Tag: maths

Questions Related to maths

If $\displaystyle\frac{a}{b}=\displaystyle\frac{7}{9},\displaystyle\frac{b}{c}=\displaystyle\frac{3}{5}$, then what is the value of $a\,\colon\,b\,\colon\,c$?

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $7:9:15$

  2. $9:7:15$

  3. $7:9:14$

  4. $1:9:15$

Show answer
Correct Option: A
Explanation:

Given, $\dfrac{a}{b}=\dfrac{7}{9}$ and$\dfrac{b}{c}=\dfrac{3}{5}$
Then $\displaystyle \frac{a}{b}\times \frac{b}{c}=\frac{7}{9}\times \dfrac{3}{5}\Rightarrow \frac{a}{c}=\frac{7}{15}$
$\Rightarrow \dfrac{b}{c}=\dfrac{3}{5}=\dfrac{9}{15}$
$\Rightarrow a:b=7:9$ and $b:c=9:15$

$\Rightarrow \dfrac{a}{b}:\dfrac{b}{c}=\dfrac{7}{9}:\dfrac{9}{15}$
In this $9$ is common. 
Then $ a:b:c=7:9:15$

If $A\,\colon\,B=\displaystyle\frac{1}{2}\colon\displaystyle\frac{1}{3},\,B\,\colon\,C=\displaystyle\frac{1}{2}\colon\displaystyle\frac{1}{3}$, then $A\,\colon\,B\,\colon\,C$ is equal to:

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $\;2\,\colon\,3\,\colon\,3$

  2. $\;1\,\colon\,2\,\colon\,6$

  3. $\;3\,\colon\,2\,\colon\,6$

  4. $\;9\,\colon\,6\,\colon\,4$

Show answer
Correct Option: D
Explanation:

$\;A\,\colon\,B=\displaystyle\frac{1}{2} \colon\displaystyle\frac{1}{3}=\displaystyle\frac{1}{2}\times6\ \colon\displaystyle\frac{1}{3}\times6=3\,\colon\,2$


$\;\;\;\;\;\;\;\;B\,\colon\,C=\displaystyle\frac{1}{2}\colon\displaystyle\frac{1}{3}=3\,\colon\,2$

By taking the L.C.M. of $2$ and $3$, i.e., $6$, we can make the value of $B$ equal in both the ratio.

$\;\;\;\;\;\;\;\;\displaystyle\frac{A}{B}=\displaystyle\frac{3}{2}=\displaystyle\frac{9}{6}$ and $\displaystyle\frac{B}{C}=\displaystyle\frac{3}{2}=\displaystyle\frac{6}{4}$

$\;\;\;\;\;\;\;\therefore\,A\,\colon\,B\,\colon\,C=9\,\colon\,6\,\colon\,4$.

The ratio of two numbers is a : b. If one of them is x then other is 

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $\displaystyle \frac{ab}{x}$

  2. $\displaystyle \frac{b}{ax}$

  3. $\displaystyle \frac{b}{a+b}x$

  4. $\displaystyle \frac{bx}{a}$

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Correct Option: D
Explanation:

Let the required number be y
$a : b : : x : y$
$a \times y = b \times x$
$\displaystyle y=\frac{bx}{a}$

If $a:b = \displaystyle \frac {2}{9} : \frac {1}{3}, b:c = \frac {2}{7}: \frac {5}{14} , d:c = \frac {7}{10} : \frac {3}{5},$ then find $a :b:c:d$.

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $2 : 12 : 28 : 30$

  2. $10 : 12 : 18 : 39$

  3. $16 : 24 : 30 : 35$

  4. $9 : 18 : 20 : 31$

Show answer
Correct Option: C
Explanation:

$\displaystyle \frac {a}{b} = \frac {2}{9} \div \frac {1}{3}=  \frac {2}{9} \times \frac {3}{1}, \frac {b}{c}= \frac {2}{7} \div \frac {5}{14} = \frac {2}{7} \times \frac {14}{5}= \frac {4}{5}$

$\displaystyle \frac {d}{c}=\frac {7}{10} \div \frac {3}{5} = \frac {7}{10} \times \frac {5}{3} = \frac {7}{6} \Rightarrow \frac {c}{d} = \frac {6}{7} \Rightarrow a = \frac {2b}{3}, c= \frac {5b}{4}, d= \frac {7c}{6}=\frac {7}{6} \times \frac {5b}{4} = \frac {35b}{24}$
$\therefore a:b:c:d = \displaystyle \frac {2b}{3}: b : \frac {5b}{4}\times 24 : \frac {35b}{24} = 16 : 24: 30 : 35.$

If $A : B = 1 : 2,$ $B : C = 3 : 4$ and $C : D = 5 : 6,$ find $D : C : B : A$.

