Tag: maths

Questions Related to maths

The value of the following is $\displaystyle \frac{(0.44)^{2}+(0.06)^{2}+(0.024)^{2}}{(0.044)^{2}+(0.006)^{2}+(0.0024)^{2}}$

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. $0.100$

  2. $0.01$

  3. $100$

  4. $1$

Show answer
Correct Option: C

$\displaystyle \frac{(0.22)^{3}+(0.11)^{3}+(0.32)^{3}}{(0.66)^{3}+(0.96)^{3}+(0.33)^{3}}-\frac{(0.32)^{3}+(0.45)^{3}-(0.77)^{3}}{81(0.32)(0.45)(0.77)}$ equals

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. 1

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

  3. 0

  4. -1

Show answer
Correct Option: C
Explanation:

$\frac{(0.22)^{3}+(0.11)^{3}+(0.32)^{3}}{(0.66)^{3}+(0.96)^{3}-(0.33)^{3}}+\frac{(0.32)^{3}+(0.45)^{3}-(0.77)^{3}}{81(0.32)(0.45)(0.77)}$
$=\frac { 8(0.11)^{ 3 }+(0.11)^{ 3 }+(0.32)^{ 3 } }{ 216(0.11)^{ 3 }+27(0.32)^{ 3 }+27(0.11)^{ 3 } } -\frac { (0.32)^{ 3 }+(0.45)^{ 3 }-(0.32+0.45)^{ 3 } }{ 81(0.32)(0.45)(0.77) } $
$=\frac { 9(0.11)^{ 3 }+(0.32)^{ 3 } }{ 243(0.11)^{ 3 }+27(0.32)^{ 3 } } -\frac { (0.32)^{ 3 }+(0.45)^{ 3 }-(0.32+0.45)^{ 3 } }{ 81(0.32)(0.45)(0.77) } $
$=\frac { 9(0.11)^{ 3 }+(0.32)^{ 3 } }{ 27{ 9(0.11)^{ 3 }+(0.32)^{ 3 }}  } -\frac { (0.32)^{ 3 }+(0.45)^{ 3 }-[(0.32)^{ 3 }+(0.45)^{ 3 }+3(0.32)(0.45)(0.32)+(0.45) }{ { 81(0.32)(0.45)(0.77) } } $
$=\frac{1}{27}-\frac{1}{27}$
$=0$

What is the value of $(7.5 \times 7.5 + 37.5 + 2.5 \times 2.5) ?$

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. $30$

  2. $60$

  3. $80$

  4. $100$

Show answer
Correct Option: D
Explanation:

$(7.5 \times 7.5 + 37.5 + 2.5 \times 2.5) $

$=(7.5)^{2}+2\times 7.5\times 2.5+(2.5)^{2}$
$ =(7.5+2.5)^{2}$         ....[Using $a^2+2ab+b^2 = (a+b)^2]$
$=10^{2}$
$=100$

If k is an integer, and if $0.02468 \times 10^k$ is greater than 10,000, what is the least possible value of k?

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. 7

  2. 4

  3. 6

  4. 5

Show answer
Correct Option: C
Explanation:

Multiplying 0.02468 by a positive power of 10 will shift the decimal point to the right. Simply shift the decimal point to the right until the result is greater than 10,000. Keep track of how many times you shift the decimal point. Shifting the decimal point 5 times results in 2,468. This is still less than 10,000. Shifting one more place yields 24,680, which is greater than 10,000.

The value of $0.768 \times 0.768 - 2 \times 0.768 \times 0.568 + 0.568 \times 0.568$ is:

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. $0.4$

  2. $0.04$

  3. $0.004$

  4. $0.0004$

Show answer
Correct Option: B
Explanation:
Given,

$0.768 \times 0.768 - 2\times 0.768 \times 0.568 + 0.568 \times 0.568$


$= ( 0.768 )^2 - 2 \times  0.768 \times 0.568 + ( 0.568 )^2$

$Using\  identity \ \because { a^2 - 2 ab + b^2 = ( a - b )^2 }$

$= ( 0.768 - 0.568 )^2$

$= ( 0.2 )^2$

$= 0.04.$

$\therefore The\  option\ B\  is \ correct .$


The value of $\left( {0.3} \right)\left( {0.3} \right) - 2\left( {0.3} \right)\left( {0.2} \right) + \left( {0.2} \right)\left( {0.2} \right)$

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. $0.1$

  2. $0.01$

  3. $1$

  4. $0.1 \times 0.1$

Show answer
Correct Option: B,D
Explanation:
$0.3 \times 0.3 - 2\times 0.3 \times 0.2 + 0.2 \times 0.2$

$= ( 0.3 )^2 - 2 \times  0.3 \times 0.2 + ( 0.2 )^2$

$Using\  identity \ \because { a^2 - 2 ab + b^2 = ( a - b )^2 }$

$= ( 0.3 - 0.2 )^2$

$= ( 0.1)^2$

$= 0.1\times0.1$

$= 0.01.$

$\therefore The\  option\ B\ and\ D\ both \ are\  \ correct .$

The difference between the numerator and the denominator of a fraction is $5$ If $5$ is added to the denominator the fraction is decreased by $\displaystyle 1\frac{1}{4}$ then the value of the fraction will be equal to:

maths mixture types of ratios ratios in proportion mathematical logic

  1. $\displaystyle \frac{1}{6}$

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

  3. $\displaystyle 3\frac{1}{4}$

  4. $6$

Show answer
Correct Option: B
Explanation:

Let the numerator be $x+5$ and denominator be $x$


Given condition is :$\dfrac {x+5}{x}-\dfrac {x+5}{x+5}=\dfrac {5}{4}$

$\Rightarrow \dfrac {x+5}{x}=\dfrac {9}{4}=2\dfrac {1}{4}=$fraction value 

$\therefore $fraction value$ =2\dfrac {1}{4}$

Seats for Mathematics, Physics and Biology in a school are in the ratio $5 : 7 : 8$.  There is a proposal to increase these seats by $40\%$, $50\%$ and $75\%$ respectively. What will be the ratio of increased seats?

maths mixture types of ratios ratios in proportion mathematical logic

  1. $2 : 3 : 4$

  2. $6 : 7 : 8$

  3. $6 : 8 : 9$

  4. None of these

Show answer
Correct Option: A
Explanation:

Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.
Number of increased seats are ($140\%$ of $5x$), ($150\%$ of  $7x$) and ($175\%$ of $8x$).
 $\rightarrow \left ( \dfrac{140}{100} x\times  5x\right ),\left ( \dfrac{150}{100} x\times 7x\right ), and \left ( \dfrac{175}{100} \times  8x\right )$
$\rightarrow 7x, \dfrac{21x}{2}$ and $14x$
$\therefore $ The required ratio$ = 7x:$ $\dfrac{21x}{2}$$: 14x$
$\rightarrow$ $14x : 21x : 28x$
$\rightarrow$ $2 : 3 : 4$

The present population of a city is 8000 if it increases by $10\%$ during the first year and by $20\%$ during the second year , then population after two years will be 

maths mixture types of ratios ratios in proportion mathematical logic

  1. 12400

  2. 14400

  3. 10560

  4. None of these

Show answer
Correct Option: C
Explanation:

Population after two years,
$=8000\times \dfrac{110}{100}\times \dfrac{120}{100}=10560$

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

maths mixture types of ratios ratios in proportion mathematical logic

  1. $8 : 1$

  2. $16 : 1$

  3. $32 : 1$

  4. $60 : 1$

Show answer
Correct Option: C
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 $\cfrac{32a}{a} = 32:1$.