Tag: mathematical logic

Questions Related to mathematical logic

Which one of the following statements is a tautology?

logical equivalence mathematical logic discrete mathematics business maths maths

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

  2. $p\vee (p\rightarrow q)$

  3. $ p\vee (q\rightarrow p)$

  4. $p\rightarrow (p\rightarrow q)$

Show answer
Correct Option: B
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

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

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

logical equivalence mathematical logic discrete mathematics business maths maths

  1. a fallacy

  2. a tautology

  3. neither tautology nor a fallacy

  4. none of these

Show answer
Correct Option: A
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

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

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.

maths mixture types of ratios ratios in proportion mathematical logic

  1. $20 \%$

  2. $0 \%$

  3. $10 \%$

  4. $5 \%$

Show answer
Correct Option: B
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 $0$

A year ago, the cost of Maruti and Figo are in the ratio of $3 : 4$. The ratio of present and past year costs of Maruti and Figo are $3 : 2$. Find the ratio of present costs

maths mixture types of ratios ratios in proportion mathematical logic

  1. $\dfrac98$

  2. $\dfrac38$

  3. $\dfrac58$

  4. $\dfrac97$

Show answer
Correct Option: A

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 ?

maths mixture types of ratios ratios in proportion mathematical logic

  1. $3200$

  2. $3600$

  3. $4000$

  4. $4400$

Show answer
Correct Option: B

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 ?

maths mixture types of ratios ratios in proportion mathematical logic

  1. $18:23:21$

  2. $16:28:30$

  3. $20:12:36$

  4. $22:21:12$

Show answer
Correct Option: A
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$