Mathematics · Quantitative Aptitude

Ratios and Proportions

309 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 direct proportion and inverse proportion rule of three types of proportions direct proportion

A sum is divided among four persons in the ratio $3\,\colon\,4\,\colon\,5\,\colon\,8$. If the second largest share is  Rs$\,2500$, what is the total sum?

  1. Rs $10000$
  2. Rs $15000$
  3. Rs $1000$
  4. Rs $1500$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The share is divided in the ratio $3:4:5:8$

$\therefore$ the second largest share is $5$.
Second largest share$\,=\displaystyle\frac{5}{(3+4+5+8)} \times$ Total share 
 $=\displaystyle\frac{5}{20}\times $ Total sum $=\displaystyle\frac{1}{4}\times $  Total sum

Given, second largest share $=Rs.2500$ 
 $\therefore \displaystyle\frac{1}{4}\times$ Total sum $Rs.2500$
 $\Rightarrow$ Total sum $=Rs.10,000$.

Multiple choice maths direct proportion and inverse proportion rule of three types of proportions direct proportion

Share of A, B and C respectively, are ____________, if Rs. $5460$ is divided in $\displaystyle\frac{1}{2}:\frac{1}{3}:\frac{1}{4}$.

  1. Rs. $1680$, Rs. $2520$, Rs. $1260$
  2. Rs. $2520$, Rs. $1680$, Rs. $1260$
  3. Rs. $1260$, Rs. $2100$, Rs. $2520$
  4. Rs. $2520$, Rs. $1260$, Rs. $1680$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let A's share $=Rs.\left(\displaystyle\frac{x}{2}\right)$
B's share $=Rs.\left(\displaystyle\frac{x}{3}\right)$
And C's share $=Rs.\left(\displaystyle\frac{x}{4}\right)$
According to equation,
$\displaystyle\frac{x}{2}+\frac{x}{3}+\frac{x}{4}=5460$
$\Rightarrow \displaystyle\frac{6x+4x+3x}{12}=5460$
$\Rightarrow 13x=5460\times 12\Rightarrow x=\displaystyle \frac{5460\times 12}{13}=5040$
$\therefore$ A's share $=Rs. \left(\displaystyle\frac{5040}{2}\right)=Rs. 2520$
B's share$=Rs.\left(\displaystyle\frac{5040}{3}\right)=Rs. 1680$
And C's share$=Rs. \left(\displaystyle\frac{5040}{4}\right)=Rs. 1260$.
Multiple choice maths direct proportion and inverse proportion rule of three types of proportions direct proportion

If $20: 28= x:7=10:y$.
The values of $x$ and $y$ in the box respectively are __________.

  1. $5, 14$
  2. $14, 5$
  3. $8, 10$
  4. $10, 8$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
We have, $20:28=x:7=10:y$
Taking first two ratios, we have
$20:28=x:7$
$\Rightarrow 20\times 7=x\times 28$
$\Rightarrow x=\displaystyle\frac{20\times 7}{28}=5$
Again taking last and first ratio, we get
$20:28=10:y\Rightarrow 20\times y=28\times 10$
$\Rightarrow y=\displaystyle\frac{10\times 28}{20}=14$.
Multiple choice properties of proportion ratio and proportions ratio and proportion maths

If $\dfrac {x}{y}=\dfrac {3}{4}$ and $\dfrac {x}{2z}=\dfrac {3}{2}$, then $\dfrac {2x+z}{x-2z}+\left (\dfrac {6}{7}+\dfrac {y-x}{y+x}\right )$ will be equivalent to

  1. 8

  2. 9

  3. 11

  4. 12
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\because \dfrac {x}{y}=\dfrac {3}{4}\Rightarrow \dfrac {y-x}{y+x}=\dfrac{4-3}{4+3}=\dfrac {1}{7}$
$\because \dfrac {x}{2z}=\dfrac {3}{2}\Rightarrow 2x=6z$ or, $x=3z$
$\therefore \dfrac {2x+z}{x-2z}+\left (\dfrac {6}{7}+\dfrac {y-x}{y+x}\right )=\dfrac {6z+z}{3z-2z}+\left (\dfrac {6}{7}+\dfrac {1}{7}\right )$
$=\dfrac {7z}{z}+\dfrac {7}{7}=7+1=8$.

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

After applying invertendo to $3:7::2:9$ we get

  1. $3:7::2:9$
  2. $3:2::7:9$
  3. $9:7::2:3$
  4. $7:3::9:2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $3:7::2:9$
After applying invertendo we get
$7:3::9:2$
Option D is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

After applying invertendo to $31:15::18:19$  we get $15:a::b:18$ 
What is the value of $a+b$

  1. $40$
  2. $50$
  3. $30$
  4. $60$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $31:15::18:19$
After applying invertendo we get
$15:31::19:18\equiv  15:a::b:18\ a+b=31+19=50$
Option B is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

Find the value of $a$ and $b$ respectively
After applying invertendo to $2:5::3:9$  we get $a:2::9:b$ 

  1. $9$ and $2$
  2. $2$ and $9$
  3. $3$ and $5$
  4. $5$ and $3$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $2:5::3:9$
After applying invertendo we get
$5:2::9:3\equiv a:2::9:b$
$\Rightarrow a=5,b=3$
Option D is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

After applying invertendo to $30:50::80:20$  we get $50:a::b:80$ 
Find the value of $a-b$

  1. $10$
  2. $20$
  3. $30$
  4. $40$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $30:50::80:20$
After applying invertendo we get
$50:30::20:80\equiv 50:a::b:20\ a-b=30-20=10$
Option A is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

After applying invertendo to $1:2::3:4$ we get

  1. $1:2::3:4$
  2. $2:1::4:3$
  3. $1:3::2:4$
  4. $4:2::3:1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $1:2::3:4$
After applying invertendo we get
$2:1::4:3$
Option B is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

After applying invertendo to $9:12::13:40$ we get

  1. $9:13::12:40$
  2. $40:12::13:9$
  3. $9:12::13:40$
  4. $12:9::40:13$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $9:12::13:40$
After applying invertendo we get
$12:9::40:13$
Option D is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

A naughty student breaks the pencil in such a way that the ratio of two broken parts is same as that of the original length of the pencil to one of the larger part of the pencil, The ratio of the other part to the original length of pencil is:

  1. $1 :2 \sqrt{5}$
  2. $2 : (3+\sqrt{5})$
  3. $2 : \sqrt{5}$
  4. $can't\ be\ determined $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Suppose that,

The length of larger part be$=x$

And the smaller part be $=y$


So, their ratio is $x$ is to

$ \dfrac{x}{y}=\dfrac{kx+ky}{kx} $

$ \Rightarrow \dfrac{x}{y}=\dfrac{x+y}{x} $

$ \Rightarrow {{x}^{2}}=xy+{{y}^{2}} $

$ \Rightarrow {{x}^{2}}-{{y}^{2}}=xy $


Let $y=1$

$ {{x}^{2}}-{{1}^{2}}=x $

$ \Rightarrow {{x}^{2}}-x-1=0 $

$ \Rightarrow x=\dfrac{1\pm \sqrt{5}}{2}\,\,\,\,\left( \text{On}\,\text{solving}\,\text{by}\,\text{second}\,\text{degree}\,\text{equation}\,\text{rule} \right) $


Negative value cannot be considered

So,

$x=\dfrac{1+\sqrt{5}}{2}$ so, the ratio

$ \dfrac{x}{y}=\dfrac{\dfrac{1+\sqrt{5}}{2}}{1} $

$ \Rightarrow x:y=\left( 1+\sqrt{5} \right):2 $


Therefore,

$ \dfrac{y}{x+y}=\dfrac{2}{\left( 1+\sqrt{5}+2 \right)} $

$ \Rightarrow \dfrac{y}{x+y}=\dfrac{2}{\left( 3+\sqrt{5} \right)} $


Hence, this is the answer.

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

Three numbers $A,B$ and $C$ are in the ratio $12\colon15\colon25$. If the sum of these numbers is $312$, find the ratio between the difference of $A$ and $B$ and the difference of $C$ and $B$.

  1. $\;3\colon7$
  2. $\;10\colon3$
  3. $\;3\colon10$
  4. $\;7\colon3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The ratio of $A,B,C =12:15:25$
Then, total of the ratio is $12+15+25=52$
Then, $ A=\displaystyle \frac{12}{52}\times 312=72$

$B=\displaystyle \frac{15}{52}\times 312=90$

$C=\displaystyle \frac{25}{52}\times 312=150$
Then, $B-A=18$ and $C-B=60$
Hence, the required ratio $=18:60\Rightarrow 3:10$