Mathematics · Quantitative Aptitude

Ratios and Proportions

302 Questions

Ratios and proportions deal with comparing two or more quantities and finding their relationships. The questions involve calculating compound ratios, duplicate ratios, and solving proportional equations. This topic is a crucial part of the mathematics and quantitative aptitude sections in competitive exams.

compound ratiosduplicate ratiosproportion equationssimple ratio calculationscombining multiple ratios

Ratios and Proportions Questions

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

Given that  $n$  $AM's$  are inserted between two sets of numbers  $a , 2 b$  and  $2 a , b$  where  $a , b \in R .$  Suppose further that  $mth$  mean between these sets of numbers is same, then the ratio  $a : b$  is equal to

  1. $( n - m + 1 ) : m$
  2. $( n - m + 1 ) : n$
  3. $n : ( n - m + 1 )$
  4. $m : ( n - m + 1 )$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the common difference be d. As there are $n\;AM's$ between $a$ and $b$ and total number of terms in the sequence is $=n+2$ 

$\Rightarrow nth \;term\;b=a+\left( n-1\right)d$
      $d=\dfrac{\left( b-a\right)}{n+1}$
so, 2nd term that is first term $A\left( 1\right) =a+\left[ \dfrac{\left( b-a\right)}{\left( n+1\right)}\right]$
3rd term that is second mean $A\left(2\right)=a+\left[ 2\times \dfrac{ \left( b-a\right)}{\left( n+1\right)}\right]$
In the way $r^{th}$ mean $=a+\left[ r\times \dfrac{ \left( b-a\right) }{\left(n+1\right)}\right]$
In the first sequence first term is a $n^{th}$ term $=2b$ and $n\;AM's$ between them. 
As such from above concept  -
$\Rightarrow r^{th}$ term is $a+\left[ r\times \dfrac{\left( 2b-a\right)}{\left( n+1\right)}\right]$
       $m^{th}$ term is $a+\left[ m\times \dfrac{\left( 2b-a\right)}{\left( n+1\right)}\right]$
$ii)$ Similarly for second sequence -
    $m^{th}$ mean $=2a+\left[ m\times \dfrac{\left( b-2a\right)}{\left( n+1\right)}\right]$
$iii).$ Since the $m^{th}$ mean, are equal equation like the above. 
$=a+\left[ m\times \dfrac{\left( 2b-a\right)}{\left( n+1\right)}\right]$
$=2a+m\times \dfrac{\left( b-a\right) }{\left( n+1\right)}$
$=\left[ m\times \dfrac{\left( 2b-a\right)}{\left( n+1\right)}\right]-\left[ m\times \dfrac{\left( b-2a\right)}{\left( n+1\right)}\right]$
$=2a-a$
$\Rightarrow \dfrac{m}{\left( n+1\right)} \times \left( 2b-a-b+2a\right)=a$
$\dfrac{a+b}{a}=\dfrac{n+1}{m}$
subtracting on both sides 
$\Rightarrow \dfrac{a+b}{a}-1=\dfrac{n+1}{m}-1$
$\Rightarrow \dfrac{a+b-a}{a}=\dfrac{n+1-m}{m}$

$\Rightarrow \dfrac{b}{a}=\dfrac{n+1-m}{m}$

$\Rightarrow \dfrac {a}{b}=\dfrac{m}{n-m+1}$
Hence, the answer is $\dfrac{m}{n-m+1}.$

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

A _________ is a sequence of numbers where each term in the sequence is found by multiplying the previous term with a unchanging number called the common ratio.

  1. geometric progression

  2. arithmetic series

  3. arithmetic progression

  4. harmonic progression
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A geometric progression is a sequence of numbers where each term in the sequence is found by multiplying the previous term with a with a unchanging number called the common ratio.
Example: $2, 6, 18, 54, 108....$
This geometric sequence has a common ratio $3$.

