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

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

  1. $0.100$
  2. $0.01$
  3. $100$
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The expression is (0.44^2 + 0.06^2 + 0.024^2) / (0.044^2 + 0.006^2 + 0.0024^2). Since 0.44 = 10 * 0.044, 0.06 = 10 * 0.006, and 0.024 = 10 * 0.0024, the numerator is 10^2 times the denominator, which equals 100.

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

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

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

  3. 0

  4. -1
Reveal answer Fill a bubble to check yourself
C Correct answer
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) } } $
$=\frac{1}{27}-\frac{1}{27}$
$=0$

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

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

  1. $30$
  2. $60$
  3. $80$
  4. $100$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $a^2+2ab+b^2 = (a+b)^2]$
$=10^{2}$
$=100$

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

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?

  1. 7

  2. 4

  3. 6

  4. 5
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

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

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

  1. $0.4$
  2. $0.04$
  3. $0.004$
  4. $0.0004$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given,

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


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

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

$= ( 0.768 - 0.568 )^2$

$= ( 0.2 )^2$

$= 0.04.$

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


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

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

  1. $0.1$
  2. $0.01$
  3. $1$
  4. $0.1 \times 0.1$
Reveal answer Fill a bubble to check yourself
B,D Correct answer
Explanation
$0.3 \times 0.3 - 2\times 0.3 \times 0.2 + 0.2 \times 0.2$

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

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

$= ( 0.3 - 0.2 )^2$

$= ( 0.1)^2$

$= 0.1\times0.1$

$= 0.01.$

$\therefore The\  option\ B\ and\ D\ both \ are\  \ correct .$
Multiple choice maths mixture types of ratios ratios in proportion mathematical logic

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:

  1. $\displaystyle \frac{1}{6}$
  2. $\displaystyle 2\frac{1}{4}$
  3. $\displaystyle 3\frac{1}{4}$
  4. $6$
Reveal answer Fill a bubble to check yourself
B Correct answer
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}$

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

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?

  1. $2 : 3 : 4$
  2. $6 : 7 : 8$
  3. $6 : 8 : 9$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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$
$\rightarrow$ $14x : 21x : 28x$
$\rightarrow$ $2 : 3 : 4$

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

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 

  1. 12400

  2. 14400

  3. 10560

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

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 $\cfrac{32a}{a} = 32:1$.