Ratios & Proportion
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Q.

There are certain numbers of toys in the box. They are divided into such a way that the person who gets \(\dfrac{1}{4}\) of the whole gets thrice of what the others get on an average.

Find the number of people amongst whom the toys are distributed?

 A.

8

 B.

10

 C.

12

 D.

9

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Solution:
Option(B) is correct

If the person who gets \(\dfrac{1}{4}\) of the whole gets thrice of what the others get on an average, each one will get = \(\dfrac{1}{3}\times \dfrac{1}{4}=\dfrac{1}{12}\) of the whole.

Therefore, if there are $k$ persons other than the person who gets one-fourth, then

\(\begin{align*} \dfrac{1}{4}+\dfrac{k}{12}&=1\\ k&=9 \end{align*}\)

Hence, the total number of people = 10.

Edit:  For an alternative solution, using the verbal method, check comment by Karan Sharma.

Edit 2: For yet another alternative approach, check comment by Naga.

Edit 3: Building onto Naga's solution, Ira explains further to reach the solution.

Edit 4: Yet another solution from Amrendra.


(6) Comment(s)


Amrendra
 ()

let $x$ be the total number of toys, $a$ be the average number of toys that everyone gets and $n$ be the number of people.

now,

$(1/4)*x=3*a$

or

$a=(1/12)*x$

Therefore, the total no. of toys are:

$(x/4) + n*a = x$

solving, we have

$n=9$

Therefore,

Total no. of persons $=9+1=10$



Ira
 ()

$\dfrac{1}{4}= 3 \times \left(\dfrac{3}{4}\right) \times \left(\dfrac{1}{x-1}\right)$

The above equation (from Naga's answer) has been derived like this:

The statement says that $\frac{1}{4}$ equals 3 times 'the average'.

$\text{The average} = \dfrac{\text{Toys that the remaining men get}}{\text{No. of remaining men}}$

$= \dfrac{\frac{3}{4}}{x-1}$

$= \dfrac{3}{4} \times \dfrac{1}{x-1}$

Now, putting this average back to the statement 1:

$\therefore \dfrac{1}{4} = 3 \times \dfrac{3}{4} \times \dfrac{1}{x-1}$

Solving this,

we get $x = 10$



Naga
 ()

you get the same answer with another approach!!!!

let's assume we have 1 toy and $x$ men.

Then out of $x$ men, 1 gets $\frac{1}{4}^{th}$ of the toy.

Hence the remaining $\frac{3}{4}^{th}$ is to be distributed among $x-1$ men.

Now, according to the given statement,

$\dfrac{1}{4}= 3 \times \left(\dfrac{3}{4}\right) \times \left(\dfrac{1}{x-1}\right)$

$\therefore \dfrac{x-1}{4} = \dfrac{9}{4}$

$\therefore x-1=9$

$\therefore x=10$



Navjot Kaur
 ()

please explain the answer in detail



Karan Sharma
 ()

let's make it simple people...

I initially tried this question. by letting total no. of toys equal to 100.Now, one person gets one-fourth of the total.

This means one person gets 25. Right, so now, every other person gets one-third of 25.

Oops...

Now, 25 is not perfectly divisible by 3. so I decided to take total no. of toys equal to 120.Now, that means one person gets one-fourth of 120 which is equal to 30.

Now, we are left with 90 toys which we have to divide among the rest. Now we know that every other person gets 3 times less than the person who got 30.

So, that means everyone else got $\frac{30}{3}$ which is 10.

So, with 90 toys divided by giving 10 toys to everyone means there is total of 9 persons with one getting 30 makes a total of 10, which is the correct answer.



Anu
 ()

please explain the answer to this question