1. 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?
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
2. 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.
• 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.
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?
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.
4. In which ratio must water be mixed with milk costing Rs. 12 per
litre to obtain a mixture worth of Rs. 8 per litre?
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
5. 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
To find the price per kg of the mixed variety, use the formula for the weighted average:
Price of mixture = (P1×Q1)+(P2×Q2) / Q1+Q2
Where:
• P1=15 (Price of Type 1 rice)
• P2=20(Price of Type 2 rice)
• Q1=2 (Quantity of Type 1 rice)
• Q2=3 (Quantity of Type 2 rice)
Now, substitute the values:
Price of mixture = (15×2)+(20×3) / 2+3 = 90/5 = 18
So, the price per kg of the mixed rice is Rs. 18.