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

A sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence is known as:

  1. geometric series

  2. arithmetic progression

  3. harmonic sequence

  4. geometric sequence
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A sequence of number, ${ a } _{ 1 }+{ a } _{ 2 }+......{ a } _{ n }$ quotient of any two successive number is a constant,

$\cfrac { { a } _{ 2 } }{ { a } _{ 1 } } =\cfrac { { a } _{ 3 } }{ { a } _{ 2 } } =........=\cfrac { { a } _{ n } }{ { a } _{ n-1 } } =$common ratio $(r)$
So we can write
${ a } _{ 1 }+{ a } _{ 1 }r+{ a } _{ 2 }r+{ a } _{ 3 }r.......{ a } _{ n-1 }r\ ={ a } _{ 1 }+{ a } _{ 1 }r+{ a } _{ 1 }{ r }^{ 2 }..........{ a } _{ n-2 }{ r }^{ 2 }$
and in the end in terms of ${ a } _{ 1 }$
$={ a } _{ 1 }+{ a } _{ 1 }r+{ a } _{ 1 }{ r }^{ 2 }+{ a } _{ 1 }{ r }^{ 3 }.........{ a } _{ 1 }{ r }^{ n-1 }$
We can clearly say this series is in $GP$.
Answer $(D)$

Multiple choice

What is the golden ratio in composition?

  1. A ratio of 1:1.618.

  2. A ratio of 1:2.

  3. A ratio of 2:3.

  4. A ratio of 3:4.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The golden ratio is a ratio of 1:1.618. It is often considered to be the most aesthetically pleasing ratio and is often used in composition to create a sense of balance and harmony.

Multiple choice

What is the golden ratio?

  1. $$\phi = \frac{1 + \sqrt{5}}{2}$$
  2. $$\phi = \frac{1 - \sqrt{5}}{2}$$
  3. $$\phi = \frac{\sqrt{5} + 1}{2}$$
  4. $$\phi = \frac{\sqrt{5} - 1}{2}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The golden ratio is $$\phi = \frac{1 + \sqrt{5}}{2}$$

Multiple choice

What is the relationship between the golden ratio and the Fibonacci sequence?

  1. The limit of the ratio of consecutive Fibonacci numbers is the golden ratio.

  2. The golden ratio is the average of two consecutive Fibonacci numbers.

  3. The golden ratio is the square root of the sum of two consecutive Fibonacci numbers.

  4. The golden ratio is the product of two consecutive Fibonacci numbers.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The limit of the ratio of consecutive Fibonacci numbers is the golden ratio.

Multiple choice

The golden ratio, also known as the divine proportion, is a specific ratio of two quantities that is often considered to be aesthetically pleasing. This ratio is approximately equal to:

  1. 1.618

  2. 2.718

  3. 3.141

  4. 4.236
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The golden ratio is approximately equal to 1.618. It is often used in Indian art to create harmonious and balanced compositions.

Multiple choice

What is the formula for calculating the Odds Ratio?

  1. (a * d) / (b * c)

  2. (a + b) / (c + d)

  3. (a - b) / (c - d)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for calculating the Odds Ratio is (a * d) / (b * c), where 'a' represents the number of individuals with both exposure and outcome, 'b' represents the number of individuals with exposure but without outcome, 'c' represents the number of individuals without exposure but with outcome, and 'd' represents the number of individuals without exposure and without outcome.

Multiple choice

What is the Golden Ratio and how is it used in composition?

  1. A mathematical ratio of 1:1.618

  2. A compositional guideline for placing the subject in the frame

  3. A method for creating a sense of balance and harmony

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Golden Ratio, also known as the Divine Proportion, is a mathematical ratio of approximately 1:1.618. It is often used in composition to create a sense of balance and harmony. The Golden Ratio can be applied to the placement of the subject, the arrangement of elements within the frame, and even the cropping of the image.

Multiple choice

What is the aspect ratio of a traditional 4:3 television screen?

  1. 1.85:1

  2. 2.35:1

  3. 16:9

  4. 4:3
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Traditional 4:3 television screens have an aspect ratio of 4:3, which was widely used before the adoption of widescreen formats.

Multiple choice

What is the standard aspect ratio for DVDs?

  1. 4:3

  2. 16:9

  3. 2.35:1

  4. 2.40:1
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The standard aspect ratio for DVDs is 16:9, which is also known as widescreen. However, some DVDs may be encoded in a different aspect ratio, such as 4:3 or 2.35:1.