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $11 : 16 : 24 : 19$

  2. $48 : 40 : 30 : 15$

  3. $14 : 23 : 14 : 19$

  4. $32 : 17 : 16 : 15$

Show answer
Correct Option: B
Explanation:

$\displaystyle \frac {A}{B} = \frac {1}{2} \Rightarrow A = \frac {1}{2} B , \frac {B}{C} = \frac {3}{4} \Rightarrow C = \ \frac {4B}{3} $

$\displaystyle \frac {C}{D} = \frac {5}{6} \Rightarrow  D = \frac {6C}{5} = \frac {6}{5} \times \frac {4B}{3}= \frac {8B}{5}$
$\displaystyle \therefore D:C:B:A = \frac {8B}{5} : \frac {4B}{3} : B : \frac {B}{2}$
$= \displaystyle \frac {8B}{5} \times 30 : \frac {4B}{3} \times 30 : B \times 30 : \frac {B}{2} \times 30 = 48:40:30:15$

When Rs. $4572$ is divided among $A, B$ and $C$ such that three times of $A'$s share is equal to $4$ times of $B'$s share is equal to $6$ times $C'$s share . What is $A'$s share ?

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. Rs. $4689$

  2. Rs. $2689$

  3. Rs. $4032$

  4. Rs. $2032$

Show answer
Correct Option: D
Explanation:

Given, $3A = 4B = 6C$ 

$ \displaystyle \Rightarrow \dfrac {A}{B} = \dfrac {4}{3}$ and$  \dfrac {B}{C} = \dfrac {6}{4} = \dfrac {3}{2}$
Therefore, $ \displaystyle A:B:C = 4:3:2$

$ \Rightarrow A = 4k, B = 3k, C =2K$
Thus $\displaystyle 4k + 3k + 2K = 4572$
$ \Rightarrow 9k = 4572 $
$\Rightarrow k = 508$
Hence, $ \displaystyle A's$ share $= 4 \times 508 =$ Rs. $2032.$

A bag contains $50$ paise, $1$ rupee and $2$ rupee coins in the ratio of $2:3:4$. If the total amount is $Rs. 240,$ what is the total number of coins ?

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $210$

  2. $180$

  3. $270$

  4. $171$

Show answer
Correct Option: B
Explanation:

Let the number of $50 p$ coins be $2x$, $1$ rupee coins be $3x$ and $2$ rupee coins be $4x$.
Total amount=240 

Then, $ \displaystyle 2x \times \frac {1}{2} + 3x \times 1 + 4x \times 2 = 240 \Rightarrow x + 3x + 8x = 240 \Rightarrow 12x = 240 \Rightarrow x = 20$
$\displaystyle \therefore Total  \ number \  of \  coins  = 2x + 3x + 4x = 9x = 9 \times 20 = 180 $

If $A : B = \displaystyle \frac{1}{2} : \dfrac {3} {8},   B : C = \frac{1}{3} : \frac{5}{9} $ and $\displaystyle C : D = \frac{5}{6} : \frac{3}{4}$, then the ratio $A : B : C : D$ is

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $6 : 4 : 8 : 10$

  2. $6 : 8 : 9 : 10$

  3. $8 : 6 : 10 : 9$

  4. $4 : 6 : 9 : 10$

Show answer
Correct Option: C
Explanation:

$A:B = \displaystyle \frac{1}{2} : \frac{3}{8} = \frac{1}{2} \div \frac{3}{8} = \frac{1}{2} \times \frac{8}{3} = \frac{4}{3} = 4 : 3 = 8:6$


$B:C = \displaystyle \frac{1}{3} : \frac{5}{9} = \frac{1}{3} \div \frac{5}{9} = \frac{1}{3} \times \frac{9}{5} = \frac{3}{5} = 3 : 5 = 6 : 10$

$C:D = \displaystyle \frac{5}{6} : \frac{3}{4} = \frac{5}{6} \div \frac{3}{4} = \frac{5}{6} \times \frac{4}{3} = \frac{10}{9}  = 10 : 9$

$\therefore A : B : C : D = 8 : 6 : 10 : 9.$

The ratio of $A$ to $B$ is $4 : 5$ and that of $B$ to $C$ is $2 : 3.$ If $A$ equals $800,$ then $C$ equals

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $1000$

  2. $1200$

  3. $1500$

  4. $2000$

Show answer
Correct Option: C
Explanation:

$\displaystyle \frac{A}{B} = \frac{4}{5} $ and $\displaystyle \frac{B}{C} = \frac{2}{3}$
To make both the values of B equal, take LCM of both values of B, i.e., 5 and 2 which is equal to 10.
$\therefore \displaystyle \frac{A}{B} = \frac{4}{5} = \frac{8}{10}$ and $\displaystyle \frac{B}{C} = \frac{2}{3} = \frac{10}{15}$
$\Rightarrow A : B : C =8 : 10 : 15$
$\Rightarrow A = 8x, B = 10 x, C = 15 x$
Given, $A = 800 \Rightarrow 8x = 800$
$\Rightarrow x = 100       \Rightarrow C = 1500$

If $x : y = 3 : 1$ then find the ratio $\displaystyle x^{3}-y^{3}:x^{3}+y^{3}$

maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

  1. $13 : 14$

  2. $12 : 13$

  3. $11: 14$

  4. $10 : 13$

Show answer
Correct Option: A
Explanation:

Let $x = 3k$ and $y = k$
Then $\displaystyle \frac{x^{3}-y^{3}}{x^{3}+y^{3}}$


$=\dfrac{(3k)^{3}-k^{3}}{(3k)^{3}+k^{3}}$


$=\dfrac{27k^{3}-k^{3}}{27k^{3}+k^{3}}$

$=\dfrac{26k^{3}}{28k^{3}}$

$=\dfrac{13}{14}$

$=13:14$