Unit2 - Subjective Questions

Definition of Ratio

A ratio is a comparison of two quantities of the same kind, expressed as or . It indicates how many times one quantity contains the other.

Types of Ratios

1. Duplicate Ratio:

   • Definition: The duplicate ratio of is .
   • Example: The duplicate ratio of is .

2. Sub-duplicate Ratio:

   • Definition: The sub-duplicate ratio of is .
   • Example: The sub-duplicate ratio of is .

3. Triplicate Ratio:

   • Definition: The triplicate ratio of is .
   • Example: The triplicate ratio of is .

4. Sub-triplicate Ratio:

   • Definition: The sub-triplicate ratio of is .
   • Example: The sub-triplicate ratio of is .

Concept of Percentage

Percentage is a way of expressing a number or ratio as a fraction of 100. It is denoted by the symbol '%'. For example, 25% means 25 out of 100, or . Percentages are dimensionless numbers used to represent proportions or fractions in a standardized way, making comparisons easier.

Usage in Business Context

Percentages are widely used in business for various calculations, including:

• Profit Margin: To express profit as a proportion of revenue or cost.
• Discount: To show the reduction in price as a proportion of the original price.
• Interest Rates: To indicate the cost of borrowing or the return on investment.
• Growth Rates: To measure the increase or decrease in sales, production, or other metrics over time.
• Taxes: To calculate sales tax, income tax, etc.

Example: Calculating Profit Margin

Scenario: A retailer buys a product for $80 and sells it for $120.

Calculation:

1. Calculate the Profit:
   Profit = Selling Price - Cost Price
   Profit = $120 - $80 = $40

2. Calculate the Profit Margin (as a percentage of selling price):
   Profit Margin Percentage =
   Profit Margin Percentage =
   Profit Margin Percentage =

This means that for every dollar of sales, the retailer makes approximately 33.33 cents in profit.

Distinction between Ratio and Proportion

Fundamental Properties of Proportion

If (or ), then the following properties hold:

1. Cross-Multiplication Property (Product of Extremes and Means):

   • Statement: The product of the extremes is equal to the product of the means.
   • Explanation: In a proportion , and are called 'extremes', and and are called 'means'. This property states that .
   • Example: Given the proportion . Here, . According to the property, . This is true.

2. Invertendo Property:

   • Statement: If four quantities are in proportion, then their inverse ratios are also in proportion.
   • Explanation: If , then .
   • Example: Given the proportion , which means . By Invertendo, . Both sides simplify to $1.5$, thus the property holds.

(Other properties include Alternando, Componendo, Dividendo, Componendo and Dividendo)

Direct Variation

• Definition: Direct variation describes a relationship where two quantities change in the same direction. If one quantity increases, the other quantity also increases proportionally, and if one decreases, the other decreases proportionally.
• Mathematical Representation: If varies directly as , it is written as or , where is the constant of proportionality.
• Constant of Proportionality (): In direct variation, . It represents the constant ratio between the two quantities.
• Real-world Example: The cost of buying apples varies directly with the number of apples bought. If 1 apple costs $2, then 5 apples cost $10. Here, per apple. As the number of apples increases, the total cost increases.

Inverse Variation

• Definition: Inverse variation describes a relationship where two quantities change in opposite directions. If one quantity increases, the other quantity decreases proportionally, and vice versa.
• Mathematical Representation: If varies inversely as , it is written as or , where is the constant of proportionality.
• Constant of Proportionality (): In inverse variation, . It represents the constant product of the two quantities.
• Real-world Example: The time taken to complete a certain amount of work varies inversely with the number of workers. If 2 workers can complete a task in 6 hours, then 4 workers can complete the same task in 3 hours. Here, worker-hours. As the number of workers increases, the time taken to complete the work decreases.

Key Differences

Derivation of Relationship between Direct Variation and Proportion

Given: varies directly as .

Step 1: Express direct variation mathematically.
According to the definition of direct variation, if varies directly as , we can write the relationship as:

This means that is equal to multiplied by a constant, known as the constant of proportionality.

where is the constant of proportionality.

Step 2: Express the constant of proportionality.
From equation (1), we can isolate :

This implies that the ratio is constant for any corresponding values of and .

Step 3: Consider two sets of corresponding values.
Let be one set of corresponding values for and , and be another set of corresponding values.

Step 4: Apply the constant of proportionality.
Since is a constant, it must be the same for all corresponding pairs of and .
For the first set of values , we have:

For the second set of values , we have:

Step 5: Equate the expressions for .
Since both expressions (3) and (4) are equal to the same constant , we can equate them:

Step 6: Rearrange the equation to the desired form.
To prove , we can rearrange the equation from Step 5 by cross-multiplication or by multiplying both sides by :

Conclusion:
This proves that if varies directly as , then the ratio of two values of is equal to the ratio of the corresponding two values of . This relationship forms the basis of direct proportion.

