Unit 4: Ratio, Proportion, Alligation and Mixture - Subjective Questions

Definition: A ratio compares two quantities of the same kind by division. If the quantities are and , their ratio is written as or , where .

Conditions:

• The quantities must be expressed in the same unit.
• The second term of the ratio must not be zero.
• A ratio has no unit after like quantities are compared.

Simplification:

Convert both quantities to grams:

Therefore,

Hence, the simplified ratio is .

Four quantities and are in proportion if the ratio of the first two equals the ratio of the last two:

Equivalently, their cross-products must be equal:

For the given numbers:

Checking the cross-products:

Since the cross-products are equal, the numbers are in proportion.

• Extremes: and
• Means: and

Direct proportion: Two quantities are directly proportional when an increase in one produces a proportional increase in the other. Thus,

Inverse proportion: Two quantities are inversely proportional when an increase in one produces a proportional decrease in the other. Thus,

The number of workers and the required time are inversely proportional. Let be the required number of days:

Therefore, workers will complete the task in days.

The compound ratio is obtained by multiplying all antecedents together and all consequents together.

For , , and :

Thus, the compound ratio is .

For a ratio :

• Its duplicate ratio is .
• Its triplicate ratio is .

For example, for :

• Duplicate ratio
• Triplicate ratio

Let the present ages of A and B be years and years, respectively.

After years, their ages will be and . According to the question:

Cross-multiplying:

Therefore:

• A's present age years
• B's present age years

Hence, their present ages are years and years.

Let the son's present age be years. Then the father's present age is years.

Twelve years ago:

• Son's age
• Father's age

According to the condition:

Therefore:

• Son's present age years
• Father's present age years

Verification: Twelve years ago, their ages were and years, and

Hence, the present ages are years and years.

Let the daughter's present age be years. The mother's present age is therefore years.

Five years ago:

• Daughter's age
• Mother's age

Using the given condition:

Thus:

• Daughter's present age years
• Mother's present age years

Therefore, their present ages are years and years.

Let the ages of P and Q ten years ago be and , respectively.

Their present ages are:

• P:
• Q:

Ten years from now, their ages will be and . Therefore:

Cross-multiplying:

Hence:

• P's present age years
• Q's present age years

Therefore, their present ages are years and years.

In a partnership, profit or loss is divided according to the effective use of each partner's capital.

If a partner invests capital for time , the partner's share is proportional to:

For partners A, B, and C, the partnership ratio is:

Important points:

• Capital must be measured in the same monetary unit.
• Time must be measured in the same unit, usually months.
• If capital changes during the year, each investment period is calculated separately.
• A working partner's salary or commission is deducted before the remaining profit is divided, if the agreement specifies it.

Since both partners invest for the same period, their profit-sharing ratio equals their capital ratio:

The total number of ratio parts is:

Therefore:

Hence:

• A receives .
• B receives .

Calculate capital-months for each partner.

A's investment:

• for the first months
• for the next months

Therefore:

B's investment:
B joins after months, so B invests for months:

Thus, the profit-sharing ratio is:

Hence, A and B share the profit in the ratio .

First deduct B's salary from the total profit:

Calculate the capital-month ratio:

Thus:

The total number of parts is .

Profit shares from are:

B also receives the salary of .

Therefore:

• A receives .
• B receives .
• C receives .

Alligation is a method used to determine the ratio in which two or more ingredients of different costs, strengths, or concentrations must be mixed to obtain a required mean value.

If the cheaper value is , the dearer value is , and the mean value is , where , then:

This can be represented as cross-differences:

• Difference associated with the cheaper ingredient:
• Difference associated with the dearer ingredient:

The rule assumes that the weighted average of the ingredient values equals the required mean value.

Using alligation:

• Cheaper price
• Dearer price
• Mean price

Therefore:

Thus, the required ratio is .

Verification:

Hence, rice costing and per kg must be mixed in the ratio .

Using alligation:

• Lower concentration
• Higher concentration
• Required concentration

The ratio of lower-concentration solution to higher-concentration solution is:

The ratio parts of the solution correspond to litres. Therefore, one part is:

The required amount of the solution is:

Hence, litres of the acid solution are required.

First determine the cost price of the mixture. If the selling price is with a profit of , then:

Now apply alligation:

• Cheaper tea
• Dearer tea
• Mean cost

Therefore:

Verification:

Hence, the two varieties must be mixed in the ratio .

Initially, the mixture contains:

Since the removed mixture has the same ratio, litres removed contains:

Quantities left after removal:

• Milk litres
• Water litres

After adding litres of water:

• Final milk litres
• Final water litres

Hence, the vessel finally contains litres of milk and litres of water.

Initially, the quantity of the original liquid is litres.

After removing litres, the fraction of the original liquid remaining is:

Therefore, after the first replacement, the original liquid remaining is:

During each subsequent operation, the same fraction of the liquid present remains. Thus, after two operations:

Continuing this process, after operations:

If the vessel was initially filled entirely with the original liquid, the amount of the replacement liquid after operations is:

The formula applies when the mixture is uniform before every removal.

Use the repeated replacement formula:

Here:

• litres
• litres

Therefore:

The quantity of water is:

Hence, after three operations, the vessel contains litres of milk and litres of water.

Initially, the amount of acid is:

The amount of water is:

In each operation, the fraction of the mixture retained is:

After two operations, the acid remaining is:

Since the total volume remains litres, the final amount of water is:

Therefore, the final mixture contains:

• Acid: litres
• Water: litres

The final ratio of acid to water is: