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 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 $=18:60\Rightarrow 3:10$

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

If $a : b = c : d = e : f$, then the value of each ratio is $(a + c + e) : (b + d + f)$
This property is called as 

  1. Componendo property

  2. Convertendo property

  3. Addendo property

  4. Dividendo property
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=k$

$a=bk,c=dk,e=fk\ \Rightarrow \dfrac{a+c+e}{b+d+f}=\dfrac{bk+dk+fk}{b+d+f}=k\ \Rightarrow a:b=c:d=e:f=(a+c+e):(b+d+f) $
addendo property

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

Which of the following ratios is equal to $13:4$ in its simplest form?

  1. $18:8$
  2. $105:36$
  3. $91:28$
  4. $144:250$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A.  $18 : 8 = \dfrac{18}{8}$

      Cancelling both numerator and denominator by $2$, the ratio becomes $9 : 4$
      Hence this option is wrong.

B. $105 : 36 = \dfrac{105}{36}$
    Cancelling both numerator and denominator by $3$, the ratio becomes $35 : 12$
     Hence this option is wrong.

C.  $91 : 28 = \dfrac{91}{28}$
      Cancelling both numerator and denominator by $7$, the ratio becomes $13 : 4$
      Therefore this is correct option. 

D.  $144 : 250 = \dfrac{144}{250}$
      Cancelling both numerator and denominator by $2$, the ratio becomes $72 : 125$
      Hence this option is also wrong. 

Multiple choice maths properties of parallel lines and their transversal how to check for similarity in triangles? criteria for similarity of triangles criteria for triangle similarity

If the ratio of the corresponding sides of two similar triangles is 2:3 then the ratio of their corresponding altitude is

  1. 3 : 5

  2. 16 : 81

  3. 4 : 9

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

If two triangles are similar ,then the ratio of their corresponding sides and altitude are also same. Therefore the ratio of the altitude is 2:3.

Multiple choice maths properties of parallel lines and their transversal how to check for similarity in triangles? criteria for similarity of triangles criteria for triangle similarity

If the ratio of the corresponding sides of the two similar triangles is 2 : 3 then the ratio of their corresponding attitudes is

  1. $2 : 3$
  2. $4 : 9$
  3. $16 : 81$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If two triangles are same than their sides and corresponding attitudes are also same then the ratio of the corresponding altitude is 2:3. 

Multiple choice maths three dimensional geometry - ii point of intersection of a line and a plane line and a plane three dimensional geometry

The ratio in which the joint of $(2, 1, 5), (3, 4, 3)$ is divided by the plane $2x + 2y - 2z - 1 = 0$

  1. $5 : 12$
  2. $12 : 5$
  3. $5 : 7$
  4. $7 : 5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let P point lie on a plane $2x+2y-2z-1=0 $and divide $(2, 1, 5), (3, 4, 3)$ in a ratio of  $\lambda :1$
by section formula
P=($\frac { 3\lambda +2 }{ \lambda +1 } ,\frac { 4\lambda +1 }{ \lambda +1 } ,\frac { 3\lambda +5 }{ \lambda +1 } $)
put co-ordinate of P in plane equation
2$\left(\cfrac { 3\lambda +2 }{ \lambda +1 } +\cfrac { 4\lambda +1 }{ \lambda +1 } -\cfrac { 3\lambda +5 }{ \lambda +1 } \right)$=1
by solving above equation,
we get,
 $\lambda=\dfrac{5}{7}$

Multiple choice maths mixture types of ratios ratios in proportion mathematical logic

The population of a bacteria culture doubles in number every 12 minutes. The ratio of the number of bacteria at the end of 1 hour to the number of bacteria at the beginning of that hour is

  1. $8 : 1$
  2. $16 : 1$
  3. $32 : 1$
  4. $60 : 1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If a number is multiplies repeatedly by 2 for n times then the last result will be $ { 2 }^{ n }\times number $.
Let  the number of bacteria is $a$. The number of bacteria doubles every $12$   minutes, which means that it doubles $5$ times in an hour.
Therefore,
$(2)(2)(2)(2)(2)a =a\times2^5$
                            $ = 32 a$  at the end of the hour.
Thus, the ratio of the number of bacteria at the end of 1 hour to the number of bacteria at the beginning of that hour is $\cfrac{32a}{a} = 32:1$.

Multiple choice maths mixture types of ratios ratios in proportion mathematical logic

The L.C.M. of two numbers is 48. The numbers are in the ratio 2:3. Then sum of the number is:

  1. 40

  2. 50

  3. 60

  4. 35
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the numbers be $2x$ and $3x $

Then their L.C.M. = $6x$ So $6x = 48$ or $x = 8$ 
The numbers are $16$ and $24$  Hence required sum $= (16 + 24) = 40$

Multiple choice maths mixture types of ratios ratios in proportion mathematical logic

The difference of the squares is of two numbers is 80% of the sum of their squares The ratio of the larger number to the smaller number is

  1. 5 : 2

  2. 2 : 5

  3. 3 : 1

  4. 1 : 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the two numbers be x and y Then $\displaystyle x^{2}-y^{2}=80\%$ of $\displaystyle (x^{2}+y^{2})$
$\displaystyle \Rightarrow x^{2}-y^{2}=\frac{4}{5}(x^{2}+y^{2})\Rightarrow x^{2}-\frac{4}{5}x^{2}=\frac{4}{5}y^{2}+y^{2}$
$\displaystyle \Rightarrow \frac{1}{5}x^{2}=\frac{9}{5}y^{2}\Rightarrow \frac{x^{2}}{y^{2}}=\frac{9}{1}\Rightarrow \frac{x}{y}=\frac{3}{1}\Rightarrow x:y=3:1$

Multiple choice maths mixture types of ratios ratios in proportion mathematical logic

The least whole number which when subtracted from both the terms of the ratio  $6:7$  gives a ratio less than $16:21.$

  1. $2$
  2. $3$
  3. $4$
  4. $6$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the whole number is X.
Now, according to question,
(6-X) / (7-X) < 16/21
21 *(6-X) < 16 *(7-X)
126 - 21X < 112 - 16X
126 - 112 < -16X + 21X
14 < 5X
5X > 14
X > 2.8
So, Least such whole number would be 3.

Multiple choice maths mixture types of ratios ratios in proportion mathematical logic

After decreasing $60$ in the ratio $3 : 4$ we get?

  1. $45$
  2. $50$
  3. $40$
  4. $60$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$3 : 4$ is the ratio of the new quantity to the original quantity.
Let the new number be x.
$\therefore \dfrac {x}{60} = \dfrac {3}{4}$
$\therefore x = 45$
$\therefore$ The decreased quantity is $45$

Multiple choice maths mixture types of ratios ratios in proportion mathematical logic

The multiplying ratio which changes Rs $90$ into Rs $63$ is ____

  1. $5 : 9$
  2. $3 : 4$
  3. $6 : 8$
  4. $7 : 10$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Final quantity $= 63$
Original quantity $= 90$
Multiplying ratio
$\dfrac {\text {Final quantities}}{\text {Original quantity}} = \dfrac {63}{90} = \dfrac {7}{10}$