Unit4 - Subjective Questions
PEA305 • Practice Questions with Detailed Answers
Define Ratio and Proportion. Differentiate between the two and list three key properties of proportion.
Definition of Ratio:
A ratio is the comparison of two quantities of the same kind and in the same units. It indicates how many times one quantity is contained in another. It is denoted by a colon (:).
For example, the ratio of to is written as or , where is called the antecedent and is called the consequent.
Definition of Proportion:
A proportion is an equation stating that two ratios are equal. If , then and are said to be in proportion.
Mathematical representation:
Here, and are called extremes, and and are called means.
Key Properties of Proportion:
- Product Rule: Product of extremes = Product of means ().
- Invertendo: If , then .
- Componendo and Dividendo: If , then .
Two numbers are in the ratio . If 9 is subtracted from each, the new numbers are in the ratio . Find the smaller number.
Step 1: Set up the variables.
Let the two numbers be and .
Step 2: Formulate the equation based on the condition.
According to the problem, if 9 is subtracted from each, the ratio becomes .
Step 3: Solve for .
Cross-multiply:
Step 4: Find the smaller number.
The numbers are and .
The smaller number is 33.
Explain the Rule of Alligation with the help of a diagram. When is this rule applicable?
Rule of Alligation:
It is a rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a desired mean price.
Method/Diagram:
Let the cost price of the cheaper ingredient be and the cost price of the dearer (more expensive) ingredient be . Let the mean price (cost price of the mixture) be .
- Quantity of Cheaper : Quantity of Dearer =
Visual Representation:
- Cheaper Price () ---------- Dearer Price ()
- ---------- Mean Price () ----------
- --------------------
So,
Applicability:
This rule is applicable when:
- Mixing two ingredients of different prices.
- Finding the ratio of quantities when the average value is known (e.g., average age, average marks, average speed).
A bag contains 50 P, 25 P, and 10 P coins in the ratio , amounting to Rs. 206. Find the number of coins of each type.
Step 1: Define the number of coins.
Let the number of 50 P, 25 P, and 10 P coins be , , and respectively.
Step 2: Convert values to Rupee terms.
- Value of 50 P coins = Rs.
- Value of 25 P coins = Rs.
- Value of 10 P coins = Rs.
Step 3: Formulate the total amount equation.
Total Amount = Rs. 206
Step 4: Solve for .
Step 5: Calculate the number of coins.
- 50 P coins:
- 25 P coins:
- 10 P coins:
Answer: There are 200 coins of 50 P, 360 coins of 25 P, and 160 coins of 10 P.
Three partners , , and invest in a business. 's capital is equal to twice 's capital and 's capital is three times 's capital. Find the ratio of their capitals. If the total profit is Rs 11,000, find 's share.
Step 1: Determine the relationship between capitals.
Let 's capital = .
Then, 's capital = .
And, 's capital = 's capital = .
Step 2: Find the ratio of capitals.
Ratio .
Step 3: Calculate Total Ratio Units.
Total units = .
Step 4: Calculate A's share of the profit.
Total Profit = Rs. 11,000.
's share =
's share = .
Answer: The ratio of capitals is 6:3:1 and A's share is Rs. 6,600.
Explain the different types of partnership (Simple vs. Compound) and the concepts of Sleeping and Working partners.
1. Types of Partnership:
- Simple Partnership: In this type, capitals of the partners are invested for the same period of time. The profit or loss is distributed purely in proportion to their invested capitals.
- Formula:
- Compound Partnership: In this type, capitals are invested for different periods of time. The profit or loss is distributed in proportion to the product of capital and the time period for which it was invested.
- Formula:
2. Types of Partners:
- Working (Active) Partner: A partner who manages the business. They are usually paid a salary or a specific percentage of profit for their work before the remaining profit is divided based on investment.
- Sleeping (Dormant) Partner: A partner who only invests money but does not participate in the daily management of the business. They only get a share of profit proportional to their investment.
A container contains 40 liters of milk. From this container, 4 liters of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?
Concept: This is a replacement problem. The formula for the quantity of the original liquid left after operations is:
Where:
- = Initial quantity (40 liters)
- = Quantity removed and replaced (4 liters)
- = Number of times the process is performed (1 initial + 2 repeats = 3 times)
Calculation:
Answer: The container now contains 29.16 liters of milk.
Solve the following age problem: The ratio of the present ages of a father and his son is . Ten years later, the ratio of their ages will be . Find their present ages.
