Tag: forming equations from statements

Questions Related to forming equations from statements

A can do a piece of work in $24$ days. If B is $60\%$ more efficient then the number of days required by B to do the twice as large as the earlier work is-

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. $24$

  2. $36$

  3. $15$

  4. $30$

Show answer
Correct Option: D
Explanation:
According to the question-
A can do a piece of work in $24$ days.
$\therefore$ work done by A in $1$ day  $=\cfrac{1}{24}$
Given that, B is $60 \%$ more efficient, i.e., B can do $60 \%$ more work than work done by A in $1$ day-
Work done by B in $1$ day  $=\cfrac{1}{24} + \cfrac{60}{100} \times {1}{24} = \cfrac{1}{15}$
$\therefore$ number of days required by B to do the same work $= 15 $days
$\therefore$ number of days required by B to do the twice as large as the earlier work  $=2 \times 15 = 30$ days

A and B can do a job in $12$ days, B and C can do it in $16$ days. After A has worked for $5$ days and B has worked for $7$ days, C can finish the rest in $13$ days. In how many days can C do the work alone?

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. $16$ days

  2. $24$ days

  3. $36$ days

  4. $48$ days

Show answer
Correct Option: B
Explanation:

Solution:-

Let the amount of work done by $A, B,$ and $C$ per day be $x, y$ and $z$ respectively.
Case I:-

$A$ and $B$ can do the job in $12$ days.
$\therefore$ work done by $A$ and $B$ in one day  $=\cfrac{1}{12}$
$\Rightarrow \; x + y = \cfrac{1}{12}$
Case II:-
B and C can do the job in $16$ days.
$\therefore$ work done by B and C in one day  $=\cfrac{1}{16}$
$\Rightarrow \; y + z = \cfrac{1}{16}$
As given, A has worked for $5$ days and B has worked for $7$ days, C can finish the rest in $13$ days.
$5x + 7y + 13z = 1$
$\Rightarrow$ $5x + 5y + 2y + 2z + 11z = 1$
$\Rightarrow$ $5(x + y) + 2(y + z) + 11z = 1$
$\Rightarrow \; 5 \times \cfrac{1}{12} + 2 \times \cfrac{1}{16} + 11z = 1$
$\Rightarrow \; 11z = 1 - \cfrac{5}{12} - \cfrac{1}{8}$
$\Rightarrow \; 11z = \cfrac{22}{48}$
$\Rightarrow \; z = \cfrac{1}{24}$
Therefore, work done by $C$  $1$ day  $=\cfrac{1}{24}$
Hence, C alone can finish the work in $24$ days.

A thief escaped from police custody. Since he was sprinter he could clock $40\ km/hr$. The police realized it after $3$ hr and started chasing him in the same direction at $50\ km/hr$. The police had a dog which could run at $60\ km/hr$. The dog could run to the thief and then return to the police and then would turn back towards the thief. If kept on doing so till the police caught the thief. Find the total distance travelled by the dog in the direction of the thief.

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. $720 km$

  2. $600 km$

  3. $660 km$

  4. $360 km$

Show answer
Correct Option: C
Explanation:
Speed of thief $=40 \  km \ per\ hr=v _{t}$
Speed of police $=50 \ km\ per\ hr=v _{p}$
Police started after 3hrs of start of thief
$50t=40t+40(3)$
$\therefore   t=12 hr$
Time taken by police to catch thief $=12 hr$
Distance travelled by dog $=12\times 60$
$=720 km$
$\Rightarrow$   Distance travelled by police + To and from distance by Dog $=720$
Distance travelled by police $=50\times 12$
$=600 km$
$\Rightarrow$   $600+2(\text{from distance by dog})=720$
"Fro" distance (distance from thief to police)="to"distance(distance from police to thief after once it returns)
$600+2(\text{'fro' distance by dog})=720$
$\therefore$ 'fro' distance $=60$
$\therefore$ Total distance towards thief $=600+'to"\ distance$
$=720-$ 'fro' distance
$=660 km$

Grass in lawn grown equally thick and in a uniform rate. It takes $24$ days for $70$ cows and $60$ days for $30$ cows to eat the whole of the grass. How many cows are needed to eat the grass in $96$ days?

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. $20$ cows

  2. $24$ cows

  3. $28$ cows

  4. $32$ cows

Show answer
Correct Option: A
Explanation:
Let $n, g, r,$ and $c$ be the no. of cows, grass initially, the rate at which grass grow/day, and grass eaten by cow/day respectively.
Now according to the question-
It takes $24$ days for $70$ cows to eat the whole of the grass.
$g + 24r = 70 \times 24c = 1680c \longrightarrow \left( i \right)$
$g + 60r = 60 \times 30c = 1800c$
Again, given that it takes 60 days for 30 cows to eat the whole of the grass.
$\therefore \; g = 1800c - 60r \longrightarrow \left( ii \right)$
From ${eq}^{n} \left( i \right) \& \left( ii \right)$, we have
$c = \cfrac{3}{10} r \longrightarrow \left( iii \right)$
Now, to eat the grass in 96 days-
$g + 96r = 96 \times nc$
$\Rightarrow \; 96 \times nc = 1800c - 60r + 96r = 1800c + 36r = 1800c + 120c = 1920c$
$\Rightarrow \; n = 20$
Hence, $20$ cows are needed to eat the grass in $96$ days.

A large tanker can be filled by two pipes A and B in 60 minutes and 40 minutes respectively. How many minutes will it take to fill the empty tanker if only B is used in the first-half of the time and A and B are both used in the second-half of the time?

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. $15$

  2. $20$

  3. $27.5$

  4. $30$

Show answer
Correct Option: D
Explanation:

Let x minute will be taken. In one minute A can fill the $\dfrac {1}{60}$ part of tanker and in one minute B can fill the $\dfrac {1}{40}$ part.
Both can fill in t
$\dfrac {t}{60}+\dfrac {t}{40}=1$
$t=\dfrac {60\times 40}{100}$
$t=24$
both can fill in one minute $\dfrac {1}{24}$ part of tanker.
$1=\left (\dfrac {x}{2}\right )\dfrac {1}{40}+\dfrac {x}{2}\left (\dfrac {1}{24}\right )$
$1=\dfrac {x}{80}+\dfrac {x}{48}$
$x=\dfrac {80\times 48}{128}=30$

Two birds are flying in opposite directions over the edge of a circle-shaped forest of radius 4 km. Both start off from the same point simultaneously and both have to go to the same nest. Who reaches the nest first?
I. Speed of bird A is 60 km/hr and speed of bird B is 50 km/hr.
II. The nest is diametrically opposite to the starting point of the flight of the two birds, on the circumference of the forest.

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. If the question can be answered by anyone of the statements alone, but cannot be answered by using the other statement alone.

  2. If the question can be answered by using either statement alone.

  3. If the question can be answered by using both the statements together, but cannot be answered by using either statement alone.

  4. If the question. cannot be answered even by using both the statements together.

Show answer
Correct Option: D
Explanation:

In the problem statement, the distance is given. 
In the first statement, the speeds are given.
In the second statement, nothing substantial can be concluded.
Since information on the paths of flight is not given, we can not determine which bird reaches first. D,

To find the present age of Rishi, which statements can be dispensed.
I. In ten years,Richard will be twice as old as Rishi was 10 years ago.
II. Richard is now 9 years older than Rishi.
III. Five years ago, Rishi was 9 years younger than Richard.

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. Only II

  2. Only III

  3. Either I or II

  4. None of the three statements can be dispensed with

Show answer
Correct Option: C
Explanation:
There are 2 unkown variables, Rishi's age x and Richard's age y.
Each of the following statements give relation between x and y.

Statement I  says: $\quad y+10=2x.$
Statement II says: $\quad y=x+9$

Statement III says:  $y-5=(x-5)+9$= Statement II.
To solve for 2 unknown we require 2 equations.

Since, Statement II and Statement III are equivalent , either can be dispensed with.

Helen is determining how much money she and her friends will need to go for a movie.
Each person going will buy a ticket, a bag of popcorn, and a drink. Helen writes the given formula that will represent the situation.
$m = q (t + p + d)$
Which description represents the meaning of the variable $q$?

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. The price of a ticket

  2. The cost of concessions

  3. The number of people going

  4. The amount of money required

Show answer
Correct Option: C
Explanation:

From the formula $ m = q \left(t+p+d \right)$
t represent the money required for  each ticket
p represent the money required for popcorn for each person
d represent the money required for drink for each person
If we multiply this with the total no of persons going to watch movie we can get total money require.

A clock is set right at $8:00\ a.m$. The clock gains $10$ minutes in $24$ hours. What will be the right time when this clock indicates $1\ p.m$ on the following day?

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. $11.40\ p.m$

  2. $12:00\ p.m$

  3. $10:00\ p.m$

  4. $12:48\ p.m$

  5. $None\ of\ these$

Show answer
Correct Option: D
Explanation:

The clock gains $10$ mins in $24$ hours,
It will gain 1 min in every $2.4$ hours.
Difference in time the clock indicates=$1$ p.m.-$8$ a.m. (the next day)
=$29$ hours,
Time gained=$\dfrac {29}{2.4}=12$ mins.
Actual time=$12:48$ p.m.

Three years ago, the average age of the family of 5 members was 17 years. A baby having been born, the average age of the family is the same today. What is the baby today?

maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

  1. 4 years

  2. 3 years

  3. 2 years

  4. 1 year

Show answer
Correct Option: C
Explanation:

Present total ages of six members $=17\times6$ i.e., 102 years.
Present ages of 5 members $=5\times(17+3)$ i.e. 100 years.
$\implies$ Baby is 2 years old to-day.