Compound Ratio

• Definition: A compound ratio is the ratio obtained by multiplying the corresponding terms of two or more simple ratios. If we have two ratios and , their compound ratio is or . Similarly, for three ratios , the compound ratio is .
• Example: Find the compound ratio of , , and .
  • The antecedents are $2, 5, 6$.
  • The consequents are $3, 4, 7$.
  • Compound Ratio =
  • Compound Ratio =
  • Simplifying by dividing by 12: .

Continued Proportion

• Definition: Three or more quantities are said to be in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third, and so on. For three quantities to be in continued proportion, it must satisfy , which implies . In this case, is called the mean proportional between and , and .
• Example: Are the numbers $4, 12, 36$ in continued proportion?
  • We check if .
  • First ratio: .
  • Second ratio: .
  • Since , the numbers $4, 12, 36$ are in continued proportion. Here, 12 is the mean proportional between 4 and 36, and .

Problem Solution

Step 1: Define initial quantities based on the given ratio.
Let the initial quantity of milk be liters.
Let the initial quantity of water be liters.

Step 2: Account for the change in the mixture.
14 liters of water are added.
New quantity of milk = liters (remains unchanged)
New quantity of water = liters

Step 3: Formulate an equation using the new ratio.
The new ratio of milk to water is given as .
So,

Step 4: Solve the equation for .
Cross-multiply:


Subtract from both sides:

Step 5: Calculate the initial quantity of milk.
Initial quantity of milk =
Initial quantity of milk = liters.

Verification (Optional):
Initial milk = 35 L, Initial water = L. Ratio = .
After adding 14 L of water: Milk = 35 L, Water = L. Ratio = . This matches the given new ratio.

Conclusion

The initial quantity of milk in the mixture was 35 liters.

Problem Solution

Step 1: Write down the variation relationship.
Given that varies directly as and inversely as . This is a case of joint variation.
Mathematically, this can be expressed as:

Step 2: Introduce the constant of proportionality.
To convert the proportionality into an equation, we introduce a constant of proportionality, :

Step 3: Use the initial conditions to find the value of .
We are given when and . Substitute these values into equation (1):

Now, solve for :

Step 4: Rewrite the variation equation with the determined .
Now we have the specific relationship for this problem:

Step 5: Use the new conditions to find the value of .
We need to find when and . Substitute these values into equation (2):

Conclusion

When and , the value of is 10.

Definition of Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is known as the common difference, usually denoted by .

For example, the sequence is an AP with the first term and common difference .

Derivation of the Formula for the Term of an AP

Let's consider an Arithmetic Progression with:

• The first term = (or simply )
• The common difference =

We can write out the terms of the AP as follows:

• First term ():

• Second term (): To get the second term, we add the common difference to the first term.
  

• Third term (): To get the third term, we add the common difference to the second term.
  

• Fourth term (): To get the fourth term, we add the common difference to the third term.
  

Observing the pattern, we can see a relationship between the term number and the multiple of the common difference added to the first term:

• For the 1st term, is added 0 times ().
• For the 2nd term, is added 1 time ().
• For the 3rd term, is added 2 times ().
• For the 4th term, is added 3 times ().

Generalizing this pattern, for the term (), the common difference will be added times to the first term .

Thus, the formula for the term of an Arithmetic Progression is:

Where:

• = the term
• = the first term
• = the term number
• = the common difference

Derivation of the Formula for the Sum of the First Terms of an AP

Let's denote the sum of the first terms of an Arithmetic Progression as .
Let the AP be .

Step 1: Write the sum in two ways.

We can write the sum of the first terms as:

Let the last term, , be denoted by . So, .
We can also write the sum by reversing the order of the terms:

Step 2: Add equation (1) and equation (2).

Add the corresponding terms from both equations:

Notice that in each pair of brackets, the common difference cancels out:


... and so on.

So, each term in the sum on the right-hand side simplifies to .
There are such pairs, so we have terms of .

Step 3: Solve for .

Divide by 2 to find the formula for :

Step 4: Substitute the formula for the last term () into the sum formula.

We know that the last term .
Substitute this expression for into equation (3):

Conclusion

The two main formulas for the sum of the first terms of an Arithmetic Progression are:

1. When the first term () and the last term () are known:
   

2. When the first term (), the number of terms (), and the common difference () are known:
   

Concept of Arithmetic Mean (AM)

An Arithmetic Mean (AM) is a single value that represents the 'average' of a set of numbers. When a certain number of terms are inserted between two given numbers such that the entire sequence forms an Arithmetic Progression (AP), these inserted terms are called arithmetic means.

