Aptitude Round Question-Ratio & Proportion-Free Download

Conventional Method to Solve Ratio and Proportion Problems

The school book approach to this question uses the ratio and proportion formulas and starts with assuming T (or any other variable) as the total and then dividing it based on the relative division of the quantity between the quantities.

Let the total amount be T

Step 1:

As share= (3/25)T, Bs share= (5/25)T, Cs share= (8/25)T, Ds share= (9/25)T

Step 2:

D = A + 1872

Step 3:

(9/25)T = (3/25)T + 1872

Step 4:

(6/25)T = 1872

Step 5:

T = 1872 x (25/6)= 7800

Step 6:

B + C = (5/25)T + (8/25)T

Step 7:

B + C = (13/25)x 7800

Step 8:

B + C= 4056

Phew! Finally the correct answer after so many steps

Smart Method to Solve Ratio and Proportion Problems

The above method may lead you to the correct answer but its a bad idea to use it in exams where time is not your friend! A smart method is one which eliminates all extra calculations and steps.

We know that

As share= 3 parts, Bs share= 5 parts, Cs share= 8 parts, Ds share= 9 parts

Step 1:

Also we know D- A= 1872= 9 parts 3 parts= 6 parts= 1872

Step 2:

B + C = 13 parts

Step 3:

On Cross Multiplication we get B + C = 13 x (1872/6) = 4056

Rather than first get the value of whole, establish a relationship between the given ratios and eliminate all redundant steps and calculations.

Ratios and proportions and how to solve them

Let's talk about ratios and proportions. When we talk about the speed of a car or an airplane we measure it in miles per hour. This is called a rate and is a type of ratio. A ratio is a way to compare two quantities by using division as in miles per hour where we compare miles and hours.

A ratio can be written in three different ways and all are read as "the ratio of x to y"

xtoy

x :y

xy

A proportion on the other hand is an equation that says that two ratios are equivalent. For instance if one package of cookie mix results in 20 cookies than that would be the same as to say that two packages will result in 40 cookies.

201=402

A proportion is read as "x is to y as z is to w"

xy =zwwherey,w?0

If one number in a proportion is unknown you can find that number by solving the proportion.


Example

You know that to make 20 pancakes you have to use 2 eggs. How many eggs are needed to make 100 pancakes?

Eggs

pancakes

Small amount

2

20

Large amount

x

100

eggspancakes =eggspancakesorpancakeseggs=pancakeseggs

If we write the unknown number in the nominator then we can solve this as any other equation

x 100=220

Multiply both sides with 100

100x100=100220

x =20020

x =10

If the unknown number is in the denominator we can use another method that involves the cross product. The cross product is the product of the numerator of one of the ratios and the denominator of the second ratio. The cross products of a proportion is always equal

If we again use the example with the cookie mix used above

201=402

140=220=40

It is said that in a proportion if

xy =zwwherey,w?0

xw =yz

If you look at a map it always tells you in one of the corners that 1 inch of the map correspond to a much bigger distance in reality. This is called a scaling. We often use scaling in order to depict various objects. Scaling involves recreating a model of the object and sharing its proportions, but where the size differs. One may scale up (enlarge) or scale down (reduce). For example, the scale of 1:4 represents a fourth. Thus any measurement we see in the model would be 1/4 of the real measurement. If we wish to calculate the inverse, where we have a 20ft high wall and wish to reproduce it in the scale of 1:4, we simply calculate:

201:4=2014=5

In a scale model of 1:X where X is a constant, all measurements become 1/X - of the real measurement. The same mathematics applies when we wish to enlarge. Depicting something in the scale of 2:1 all measurements then become twice as large as in reality. We divide by 2 when we wish to find the actual measurement.

  • Ratio:

The ratio of two quantities a and b in the same units, is the fraction and we write it as a : b.

In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.

Eg. The ratio 5 : 9 represents

5

with antecedent = 5, consequent = 9.

9

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.

  • Proportion:

The equality of two ratios is called proportion.

If a : b = c : d, we write a :b :: c : d and we say that a, b, c, d are in proportion.

Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.

Thus, a : b :: c : d ( b x c) = (a x d).

  • Fourth Proportional:

If a : b = c : d, then d is called the fourth proportional to a, b, c.

Third Proportional:

a : b = c : d, then c is called the third proportion to a and b.

Mean Proportional:

Mean proportional between a and b is ab.

  • Comparison of Ratios:

We say that (a : b) > (c : d)

a

>

c

.

b

d

Compounded Ratio:

The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

  • Duplicate Ratios:

Duplicate ratio of (a : b) is (a2 : b2).

Sub-duplicate ratio of (a : b) is (a : b).

Triplicate ratio of (a : b) is (a3 : b3).

Sub-triplicate ratio of (a : b) is (a 1/3 : b1/3).

If

a

=

c

, then

a + b

=

c + d

. [componendo and dividendo]

b

d

a - b

c - d

  • Variations:

We say that x is directly proportional to y, ifx = ky for some constant k and we write, x y.

We say that x is inversely proportional to y, if xy = k for some constant k and

we write, x

1

.

y

1.

A and B together have Rs. 1210. If of A's amount is equal to of B's amount, how much amount does B have?

A.

Rs. 460

B.

Rs. 484

C.

Rs. 550

D.

Rs. 664

Answer: Option B

Explanation:


4

A

=

2

B

15

5

A =


2

x

15

B

5

4

A =

3

B

2


A

=

3

B

2

A : B = 3 : 2.

B's share = Rs.


1210 x

2


= Rs. 484.

5

2.

Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:

A.

2 : 5

B.

3 : 5

C.

4 : 5

D.

6 : 7

Answer: Option C

Explanation:

Let the third number be x.

Then, first number = 120% of x =

120x

=

6x

100

5

Second number = 150% of x =

150x

=

3x

100

2

Ratio of first two numbers =


6x

:

3x


= 12x : 15x = 4 : 5.

5

2

3.

A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B's share?

A.

Rs. 500

B.

Rs. 1500

C.

Rs. 2000

D.

None of these

Answer: Option C

Explanation:

Let the shares of A, B, C and D be Rs. 5x, Rs. 2 x, Rs. 4x and Rs. 3x respectively.

Then, 4x - 3x = 1000

x = 1000.

B's share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.

4.

Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?

A.

2 : 3 : 4

B.

6 : 7 : 8

C.

6 : 8 : 9

D.

None of these

Answer: Option A

Explanation:

Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8 x respectively.

Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).



140

x 5x


,


150

x 7x


and


175

x 8x


100

100

100

7x,

21x

and 14x.

2

The required ratio = 7x :

21x

: 14x

2

14x : 21x : 28x

2 : 3 : 4.

5.

In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quanity of water to be further added is:

A.

20 litres

B.

30 litres

C.

40 litres

D.

60 litres

Answer: Option D

Explanation:

Quantity of milk =


60 x

2

litres = 40 litres.

3

Quantity of water in it = (60- 40) litres = 20 litres.

New ratio = 1 : 2

Let quantity of water to be added further be x litres.

Then, milk : water =


40


.

20 + x

Now,


40


=

1

20 + x

2

20 + x = 80

x = 60.

Quantity of water to be added = 60 litres.

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