Question

Tea worth Rs.126 per kg and Rs.135 per kg are mixed with a third variety in the ratio 1 :1: 2. If the mixture is worth Rs.153 per kg then find the price of the third variety per kg.

Answer 1

Hint: At first find the average of Rs 126 and Rs 135 price as it represents the mixture then take the average as one kind and let another kind be Rs $x$ .The average price is Rs 153. So then apply the rule of allegation to find the value of the third type of tea.

Complete step-by-step answer:
In the question we are given three varieties of tea such that two of them are worth Rs 126 and Rs 135 but the price of third is unknown. Then we are told in the question that all are mixed and worth Rs 153 per kg in the proportion of 1: 1:; 2. Now we have to find the price of third variety of tea
Now as we know that the first and second varieties are mixed in equal proportions so their average price will be $Rs\dfrac{126+135}{2}=Rs130.50$ .
The mixture of only first and second will cost Rs 130.50
Now we can take a mixture of tea of worth Rs 130.50 and that of a third variety is formed but in this case it will be 2: 2 or 1: 1 as the Rs 130.50 comprises a mixture of both Rs 126, Rs 135.
Let the third variety price be $x$
Now we will apply rule of allegation which is:
Let us suppose that two ingredients of concentrations ‘a’ and ‘b’ respectively have been mixed in some proportion. Let ‘a’ be a cheaper component and ‘b’ be the dearer or the costlier component.
The rule of allegation states that:
$\dfrac{\text{Quantity of cheaper substance}}{\text{Quantity of dearer substance}}=\dfrac{\text{C}\text{.P of dearer substance - Mean Price}}{\text{Mean price - C}\text{.P of cheaper substance}}$
Let ‘c’ be the cost price or C.P of a unit quantity of a cheaper substance, ‘m’ be the mean price, ‘d’ is cost of a unit quantity of the dearer substance, then we can write:
(Quantity of Cheaper Substance): (Quantity of dearer substance) = (d – m): (m – c)
Here in the question the quantity ratio of both the cheaper and dearer one is 1: 1 so ratio is 1.
Here the mean price is Rs 153, $x$ be the price of the dearer one which is to be found out and Rs 130.50 is the cost of the cheaper one.
So,
$1=\dfrac{x-153}{153-130.50}$
Which can be further simplified as,
$1=\dfrac{x-153}{22.5}$
On cross multiplication we get,
$x-153=22.5$
So $x$ is equal to Rs 175.50.

Note: Students should know the rule of allegation before doing this problem. They should also be careful about the errors in calculations.