Tag: maths

Questions Related to maths

A fraction $\displaystyle\frac{x}{y}$ can be expressed as a terminating decimal if y has no prime factors other than _________.

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

  1. $2, 3$

  2. $3, 5$

  3. $2, 5$

  4. $2, 3, 5$

Show answer
Correct Option: C
Explanation:

If the denominator has a prime factor $2$ or $5$ then it is a terminating decimal

Hence the correct answer is option C

The value of $\left(\displaystyle 1-\frac{1}{3}\right)\left(\displaystyle 1-\frac{1}{4}\right)\left(\displaystyle 1-\frac{1}{5}\right)\left(\displaystyle 1-\frac{1}{6}\right).....\left(\displaystyle 1-\frac{1}{n}\right)$ is _________.

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

  1. $\displaystyle\frac{1}{n}$

  2. $\displaystyle\frac{2}{n}$

  3. $\displaystyle\frac{2(n-1)}{4}$

  4. $\displaystyle\frac{2}{n(n+1)}$

Show answer
Correct Option: B
Explanation:
We need to find value of $\left(1-\dfrac {1}{3}\right)\left (1-\dfrac {1}{4}\right)\left (1-\dfrac {1}{5}\right)\left(1-\dfrac {1}{6}\right).....\left (1-\dfrac {1}{n}\right)$
Solving each bracket, we get 
$\dfrac{2}{3} \times \dfrac{3}{4} \times \dfrac{4}{5} . . . . ...\dfrac{n-3}{n-2}\times \dfrac{n-2}{n-1} \times \dfrac{n-1}{n}$
Starting from $1^{st}$ term each denominator is cancelled out by next numerator.
Finally, we get $\dfrac{2}{n}$

The product of the $9$ fractions $\left(\displaystyle 1-\frac{1}{2}\right)\left(\displaystyle 1-\frac{1}{3}\right)\left(\displaystyle 1-\frac{1}{4}\right)..........\left(\displaystyle 1-\frac{1}{10}\right)=$____________.

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

  1. $\displaystyle\frac{10}{11}$

  2. $\displaystyle\frac{1}{9}$

  3. $\displaystyle\frac{1}{10}$

  4. $\displaystyle\frac{1}{2}$

Show answer
Correct Option: C
Explanation:
We need to simplify $\left (1-\dfrac {1}{2}\right)\left (1-\dfrac {1}{3}\right)\left (1-\dfrac {1}{4}\right).....\left (1-\dfrac {1}{10}\right)$
Simplifying each bracket, we get
$\dfrac{1}{2} \times \dfrac{2}{3} \times \dfrac{3}{4} . . . . ...\dfrac{7}{8}\times \dfrac{8}{9} \times \dfrac{9}{10}$
As we can see 
Starting from $1^{st}$ term, each denominator is cancelled out by next numerator.
Finally we get $\dfrac{1}{10}$

The standard form of $\displaystyle\frac{192}{-168}$ is _________.

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

  1. $\displaystyle\frac{-2}{3}$

  2. $\displaystyle\frac{-8}{7}$

  3. $\displaystyle\frac{-1}{7}$

  4. $\displaystyle\frac{-6}{7}$

Show answer
Correct Option: B
Explanation:

the standard form of $ - \dfrac{192}{168}$

$=-\dfrac{2 \times 3\times 4\times 8}{2\times 3\times 4\times 7}$
$-\dfrac{8}{7}$

Which of the following sum is in the simplest form?

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

  1. $\dfrac{4}{9}+\dfrac{-5}{9}$

  2. $\dfrac{-2}{5}+\dfrac{13}{20}$

  3. $\dfrac{-5}{12}+\dfrac{11}{-12}$

  4. $\dfrac{-7}{8}+\dfrac{1}{12}+\dfrac{2}{3}$

Show answer
Correct Option: A
Explanation:
A. $\dfrac{4}{9} + \dfrac{-5}{9} = \dfrac{-1}{9}$ Simplest Form

B. $\dfrac{-2}{5} + \dfrac{13}{20} = \dfrac{-8+13}{20} = \dfrac{5}{20}$ Not the simplest form

C. $\dfrac{-5}{12} + \dfrac{11}{-12} = \dfrac{-5-11}{12} = \dfrac{-16}{12}$ Not the simplest form

D. $\dfrac{-7}{8} + \dfrac{1}{12} + \dfrac{2}{3} = \dfrac{-21+2+16}{24} = \dfrac{-3}{24}$ Not the simplest form

$\Rightarrow$ Only A gives answer in simplest form

Which of the following is not equivalent to $\dfrac{4}{8}$ ?

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

  1. $\cfrac { 1 }{ 2 } $

  2. $\cfrac { 16 }{ 32 } $

  3. $1$

  4. $\cfrac { 12 }{ 24 } $

Show answer
Correct Option: C
Explanation:

we have, $\dfrac{4}{8}=$ $\dfrac{1}{2}=$$\dfrac{16}{32}=$$\dfrac{12}{24}$

Among the given values, the one which is not equivalent to $\dfrac{4}{8}$ is 1.

Simplify: $\dfrac{5}{11} + 4\dfrac{3}{9} $

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

  1. $\dfrac{158}{33}$

  2. $\dfrac{168}{33}$

  3. $\dfrac{178}{33}$

  4. $\dfrac{148}{33}$

Show answer
Correct Option: A
Explanation:
$\displaystyle \frac{5}{11}+4\frac{3}{9}=\frac{5}{11}+\frac{36+3}{9}=\frac{5}{11}+\frac{39}{9}$

$\displaystyle =\frac{5\times 9+11\times 39}{9\times 11}=\frac{45+429}{99}=\frac{474}{99}=\frac{158}{33}$

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:

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