# Difficult Alligations or Mixtures Solved QuestionAptitude Discussion

 Q. In what ratio must a person mix three kinds of wheat costing him Rs 1.20,Rs 1.44 and Rs 1.74 per Kg so that the mixture may be worth Rs 1.41 per Kg?
 ✔ A. 11 : 77 : 7 ✖ B. 25 : 45 : 8 ✖ C. 27 : 23 : 6 ✖ D. 11 : 45 : 7

Solution:
Option(A) is correct

Step1:

Mix wheat of first and third kind to get a mixture worth Rs 1.41 per Kg.
C.P of 1 Kg wheat of 1st kind 120p
C.P of 1 Kg wheat of 3rd kind 174p
Mean Price 141p

C.P of 1 Kg                        C.P of 1 Kg

Wheat of 1st                 wheat of 3rd

kind (120p)                    kind (174p)

\                             /

Mean Price

(141p)

/                             \

33                                    21

By Alligation rule:

$\dfrac{\text{Quantity of 1st kind of wheat}}{\text{Quantity of 3rd kind of wheat}} = \dfrac{33}{21}=\dfrac{11}{7}$

So they must be mixed in the ratio $11 : 7$

Step 2:

Mix wheat of 1st kind and 2nd kind to obtain a mixture worth of Rs. 1.41per Kg

C.P of 1 Kg                        C.P of 1 Kg

Wheat of 1st                 wheat of 3rd

kind (120p)                    kind (144p)

\                             /

Mean Price

(141p)

/                             \

3                                     21

By alligation rule:

$\dfrac{\text{Quantity of 1st kind of wheat}}{\text{Quantity of 2nd kind of wheat}}=\dfrac{3}{21}=\dfrac{1}{7}$
So they must be mixed in the ratio 1 : 7

Thus,
\begin{align*}
&\dfrac{\text{Quantity of 2nd kind of wheat}}{\text{Quantity of 3rd kind of wheat}}=\\
&\left(\dfrac{\text{Quantity of 1st kind of wheat}}{\text{Quantity of 3rd kind of wheat}}\right)\times \left(\dfrac{\text{Quantity of 2nd kind of wheat}}{\text{Quantity of 1st kind of wheat}}\right)\\
& \Rightarrow \dfrac{\text{Quantity of 2nd kind of wheat}}{\text{Quantity of 3rd kind of wheat}}=\left(\dfrac{11}{7}\times \dfrac{7}{1}\right) =\left(\dfrac{11}{1}\right)
\end{align*}

Thus, Quantities of wheat of $1^{st} : 2^{nd} : 3^{rd} = 11 : 77 : 7$

Edit: For a different and shortcut approach, check Murugan's comment.

## (20) Comment(s)

Saras
()

12:7:7

Ganesh
()

(1.2*x+1.44*y+1.74*z)/(x+y+z)=1.41

Solve the equation and we get

7x=y+11z now 12:7:7 only satisfies the equation so answer is 12:7:7

Shalini Lohia
()

12:7:7 is the correct answer. Both by following approach as well as by substitution this answer is correct.

Onkar
()

Yeah. I got that now. Thank you @Shalini :)

Onkar
()

(Approch question no 10.)

Prem
()

The answer 11:77:7 is wrong. We can check by putting back the values in question. Correct answer is 12:7:7. All given options are wrong. Please rectify.

Dhana
()

Going with a ratio of 11:77:7 yields a price of Rs. 1.43/kg which contradicts with Rs. 1.41/kg.

Current
()

Best way to solve these types are -

Considering x,y,z the three wheat breeds with 1.2 rs/kg , 1.44rs/kg, 1.74rs/kg respectively. Finding pairs which can give the given mean price of 1.41 . Only two pairs (1.2,1.44) (1.2,1.74) . Applying alligation in both cases x:y=0.03:0.21

x:z=0.33:0.21

Thus x:y:z= (0.03+0.33:0.21:0.21)

So 36:21:21

12:7:7 is only possibility

Rajesh Kumar
()

can any one tell me why we use 120 two times .. not 144 confused !!

Shivam
()

Hi Rajesh,

In order to get avg. 1.41 we need to have values such that 1.41 lies between them. We can not use a combination 1.44 and 1.74 to get 1.41. Hence two possible alligations will be 1.2 and 1.44 , 1.2 and 1.74. And every time while paring to solve alligation we have to take care of it.

Rajesh Kumar
()

#rajesh

can any one tell me why we use 120 two times to mix but not 144 ..confused !!

Anuj Bhawsar
()

why are we do we add the ratio

c:a:b=7:(11+1):7.

Simple way is to get by substitution and ans comes as 11 : 77 : 7.

Prince
()

Alligation has failed this question.

One possibility =12:7:7

Abhishek Jain
()

this is the only possibility. 11 : 77 : 7 is wrong

Murugan
()

$a:b=1:7$

$a:c=11:7$

$c:a:b=c:a::a:b$

So,

$c:a:b=7:11::7:1$

To find out the value of $a$,

$=(7*1):(7*11)::(7*11):(1*11)$

Therefore,

$=7:77::77:11$

$a:b:c=11:77:7$

Sachin Dahiya
()

Approach for this question and the last 2 question is different and also the answer shown as correct is not matching the given information. it should be 12:7:7 #Sachin

Ashok
()

why isn't that approach not working?

Naresh
()

$c:a=7:11$

a:b=1:7

So

$c:a:b=7:(11+1):7$

means,

$a:b:c = 12:7:7$

Dudeja
()

I agree with Naresh.