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Shortcut 12
Mean Price

                 \(\frac{ax+by}{a+b}=M.P \)

a : b = ratio between quantities
x, y = price of the products
M.P = mean price
Question:
Two varieties of rice with prices Rs. 30 per kg and Rs. 40 per kg are mixed in the ratio 2 : 3. Find the mean price.
Answer:
a = 2; b = 3
x = 30; y = 40
Substitute the values in the above equation.
Mean price = Rs. 36

Shortcut 13:

Allegation rule

1aa = m.p~ y
b = m.p~x      (~ means difference)
Question:
In what ratio two varieties of rice worth Rs. 40 per kg and Rs. 60 per kg should be mixed to give a variety worth Rs. 46 per kg?
Answer:
M. P = 46
x = 40
y = 60
From the above allegation rule, we get
a = 14
b = 6
The ratio between the two varieties = 14 : 6 = 7 : 3

Shortcut 14

Ratio after ‘n’ draws

         \( \lbrack1-(x/100)\rbrack^n\)

x = percentage of reduction per draw
Question:
A container has 1000 liters of wine. 100 liters of wine is drawn from the container and filled with water. This process is done two more times. What is the ratio between wine and water in the final mixture?
Answer:
Number of draws n = 3
Step 1: Find the percentage of solution drawn – ‘x’
Step 2: Find the fraction of wine after 3 draws.
[1-(10/100)]3 = 0.93 = 0.729
Step 3 : Multiply the fraction with initial quantity
0.729 x 1000 = 729
Present quantity of wine = 729
Present quantity of water = 1000 – 729 = 271
Ratio between wine and water = 729 : 271

Shortcut 15

Ratio before ‘n’ draws

                  \(\lbrack A/(A+B)\rbrack^\frac1n \)

A : B = Final ratio between quantity of the varieties.
Question:
From a mixture of wine and water, 20 liters is taken and

replaced with water. This process is repeated two more times. What is the ratio between wine and water after the first draw if current ratio is 16: 9?
Answer:
A = 16
B = 9
n = 3 – 1 = 2
Substitute the values in the above formula, we get
4 : 5
4 : 5 = quantity of wine : quantity of wine + water
Quantity of wine = 4
Quantity of water = 5 – 4 = 1
Ratio between wine and water = 4 : 1