If and are two numbers, and are arithmetic means inserted between and , then the sequence forms an AP.

For two numbers and , the single arithmetic mean is simply .

Inserting 3 Arithmetic Means Between 10 and 26

Let the two given numbers be and .
We need to insert 3 arithmetic means, let's call them , between 10 and 26.

This means the sequence will be an AP:

In this AP:

• The first term () = 10
• The last term ( or ) = 26
• The total number of terms () =

We use the formula for the term of an AP: .
Here, .

Substitute the values:

Now that we have the common difference (), we can find the arithmetic means:

The sequence is $10, 14, 18, 22, 26$.

Conclusion

The 3 arithmetic means inserted between 10 and 26 are 14, 18, and 22.

Important Properties of an Arithmetic Progression (AP)

Here are four important properties of an Arithmetic Progression:

1. Constant Common Difference:

   • Property: The defining characteristic of an AP is that the difference between any term and its preceding term is constant. This constant is called the common difference ().
   • Explanation: For any AP , we have . This property allows us to easily find any term if the first term and common difference are known.
   • Example: In AP , the common difference is , , etc.

2. Arithmetic Mean Property:

   • Property: If are in AP, then is the arithmetic mean of and , which means . More generally, any term (except the first and last) is the arithmetic mean of its preceding and succeeding terms.
   • Explanation: If are in AP, then and . Therefore, . This property is fundamental for finding missing terms or confirming if a sequence is an AP.
   • Example: In AP $5, 8, 11$, .

3. Addition/Subtraction of a Constant:

   • Property: If a constant is added to or subtracted from each term of an AP, the resulting sequence is also an AP with the same common difference.
   • Explanation: Let the AP be with common difference . If a constant is added to each term, the new sequence is . The difference between consecutive terms is . So, the common difference remains .
   • Example: AP: $2, 4, 6, 8$ (). Adding 3 to each term: $5, 7, 9, 11$. This is still an AP with .

4. Multiplication/Division by a Non-Zero Constant:

   • Property: If each term of an AP is multiplied or divided by a non-zero constant, the resulting sequence is also an AP. However, the common difference changes proportionally.
   • Explanation: Let the AP be with common difference . If each term is multiplied by a non-zero constant , the new sequence is . The new common difference is . Similarly, if divided by , the new common difference is .
   • Example: AP: $2, 4, 6, 8$ (). Multiplying by 5: $10, 20, 30, 40$. This is an AP with ().

Problem Solution

This problem describes an Arithmetic Progression (AP) because the sales increase by a fixed amount each year, which implies a constant common difference.

Let:

• = sales in the 1st year (first term of the AP)
• = annual increase in sales (common difference)
• = sales in the year (the term of the AP)

The formula for the term of an AP is .

Step 1: Formulate equations from the given information.

We are given:

• Sales in the 3rd year () = $150,000
• Sales in the 7th year () = $210,000

Using the formula :
For the 3rd year:

For the 7th year:

Step 2: Solve the system of linear equations to find and .

Subtract equation (1) from equation (2) to eliminate :

So, the annual increase in sales is $15,000.

Now substitute the value of back into equation (1) to find :

So, the sales in the 1st year were $120,000.

Conclusion

The sales in the 1st year were $120,000 and the annual increase in sales was $15,000.

Definition of Geometric Progression (GP)

A Geometric Progression (GP) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The common ratio is usually denoted by .

For example, the sequence is a GP with the first term and common ratio .

Derivation of the Formula for the Term of a GP

Let's consider a Geometric Progression with:

• The first term = (or simply )
• The common ratio =

We can write out the terms of the GP as follows:

• First term ():

• Second term (): To get the second term, we multiply the first term by the common ratio.
  

• Third term (): To get the third term, we multiply the second term by the common ratio.
  

• Fourth term (): To get the fourth term, we multiply the third term by the common ratio.
  

Observing the pattern, we can see a relationship between the term number and the exponent of the common ratio:

• For the 1st term, is raised to the power of 0 (). ()
• For the 2nd term, is raised to the power of 1 (). ()
• For the 3rd term, is raised to the power of 2 (). ()
• For the 4th term, is raised to the power of 3 (). ()

Generalizing this pattern, for the term (), the common ratio will be raised to the power of .

Thus, the formula for the term of a Geometric Progression is:

Where:

• = the term
• = the first term
• = the term number
• = the common ratio

Derivation of the Formula for the Sum of the First Terms of a GP ()

Let's denote the sum of the first terms of a Geometric Progression as .
Let the GP be .

Step 1: Write the sum .

Step 2: Multiply by the common ratio .
Multiply every term in equation (1) by :

Step 3: Subtract equation (1) from equation (2).
Subtracting (1) from (2):

Notice that most of the terms cancel out:

Step 4: Solve for .
Factor out from the right side:

Divide by (assuming ):

Alternatively, if we subtract equation (2) from (1) (this is useful when to avoid negative denominators):

If , the sum is simply .

Derivation of the Formula for the Sum to Infinity of a GP ()

For a geometric progression to have a finite sum to infinity, the absolute value of the common ratio must be less than 1 (i.e., or ).

Consider the formula for the sum of the first terms:

As , if , then approaches 0.

Let's evaluate the limit of as approaches infinity:

Since , .

Substituting this into the equation:

Conclusion

• Sum of first terms of a GP: (for ) or (for , ). If , .
• Sum to infinity of a GP: (for ).

Concept of Geometric Mean (GM)

A Geometric Mean (GM) is a type of average that is used for a set of positive numbers that are linked through multiplication (or exhibit exponential growth). When a certain number of terms are inserted between two given numbers such that the entire sequence forms a Geometric Progression (GP), these inserted terms are called geometric means.

If and are two positive numbers, the single geometric mean is .

If and are two positive numbers, and are geometric means inserted between and , then the sequence forms a GP.

Inserting two Geometric Means Between 3 and 81

Let the two given numbers be and .
We need to insert two geometric means, let's call them , between 3 and 81.

This means the sequence will be a GP:

In this GP:

• The first term () = 3
• The last term () = 81
• The total number of terms () =

We use the formula for the term of a GP: .
Here, .

Substitute the values:

Divide by 3:

To find , take the cube root of 27:

Now that we have the common ratio (), we can find the geometric means:

The sequence is $3, 9, 27, 81$.

Conclusion

The two geometric means inserted between 3 and 81 are 9 and 27.

Important Properties of a Geometric Progression (GP)

Here are four important properties of a Geometric Progression:

1. Constant Common Ratio:

   • Property: The defining characteristic of a GP is that the ratio of any term to its preceding term is constant. This constant is called the common ratio ().
   • Explanation: For any GP , we have . This property is crucial for identifying a GP and finding its terms.
   • Example: In GP , the common ratio is , , etc.

2. Geometric Mean Property:

   • Property: If are three consecutive terms in a GP, then is the geometric mean of and , which means (for positive terms). This implies .
   • Explanation: If are in GP, then and . Therefore, . For positive terms, taking the square root gives .
   • Example: In GP $2, 6, 18$, and . So, .

3. Multiplication/Division by a Non-Zero Constant:

   • Property: If each term of a GP is multiplied or divided by a non-zero constant, the resulting sequence is also a GP with the same common ratio.
   • Explanation: Let the GP be with common ratio . If each term is multiplied by a non-zero constant , the new sequence is . The new ratio between consecutive terms is . So, the common ratio remains .
   • Example: GP: $2, 4, 8, 16$ (). Multiplying by 3: $6, 12, 24, 48$. This is still a GP with .

4. Reciprocal Property:

   • Property: The reciprocals of the terms of a GP also form a GP.
   • Explanation: If is a GP with common ratio , then the sequence of reciprocals is . The ratio of consecutive terms in this new sequence is . Thus, the reciprocals form a GP with common ratio .
   • Example: GP: $2, 4, 8, 16$ (). Reciprocals: . This is a GP with common ratio .

Problem Solution

This problem involves calculating depreciation, which can be modeled as a Geometric Progression (GP) where the value decreases by a fixed percentage each period. The value at the end of each year is a certain percentage of the value at the beginning of that year.

Step 1: Identify the initial value (first term) and the common ratio.

• Initial cost (first term, ) = $100,000
• Annual depreciation rate = 15% = 0.15

When an item depreciates by 15%, its value at the end of the year is of its value at the beginning of the year. So, the value is of the previous year's value.

• Common ratio () =

We need to find the value after 3 years. Let be the initial value.

• Value after 1 year () =
• Value after 2 years () =
• Value after 3 years () =

This corresponds to the term formula if 'n' refers to the number of years passed, or if is initial and is after 3 years.

Let be the initial value (value at year 0).
Value after years is .

Step 2: Calculate the value after 3 years.
Using the formula where is the number of years:

Value after 3 years () = Initial Cost

Calculate :

Now, calculate :

Conclusion

The value of the machine after 3 years will be $61,412.50.

Comparison of Arithmetic Progression (AP) and Geometric Progression (GP)

Key Differences Summarized:

• Type of progression: AP involves addition/subtraction, while GP involves multiplication/division.
• Rate of change: AP has a constant absolute change, whereas GP has a constant relative (percentage) change.
• Sum to infinity: Only GP can have a finite sum to infinity, and only when its common ratio is between -1 and 1 (exclusive).