Step 1: Set up present ages.
Let the father's present age be and the son's present age be .
Step 2: Express ages after 10 years.
Father's age after 10 years =
Son's age after 10 years =
Step 3: Set up the ratio equation.
According to the problem:
Step 4: Solve for .
Cross-multiply:
Step 5: Calculate present ages.
Father's age = years.
Son's age = years.
Answer: Father is 70 years old, and the son is 30 years old.
How must a grocer mix two varieties of tea worth Rs. 60 per kg and Rs. 65 per kg so that by selling the mixture at Rs. 68.20 per kg he may gain 10%?
Step 1: Calculate the Cost Price (CP) of the mixture.
The Alligation rule applies to Cost Prices, not Selling Prices.
Selling Price (SP) = Rs. 68.20
Gain % = 10%
The mean cost price () is Rs. 62 per kg.
Step 2: Apply Rule of Alligation.
- Price of cheaper tea () = 60
- Price of dearer tea () = 65
- Mean Price () = 62
Ratio =
Ratio =
Ratio =
Answer: The grocer must mix the two varieties of tea in the ratio 3:2.
A and B started a business with investments of Rs. 20,000 and Rs. 30,000 respectively. After 4 months, A withdraws Rs. 5,000 and B adds Rs. 5,000. At the end of the year, the total profit is Rs. 38,400. Find the share of B.
Step 1: Calculate the Equivalent Capital for A for 1 year (12 months).
- For first 4 months:
- After withdrawal () for remaining 8 months:
- Total Equivalent Capital for A =
Step 2: Calculate the Equivalent Capital for B.
- For first 4 months:
- After addition () for remaining 8 months:
- Total Equivalent Capital for B =
Step 3: Find the ratio of profit sharing.
Step 4: Calculate B's share.
Total Profit = Rs. 38,400
Ratio sum =
B's Share =
B's Share =
Answer: B's share of the profit is Rs. 25,600.
Define Compound Ratio, Duplicate Ratio, and Sub-duplicate Ratio with examples.
1. Compound Ratio:
When two or more ratios are multiplied term-wise, the result is called the compound ratio.
If ratios are and , the compound ratio is .
Example: Compound ratio of and is .
2. Duplicate Ratio:
The duplicate ratio of is the ratio of their squares, i.e., .
Example: Duplicate ratio of is .
3. Sub-duplicate Ratio:
The sub-duplicate ratio of is the ratio of their square roots, i.e., .
Example: Sub-duplicate ratio of is .
A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
Step 1: Analyze initial state.
Total parts = parts.
Volume of Syrup = of the vessel.
Volume of Water = of the vessel.
Step 2: Analyze the process.
Let the quantity of mixture drawn off and replaced be (fraction of total volume).
Since we are replacing with water, we focus on the Syrup concentration because syrup is only removed, never added. This makes the calculation easier.
Initial concentration of Syrup = .
Final concentration of Syrup (half water, half syrup) = .
Step 3: Formulate equation for Syrup.
Quantity of Syrup Left = (Initial Syrup) - (Syrup removed in mixture)
(Total Volume Final Conc) = (Total Volume Initial Conc)
Since Total Volume is constant (taken as 1 unit):
Step 4: Solve for .
Answer: 1/5 of the mixture must be drawn off and replaced with water.
Find the Fourth Proportional to 4, 9, 12 and the Mean Proportional between 0.08 and 0.18.
Part 1: Fourth Proportional
Let the fourth proportional to 4, 9, 12 be .
This implies .
Using Product of Extremes = Product of Means:
Part 2: Mean Proportional
Let the mean proportional between 0.08 and 0.18 be .
This implies .
Answer: Fourth proportional is 27; Mean proportional is 0.12.
Three vessels whose capacities are in the ratio are completely filled with milk mixed with water. The ratio of milk and water in the mixture of vessels are , , and respectively. Taking of the first, of the second and of the third mixtures, a new mixture is kept in a new vessel. Find the percentage of water in the new mixture.
Step 1: Standardize concentrations.
- Vessel 1 (Ratio 5:2): Milk = 5/7, Water = 2/7.
- Vessel 2 (Ratio 4:1): Milk = 4/5, Water = 1/5.
- Vessel 3 (Ratio 4:1): Milk = 4/5, Water = 1/5.
Step 2: Determine quantities taken.
The capacities are in ratio . Let's assume capacities are 3, 2, and 1 liters. (Or units).
- Taken from Vessel 1: unit.
