Tag: fraction

Questions Related to fraction

Simplify: $\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} - \dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} -\dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}}$

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

  1. $0$

  2. $1$

  3. $2$

  4. $3$

Show answer
Correct Option: B
Explanation:

$\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} - \dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} -\dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}}\=\dfrac{7\sqrt{3}(\sqrt{10} - \sqrt{3})}{(\sqrt{10} + \sqrt{3})(\sqrt{10} - \sqrt{3})} - \dfrac{2\sqrt{5}(\sqrt{6} - \sqrt{5})}{(\sqrt{6} + \sqrt{5})(\sqrt{6} - \sqrt{5})} -\dfrac{3\sqrt{2}(\sqrt{15} - 3\sqrt{2})}{(\sqrt{15} - 3\sqrt{2})(\sqrt{15} + 3\sqrt{2})}\=\dfrac{7\sqrt{3}(\sqrt{10} - \sqrt{3})}{10-3} - \dfrac{2\sqrt{5}(\sqrt{6} - \sqrt{5})}{6-5} -\dfrac{3\sqrt{2}(\sqrt{15} - 3\sqrt{2})}{15-18}\=\dfrac{7\sqrt{3}(\sqrt{10} - \sqrt{3})}{7} - \dfrac{2\sqrt{5}(\sqrt{6} - \sqrt{5})}{1} +\dfrac{3\sqrt{2}(\sqrt{15} - 3\sqrt{2})}{3}$

$=\dfrac{21\sqrt{30}-63-425\sqrt{30}+210+21\sqrt{30}-18*7}{21}\=\dfrac{21}{21}=1$

Simplify:
$\dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} + \dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$

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

  1. $4\sqrt{6}$

  2. $10$

  3. $2$

  4. $\dfrac{4\sqrt{6}}{5}$

Show answer
Correct Option: B
Explanation:

$\dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} + \dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}=\dfrac{(\sqrt{3} + \sqrt{2})^2+(\sqrt{3} - \sqrt{2})^2}{3-2}=\dfrac{3+2+3+2}{1}=10$

Reduce the following fractions to their lowest forms.
a. $\dfrac{36}{144}$


b. $\dfrac{65}{117}$

c. $\dfrac{180}{120}$

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

  1. $a=\dfrac{1}{4}, b=\dfrac{65}{117}, c=\dfrac{2}{3}$

  2. $a=\dfrac{1}{4}, b=\dfrac{5}{9}, c=\dfrac{3}{2}$

  3. $a=\dfrac{1}{4}, b=\dfrac{65}{117}, c=\dfrac{3}{2}$

  4. $a=\dfrac{3}{4}, b=\dfrac{65}{117}, c=\dfrac{2}{3}$

Show answer
Correct Option: B
Explanation:

$\\(a.)(\frac{36}{144})=(\frac{3}{12})=(\frac{1}{4})\\(b.)(\frac{65}{117})=(\frac{13\cdot 5}{13\cdot 9})=(\frac{5}{9})\\(c.)(\frac{180}{120})=(\frac{60\cdot 3}{60\cdot 2})=(\frac{3}{2})$

The Simplified form of $0.35$ is

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

  1. $\dfrac {7}{20}$

  2. $\dfrac {4}{20}$

  3. $\dfrac {35}{100 }$

  4. $None$

Show answer
Correct Option: A
Explanation:

$0.35$

$=\dfrac{35}{100}=\dfrac{7}{20}$

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

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

  1. $( n - m + 1 ) : m$

  2. $( n - m + 1 ) : n$

  3. $n : ( n - m + 1 )$

  4. $m : ( n - m + 1 )$

Show answer
Correct Option: D
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 $\dfrac{m}{n-m+1}.$

The lowest form of $3.5$ is 

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

  1. $\frac{7}{20}$

  2. $\frac{4}{20}$

  3. $\frac{35}{100}$

  4. $None$

Show answer
Correct Option: A

Express in simpest from

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

  1. $-247/228$

  2. $-68/119$

  3. $87/116$

  4. $299/161$

Show answer
Correct Option: B

If $\displaystyle\,5\,\dfrac{7}{x}\,\times\,y\,\dfrac{1}{13}\,=\,12$, where fractions are in their lowest terms, then $x - y$ is equal to 

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

  1. $2$

  2. $4$

  3. $7$

  4. $9$

Show answer
Correct Option: C
Explanation:

$\displaystyle 5\,\frac{7}{x}\,\times\,y\,\frac{1}{13}\,=\,12$
By Hit and Trial method. 
Let $x = 9, y = 2$
Where the fractions are in their lowest terms, then x should be maximum possible single digit and $y$ is minimum possible single digit.
Putting this value in equ. (1)
$\displaystyle \,5\,\times\,\frac{7}{9}\,\times\,2\,\times\,\frac{1}{13}\,=\,\frac{52}{9}\,\times\,\frac{27}{13}\,=\,12
$
$\therefore \,x\,-\,y\,=7$
Hence, option 'C' is correct.

Simplest form of the ratio 140 : 24 is__

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

  1. 1 : 3

  2. 70 :12

  3. 6 : 35

  4. 35 : 6

Show answer
Correct Option: D
Explanation:

We have, $ 140:24 $
Dividing by $ 4 $ we get $ 35:6 $
As there is no other common factor  to divide with, so $ 35:6 $ is the simplest form.

What is the reciprocal of $-3$?

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

  1. $-3$

  2. $-\dfrac {1}{3}$

  3. $\dfrac {1}{3}$

  4. $3$

  5. Undefined

Show answer
Correct Option: B
Explanation:

Reciprocal of $ -3 = \dfrac {1}{-3} $ or $ \dfrac {-1}{3} $