Part 1: If the sequence is an Arithmetic Progression (AP)

Let the first term be and the common difference be .
The formula for the term of an AP is .

Given:

Subtract equation (1) from equation (2):

Now substitute into equation (1):

So, the first term is 6 and the common difference is 3.

To find the 15th term ():

Answer for AP: The 15th term is 48.

Part 2: If the sequence is a Geometric Progression (GP)

Let the first term be and the common ratio be .
The formula for the term of a GP is .

Given:

Divide equation (4) by equation (3):

To find the common ratio , take the fourth root of :

(Note: Since all terms are positive, we take the positive root for .)

Answer for GP: The common ratio is or approximately 1.1378.
(We don't need to find the 15th term for GP as only common ratio was asked.)

Problem Solution

This problem involves calculating the value of an investment with varying percentage increases over consecutive years. This is a sequential calculation, not a simple arithmetic or geometric progression with a constant difference or ratio, but rather a product of annual growth factors.

Step 1: Identify the initial investment.
Initial investment () = $5,000

Step 2: Determine the annual growth factors.

• Year 1 increase: 10% = 0.10. Growth factor () =
• Year 2 increase: 8% = 0.08. Growth factor () =
• Year 3 increase: 5% = 0.05. Growth factor () =

Step 3: Calculate the value after each year.

• Value after 1st year ():
  
  

• Value after 2nd year ():
  
  

• Value after 3rd year ():
  
  

Alternatively, we can calculate the final value directly by multiplying the initial investment by all the annual growth factors:

Conclusion

The value of the investment after 3 years will be $6,237.

Concept of Joint Variation

Joint variation describes a relationship where one quantity varies directly as the product of two or more other quantities. It combines direct variation principles for multiple variables. If a quantity varies jointly as and , it means , or , where is the constant of proportionality.

It can also be combined with inverse variation. For instance, if varies jointly as and inversely as , then , or .

Application to the Signboard Painting Problem

Let:

• = Cost of painting the signboard
• = Radius of the signboard
• = Thickness of the paint applied

From the problem statement:

• The cost of painting () varies directly as the square of its radius (). This means .
• The cost of painting () varies inversely as the thickness of the paint applied (). This means .

Combining these two variations, we get a joint variation:

Equation Relating the Variables:
To convert this proportionality into an equation, we introduce a constant of proportionality, :

Role of the Constant of Proportionality ()

In this context, is the constant of proportionality. Its role is crucial:

1. Unit Conversion and Scaling Factor: The constant adjusts the units and magnitude to make the equation balanced. For example, if is in dollars, in meters, and in millimeters, will have units that convert to dollars.

2. Specific Material/Labor Costs: represents the specific cost factors that are constant for a given painting job. For instance, it could encompass:

   • The cost per unit area of paint.
   • The labor cost for applying the paint.
   • Efficiency factors related to the painting process.

3. Unique to the Situation: The value of would need to be determined experimentally or given based on specific conditions (e.g., a particular type of paint, a specific labor rate, or a particular painting company). If any of these underlying conditions change, the value of would change.

In essence, encapsulates all other factors not explicitly included in and that affect the painting cost, ensuring the mathematical relationship holds true.

Problem Solution

This problem can be modeled as a Geometric Progression (GP) where the population grows by a constant percentage each year. This is a compound growth scenario.

Step 1: Identify the initial population (first term) and the common ratio.

• Initial Population ( or ) = 1,200,000
• Annual growth rate = 2% = 0.02

The growth factor per year is .
This growth factor serves as the common ratio () for the GP.

We need to find the population after 5 years. If is the population at year 0, then the population after years () is given by:

Step 2: Calculate the population after 5 years.

Substitute the values into the formula:

Calculate :

Now, calculate :

Since population must be a whole number, we round to the nearest whole person.

Conclusion

The estimated population after 5 years will be approximately 1,324,897 people.

Concept of Direct Proportion

Two quantities are said to be in direct proportion (or vary directly) if an increase in one quantity results in a proportional increase in the other quantity, and a decrease in one quantity results in a proportional decrease in the other. In simpler terms, their ratio remains constant.

If is directly proportional to , we write , which can be expressed as an equation , where is the constant of proportionality.

Key Properties of Direct Proportion

Let be quantities such that and are related to respectively by direct proportion.

1. Constant Ratio: The ratio of the corresponding values of the two quantities is always constant (). That is, .
2. Linear Relationship: When plotted on a graph, the relationship between two directly proportional quantities is a straight line passing through the origin.
3. Proportionality of Ratios: If and are two sets of corresponding values, then , which implies .
4. Scaling: If one quantity is multiplied by a factor, the other quantity is also multiplied by the same factor.

Example in Business Decision-Making: Production Cost

Scenario: A manufacturer produces widgets. The total production cost for materials is directly proportional to the number of widgets produced.

Application:

• Assume the cost of materials for 100 widgets is $500.
• We can establish the constant of proportionality (cost per widget):
  per widget.

Business Decision:
Now, if the company plans to increase production to 150 widgets, they can easily estimate the material cost:

• Estimated Material Cost = .

This simple direct proportional relationship allows the business to:

• Budgeting: Accurately budget for material costs based on production targets.
• Pricing: Determine the material component of the cost for new products or different production volumes, which influences pricing strategies.
• Forecasting: Forecast future material expenses based on projected demand or sales figures.

This helps in efficient resource allocation and cost management.

Concept of Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic progression (AP). If an AP is given by , then the corresponding arithmetic series is .

Difference from an Arithmetic Progression

In short, an AP is a list, while an arithmetic series is the sum of that list.

Business Example: Cumulative Production

Scenario: A new factory starts production. In the first month, it produces 500 units. Due to efficiency improvements, it manages to increase its production by 50 units each subsequent month.

Problem: Calculate the total number of units produced in the first year (12 months).

Application of Arithmetic Series:

This scenario forms an AP for monthly production:

• First term () = 500 units (production in 1st month)
• Common difference () = 50 units (monthly increase)
• Number of terms () = 12 months

To find the total units produced, we need to sum the production of each month, which is an arithmetic series.

Using the formula :

Conclusion: The factory will produce a total of 9,300 units in the first year.

This calculation is vital for:

• Inventory Planning: Knowing the total output helps manage inventory levels.
• Sales Forecasting: Total production directly impacts potential sales volume.
• Resource Allocation: Understanding cumulative output helps in planning raw material procurement and workforce requirements.

Problem Solution

This problem involves an Arithmetic Progression (AP) where the monthly installments form a series. We need to find the number of months () required for the sum of the installments to reach the total loan amount.

Let:

• Loan amount = $10,000
• First installment () = $100
• Monthly increase (common difference, ) = $10
• Number of months () = ?
• Sum of installments () = $10,000

We use the formula for the sum of the first terms of an AP:

Substitute the known values:

Multiply both sides by 2:

Rearrange into a standard quadratic equation :

Divide the entire equation by 10 to simplify:

Now, solve this quadratic equation for using the quadratic formula:

Here, .

Calculate :

We consider the positive value for since the number of months cannot be negative:

Since the number of months must be an integer, it means the loan will be fully repaid in the 37th month. The 36th month will cover most of the loan, and a smaller payment will finalize it in the 37th month.

Let's check and :

So, after 36 months, $9,900 is repaid.
Remaining balance = $10,000 - $9,900 = $100.
The 37th installment would be .
So, the 37th payment will be $100 (or the full $460, depending on loan terms, but at least $100 to clear).
Therefore, it will take 37 months to fully repay the loan.

Conclusion

It will take 37 months to repay the entire loan.

Implications for the Sum to Infinity of a Geometric Progression ()

The formula for the sum to infinity of a Geometric Progression is , where is the first term and is the common ratio.

1. Case 1: (i.e., , excluding for meaningful GP)

   • Mathematical Implication: When the absolute value of the common ratio is less than 1, the term approaches 0 as approaches infinity. This means that as more terms are added, the value of each subsequent term becomes negligibly small, and the series converges to a finite sum. The sum is a finite, calculable value.
   • Financial Context: This scenario represents a situation where a value decreases over time, or contributions diminish over time. Examples include:
     • Present Value of Perpetuity: The present value of an infinite stream of payments (perpetuity) that are decreasing over time (e.g., if each subsequent payment is discounted more heavily). If payments were constant, it's also a convergent series with for the discounted values.
     • Depreciation to Zero: An asset might depreciate towards a salvage value, with the remaining depreciation decreasing each year. While the asset value itself doesn't sum to infinity, the total depreciation amount over an infinite period could be modeled if the depreciation amount decreased geometrically.

2. Case 2: (i.e., or )

   • Mathematical Implication: When the absolute value of the common ratio is greater than or equal to 1, the terms do not approach 0 as approaches infinity. Instead, either grows without bound (if or ) or oscillates (if ). Consequently, the sum of the terms will either grow infinitely large or oscillate without settling on a finite value. In these cases, the series diverges, meaning there is no finite sum to infinity.
   • Financial Context: This scenario signifies unbounded growth or value. Examples include:
     • Compound Interest: An investment growing at a fixed percentage rate (e.g., ). The future value of this investment grows infinitely large over an infinite time horizon. There's no 'sum to infinity' in the traditional sense, as the value keeps increasing.
     • Inflation: If prices increase by a constant percentage, the cost of goods over an infinite future period would be infinite.
     • Population Growth: A population growing at a constant percentage rate will tend to infinity. The sum of yearly increases would also be infinite.

