Tag: mathematical logic

Quizzes (6) Questions (99)

Questions Related to mathematical logic

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which one of the following statements is a tautology?

  1. $\left( p\vee q \right) \rightarrow q$
  2. $p\vee (p\rightarrow q)$
  3. $ p\vee (q\rightarrow p)$
  4. $p\rightarrow (p\rightarrow q)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$ p$  $q$  $p\to q$  $q\to p$ $pvq$  $p\to (p\to q)$  $(pvq)\to q$  $pv(p\to q)$   $pv(q\to q$
 T  T  T  T  T  T  T  T  T
 T  F  F  T  T  F  F  T  T
 F  T  T  F  T  T  T  T  F
 F  F  T  T  F  T  T  T  T

$\left( p\vee q \right) \rightarrow q$

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

If $p$ and $q$ are two statement, then $(p  \wedge \sim q) \wedge   (\sim p  \wedge   q)$ is

  1. a fallacy

  2. a tautology

  3. neither tautology nor a fallacy

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$p$         $q$          $p \land \sim q$       $q \land \sim p $          $(p \land \sim q ) \land(q \land \sim p )$
T T     F       F          F
T F     T       F          F
F T      F       T          F
F F     F       F          F


$\therefore\ (p \land \sim q ) \land(q \land \sim p )$ is a fallacy

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

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

A shopkeeper brought two $TV$ sets at $Rs. 10,000$ each. He sold one at a profit of $10\%$ and the another at a loss of $10\%$. Find his total profit or loss.

  1. $20 \%$
  2. $0 \%$
  3. $10 \%$
  4. $5 \%$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given that cost of two TV sets is $Rs.10,000$ each


One was sold at a profit of $10\%$ and

the other at a loss of $10\%$

Therefore, the total cost is $20,000+10000\times0.1-10000\times0.1=20,000$

Hence, the total profit or loss percentage is $0$

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

One year ago, the ratio between Ram's and Shyam's salaries was $3 : 5$. The ratio of their individual salaries of last year and present year are $2 : 3$ and $4 : 5$ respectively. If their total salaries for the present year is $Rs. 8600$, find the present salary of Ram ?

  1. $3200$
  2. $3600$
  3. $4000$
  4. $4400$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let past salaries be 3x and 5x. Present salaries are (3/2)*3x = 4.5x and (5/4)*5x = 6.25x. Total = 10.75x = 8600. x = 800. Ram's present salary = 4.5 * 800 = 3600.

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

The ratio of the number of student studying in schools A, B and C is $6:8:7$ respectively if the number of student studying in each of the schools is increased by $20\%,15\%$ and $20\%$ respectively then what will be the new ratio of the number of student in school A, B and C ?

  1. $18:23:21$
  2. $16:28:30$
  3. $20:12:36$
  4. $22:21:12$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The ratio of the number of students studying in school A, B, C.
$= 6 : 8 : 7$

Suppose number of student's in school
$A=6x$
$B=8x$
$C= 7x$

It will be Increased by
$A= 6x + 6x \times 20$ %

$= 6x+\dfrac {6x\times 20}{100}$

$= 6x+\dfrac {6x}{5}$

$= \dfrac {30x+6x}{5}$

$=\dfrac {36 x}{5}$

$B = 8x +\dfrac {8x \times 15}{100}$

$ =8x +\dfrac {6x}{5}$

$ = \dfrac {46x}{5}$

$C= 7x+\dfrac {7x\times 20}{100}$

$=\dfrac {35x +7x}{5}$

$=\dfrac {42 x}{5}$

New Ratio $= \dfrac {36 x}{5}:\dfrac {46 x}{5}: \dfrac {42 x}{5}$

$ =36:  46 : 42$

$ = 18  : 23  : 21$