Alligation or Mixture
In what ratio must a grocer mix two varieties of pulses costing Rs.15 and Rs. 20 per kg respectively so as to get a mixture worth Rs.16.50 per kg?
Explanation: We use the allegation method
1. Cost of pulses: Rs. 15/kg and Rs. 20/kg.
2. Desired cost: Rs. 16.50/kg.
Step-by-step:
• Difference between Rs. 20 and Rs. 16.50 = 3.50.
• Difference between Rs. 16.50 and Rs. 15 = 1.50.
Ratio of quantities: Using the rule of allegation, the ratio of the two varieties is inversely proportional to the differences calculated:
Ratio = Difference in cost with the second variety / Difference in cost with the first variety
Substituting the values:
Ratio = 3.50 / 1.50 = 7/3
Thus, the grocer must mix the two varieties of pulses in the ratio 7:3
Find the ratio in which rice at 7.20 a kg be mixed with rice at 5.70 a kg to produce a mixture worth Rs. 6.30 a kg.
Explanation:
• The difference between the higher price (Rs. 7.20) and the mixture price (Rs. 6.30) is 0.90.
• The difference between the lower price (Rs. 5.70) and the mixture price (Rs. 6.30) is 0.60.
In the alligation method, the ratio is inversely proportional to the differences. This means:
• The amount of Rice 1 (Rs. 7.20) is determined by the difference between Rs. 5.70 (second rice) and Rs. 6.30 (mixture), which is 0.60.
• The amount of Rice 2 (Rs. 5.70) is determined by the difference between Rs. 7.20 (first rice) and Rs. 6.30 (mixture), which is 0.90.
Thus, the ratio of quantities is 0.60 : 0.90, which simplifies to 2 : 3.
In what ratio must tea at Rs. 62 per kg be mixed with tea at Rs. 72
per kg so that the mixture must be worth Rs. 64.50 per kg?
Explanation: The ratio in the alligation method is inversely proportional to the differences.
Prices:
o Tea 1 = Rs. 62
o Tea 2 = Rs. 72
o Mixture = Rs. 64.50
2. Step 1: Find differences:
o Difference for Tea 2: 72−64.50=7.50
o Difference for Tea 1: 64.50−62=2.50
3. Step 2: Set the ratio:
The ratio of the quantities is inversely proportional to these differences. So:
o The quantity of Tea 1 (Rs. 62) will be in the same ratio as the difference for Tea 2 (7.50).
o The quantity of Tea 2 (Rs. 72) will be in the same ratio as the difference for Tea 1 (2.50).
Hence, the ratio of the quantities is:
Ratio=2.50/7.50=1:3
So, you need 3 parts of Tea 1 (Rs. 62) for 1 part of Tea 2 (Rs. 72).
Answer: (a) 3 : 1.
In which ratio must water be mixed with milk costing Rs. 12 per
litre to obtain a mixture worth of Rs. 8 per litre?
Explanation: Using the alligation method:
Prices:
o Milk = Rs. 12 per litre
o Water = Rs. 0 per litre (since water is free)
o Mixture = Rs. 8 per litre
Calculate the differences:
o Difference between Milk and Mixture: 12−8 =4
o Difference between Mixture and Water: 8−0=8
Set the ratio:
The ratio of water to milk is the inverse of these differences:
o Ratio = 4:8=1:2
Thus, water must be mixed with milk in the ratio of 1:2
The cost of Type 1 rice is Rs. 15 per kg and Type 2 rice is Rs. 20 per kg. if both Type I and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is