- Taken from Vessel 2: unit.
- Taken from Vessel 3: unit.
Step 3: Calculate total water in new mixture.
- Water from V1 (1 unit):
- Water from V2 (1 unit):
- Water from V3 (1/7 unit):
Total Water =
LCM is 35.
Total Water =
Step 4: Calculate Total Volume.
Total Volume = .
Step 5: Calculate Percentage of Water.
Answer: The percentage of water in the new mixture is 24%.
The incomes of A, B, and C are in the ratio and their spendings are in the ratio . If A saves of his income, then find the ratio of their savings.
Step 1: Set variables.
Let incomes be .
Let expenditures be .
Step 2: Use A's saving condition.
Income - Expenditure = Saving
Given: Saving of A = of Income A = .
So,
Step 3: Calculate Savings for each.
- A: Income . Exp . Saving = .
- B: Income . Exp . Saving = .
- C: Income . Exp . Saving = .
Step 4: Form Ratio.
Ratio = .
Answer: The ratio of savings is 56 : 99 : 69.
A milkman has two types of milk. In the first container, the percentage of milk is 80% and in the second container, the percentage of milk is 60%. If he mixes 28 liters of milk from the first container and 32 liters of milk from the second container, then what is the percentage of milk in the mixture?
Step 1: Calculate amount of pure milk in first part.
Volume 1 = 28 L, Concentration = 80%.
Milk = Liters.
Step 2: Calculate amount of pure milk in second part.
Volume 2 = 32 L, Concentration = 60%.
Milk = Liters.
Step 3: Calculate Total Volume and Total Milk.
Total Volume = Liters.
Total Milk = Liters.
Step 4: Calculate new percentage.
Answer: The percentage of milk in the mixture is 69.33%.
Derive the formula for replacement: , where is total capacity, is quantity replaced, and is number of operations.
Derivation:
Let the initial quantity of the pure liquid be units.
Let units be withdrawn and replaced by water.
After 1st Operation:
Amount of pure liquid withdrawn = .
Amount of pure liquid left = .
After 2nd Operation:
The ratio of pure liquid to total mixture is now .
When units of mixture are withdrawn again, the amount of pure liquid in that withdrawn portion is (concentration) = .
Remaining pure liquid:
Generalization:
Following this pattern, after operations, the quantity of pure liquid left is:
A, B, and C enter into a partnership with capitals in the ratio . After 2 months, A withdraws half of his capital. If the total profit at the end of 12 months is Rs. 378, find B's share.
Step 1: Simplify the capital ratio.
LCM of 2, 3, 4 is 12.
Ratio .
Let initial investments be .
Step 2: Calculate Weighted Capital (Ratio of Profits).
- A: Invested for 2 months, then withdrew half (so remains) for 10 months.
. - B: Invested for 12 months.
. - C: Invested for 12 months.
.
Step 3: Simplify Profit Ratio.
Divide by 6: .
Total parts = .
Step 4: Find B's Share.
Total Profit = Rs. 378.
B's Share = .
.
.
Answer: B's share is Rs. 144.
In a zoo, there are rabbits and pigeons. If heads are counted, there are 200 and if legs are counted, there are 580. How many pigeons are there? Solve using the Alligation method.
Step 1: Determine average legs per head.
Total Heads = 200 (Total animals).
Total Legs = 580.
Average legs per animal = .
Step 2: Set up Alligation.
- Pigeons have 2 legs.
- Rabbits have 4 legs.
- Mean = 2.9.
Step 3: Apply Cross difference.
- Pigeon (2) vs Mean (2.9) -> Diff = $0.9$
- Rabbit (4) vs Mean (2.9) -> Diff = $1.1$
Ratio (Pigeon : Rabbit) = .
Step 4: Calculate number of Pigeons.
Total animals = 200.
Ratio sum = .
Pigeons =
Pigeons = .
Answer: There are 110 pigeons.
Six years ago, the ratio of the ages of Kunal and Sagar was . Four years hence, the ratio of their ages will be . What is Sagar's present age?
Step 1: Set up ages based on past ratio.
Let ages 6 years ago be and .
Step 2: Express present ages.
Kunal's present age = .
Sagar's present age = .
Step 3: Express ages 4 years hence.
Kunal: .
Sagar: .
Step 4: Formulate equation.
Step 5: Solve for .
Step 6: Find Sagar's present age.
Sagar's present age = years.
Answer: Sagar's present age is 16 years.