Significance

In a financial context, the common ratio's magnitude determines whether a value stream (like cash flows) has a finite total value over an infinite period (convergence) or if its value grows indefinitely (divergence). This distinction is fundamental in valuation models, such as those used for perpetuities, annuities, or growth stocks, where determining a finite present or future value is critical.

Equation for Total Cost of Rental

Let:

• = Total cost of rental (in dollars)
• = Number of days rented (assuming a single day for simplicity, as the daily fee is fixed)
• = Kilometers driven

Given:

• Base fee per day = $30
• Cost per kilometer = $0.25

Assuming the rental is for one day, the total cost equation is:

If the number of days is also a variable (e.g., for days):

For the purpose of discussing proportion/progression, let's assume for simplicity and focus on the relationship between cost and kilometers driven.
So, .

Is this a Direct Proportion or an Arithmetic Progression?

This relationship is neither a direct proportion nor a simple arithmetic progression in its purest form.

Why it's not a Direct Proportion:

• Definition of Direct Proportion: For a direct proportion, would have to be of the form , where is a constant. This means that the ratio must be constant for all .
• In this case: The equation is . Due to the fixed base fee of $30, the ratio is not constant. For example, if , \frac{{C}}{{K}} = 3.25K=100$, $C=$55, and . The ratio changes with .
• Graphically: A direct proportion always passes through the origin . Here, if , $C=$30, so it does not pass through the origin.

Why it's not a simple Arithmetic Progression:

• An AP describes a sequence of numbers where the difference between consecutive terms is constant. Here, we have a continuous variable (kilometers driven), not discrete terms in a sequence.
• If we were to consider the cost for driving 1 km, 2 km, 3 km, etc., the costs would be:
  • 
    The increase in cost per additional kilometer is indeed constant ($0.25). So, the sequence of total costs for successive integer kilometers forms an AP. In this sense, the total cost behaves like an AP if we consider discrete, equally spaced increments of K.
• However, the equation itself is a linear equation (), which is the general form of an AP's underlying function when plotted discretely for integer inputs, but not an AP definition itself which applies to discrete sequences.

Conclusion:

The relationship is a linear relationship (a straight line with a positive y-intercept and positive slope). While the costs for integer kilometers form an arithmetic progression, the overall functional relationship is best described as linear, not strictly a direct proportion (due to the intercept) or an AP (as it's a continuous function, not just a sequence).

Problem Solution

This problem involves a sequence of increases that follow a geometric progression, which then needs to be added to the initial revenue to find the total cumulative revenue.

Step 1: Identify Initial Revenue and First Year's Increase.

• Initial Annual Revenue () = $100,000
• Increase in 1st year () = $5,000

Step 2: Determine the pattern of increases for subsequent years.
The increase in subsequent years is 20% of the previous year's increase. This forms a Geometric Progression for the increases.

• Increase in 1st year () = $5,000 (This is the first term 'a' of the GP for increases)
• Common ratio () for increases = 20% = 0.20

Step 3: Calculate increases for years 2, 3, and 4.

• Increase in 2nd year () =
• Increase in 3rd year () =
• Increase in 4th year () =

Step 4: Calculate the annual revenue for each of the first 4 years.

• Revenue in Year 1 () =
• Revenue in Year 2 () =
• Revenue in Year 3 () =
• Revenue in Year 4 () =

Step 5: Calculate the total cumulative revenue over the first 4 years.
Total Cumulative Revenue =
Total Cumulative Revenue = $105,000 + $106,000 + $106,200 + $106,240
Total Cumulative Revenue = $423,440

Alternative Approach (Sum of initial revenue + sum of increases):
Total increases over 4 years = .
Sum of annual revenues () could also be thought of as the sum of times initial revenue, plus sum of increases.
No, this is incorrect. The total cumulative revenue is simply the sum of the annual revenues.

Conclusion

The total cumulative revenue over the first 4 years will be $423,440.

Concept of Harmonic Progression (HP)

A Harmonic Progression (HP) is a sequence of numbers such that the reciprocals of its terms are in Arithmetic Progression (AP).

If form an AP, then form an HP. Conversely, if a sequence is an HP, then the sequence formed by taking the reciprocal of each term is an AP.

Example: The sequence is an HP because its reciprocals form an AP with a first term and a common difference (e.g., ).

Relation to Arithmetic Progression

The fundamental connection is that an HP is defined through an AP. There are no direct formulas for the term or the sum of terms of an HP in the same simple algebraic form as for AP or GP. To find terms or sums of an HP, one must first convert it to an AP, perform calculations, and then convert back by taking reciprocals. For example, the Harmonic Mean (HM) of two numbers and is given by , which is the reciprocal of the arithmetic mean of their reciprocals: .

Why it is Less Commonly Used in Business Mathematics

HP is less commonly used in business mathematics compared to AP and GP for several reasons:

1. Direct Interpretability: AP and GP have very intuitive interpretations in business contexts:

   • AP models situations with constant absolute changes (e.g., simple interest, fixed annual increments, straight-line depreciation).
   • GP models situations with constant relative/percentage changes (e.g., compound interest, percentage-based depreciation, exponential growth/decay).
     HP, based on reciprocals of arithmetic changes, rarely provides such direct and intuitive financial or economic meaning.

2. Complexity of Operations: Calculations involving HP are typically more cumbersome because they require converting to AP, performing operations, and then converting back. This extra step adds complexity without often yielding a clearer business insight.

3. Specific Use Cases: While AP and GP apply broadly to financial calculations, population dynamics, production schedules, and economic models, HP has very niche applications, often in specific areas like:

   • Averaging rates (e.g., average speed for journeys with varying speeds over equal distances).
   • Finance for calculating average growth rates under specific conditions (e.g., dollar-cost averaging in investments, though other means are more common).
   • Physics and engineering contexts (e.g., electrical resistance in parallel circuits).

4. Lack of Natural Business Models: Most business phenomena (costs, revenues, profits, investments) tend to exhibit linear (AP-like) or exponential (GP-like) growth/decay patterns. Relationships whose reciprocals follow an AP are not as frequently encountered or easily modeled directly in business scenarios. For example, if profits increase arithmetically, their reciprocals (1/profit) would form an HP, but 1/profit usually doesn't have an inherent business meaning.

In essence, while mathematically valid, the concept of reciprocals of arithmetic sequences does not align as naturally with the typical additive or multiplicative growth patterns observed and analyzed in business and finance.

Problem Solution

This problem can be solved using the concept of Geometric Progression (GP) where the value multiplies by a constant factor over a fixed period.

Step 1: Identify the initial investment and the growth factor per period.

• Initial Investment () = $1,000
• Growth factor per 7-year period = 2 (since it doubles)

Step 2: Determine the number of growth periods.

• Total time period = 21 years
• Length of one growth period = 7 years
• Number of growth periods () =

Step 3: Calculate the final value after periods.
If the initial investment is , and it grows by a factor over periods, the final value is given by:

Substitute the identified values:

Conclusion

The value of the investment after 21 years will be $8,000.

Problem Solution

This problem involves calculating the sum of an Arithmetic Progression (AP) because the bonus increases by a fixed amount each year.

Step 1: Identify the components of the AP.

• First year's bonus () = $500
• Annual increase (common difference, ) = $100
• Number of years (number of terms, ) = 10

Step 2: Use the formula for the sum of the first terms of an AP.
The formula is .

Substitute the known values into the formula:

Conclusion

An employee completing 10 years of service would have received a total bonus of $9,500.

Concept of 'Variable' and 'Constant' in Variation

In the context of mathematical variation (direct, inverse, or joint variation):

• Variable: A variable is a quantity that can change or assume different values. In variation problems, we typically deal with dependent and independent variables.

  • Independent Variable: A quantity whose change causes a change in another quantity (e.g., the number of units produced).
  • Dependent Variable: A quantity whose value depends on the change in one or more independent variables (e.g., total cost).
    Variables are usually represented by letters like , etc.

• Constant (of Variation/Proportionality): A constant is a fixed value that does not change. In variation, the 'constant of proportionality' (often denoted by ) is the fixed ratio or product between the varying quantities. It establishes the specific quantitative relationship between them.

  • It's a numerical factor that scales the relationship between variables.
  • It remains the same throughout a particular problem or scenario, even if the variables change.

Example where a Constant of Variation Represents a Real-world Business Metric

Consider a scenario of direct variation in manufacturing:

Scenario: The total labor cost () to produce a certain type of product varies directly as the number of units produced ().

Mathematical Relationship:

Here:

• is the dependent variable (total labor cost).
• is the independent variable (number of units produced).
• is the constant of proportionality/variation.

Real-world Business Metric represented by :
In this context, represents the average labor cost per unit produced.

For instance:

• If it costs $500 in labor to produce 100 units, then .
• This $5 per unit is the constant of proportionality. It implies that for every unit produced, an additional $5 in labor cost is incurred, assuming the labor efficiency and wage rates remain constant.

Significance: This constant is a critical business metric for:

• Cost Management: Companies use it to understand and control variable costs.
• Budgeting: It helps in forecasting labor expenses based on production targets.
• Pricing Decisions: It directly feeds into the total cost calculation, influencing how products are priced to ensure profitability.

Thus, the abstract constant of variation directly translates into a tangible and actionable business metric.