**Ratios:** A ratio is a way to compare two or more quantities. It is often expressed in
the form of a:b, where a and b are the quantities being compared. Ratios can be simplified by
dividing both terms by their
greatest common divisor.

**Proportions**A proportion is an equation that states that two ratios are equal. For
example, if
a/b = c/d ,then a,b,c, and d are said to be in proportion. To solve proportions, you can use
cross-multiplication.

**Formulas and Key Points:**

**Simplifying Ratios:**Divide both terms of the ratio by their greatest common divisor.- Example: To simplify the ratio 12:16, divide both terms by 4 to get 3:4.
**Proportion Formula:**- If a/b = c/d, then a*d = b*c
- This can be used to find the missing term in a proportion.
**Finding Ratios:**- To find the ratio between two quantities, divide each quantity by a common term and simplify.

A. 60 and 80

B. 70 and 90

C. 84 and 56

D. 90 and 50

Use the ratio to express the numbers in terms of a single variable and solve the
equation based on their sum.

Let the numbers be 3x and 4x. Then 3x + 4x = 140. Solving, 7x = 140 => x = 20. Thus,
the numbers are 3 * 20 = 60 and 4 * 20 = 80.

A. 6

B. 9

C. 8

D. 15

Use cross-multiplication to solve for a.

a/12 = 3/4.Cross-multiplying, a*4 = 12*3, which simplifies to a = 9.

A. 6

B. 7

C. 9

D. 10

Set up the proportion and solve for x.

Cross-multiplying, 5*12 = 8*x. Solving for x, 60 = 8x => x = 7.5.

A. 10 years

B. 8 years

C. 9 years

D. 12 years

Use the ratio and the elder brother's age to find the younger brother's age.

If the ratio of their ages is 2:3 and the elder brother is 15 years old, then 2/3 of
x = 15..Solving for x, x = (15*3)/2 = 22.5, so the younger brother is 15*(2/3) = 10

A. -4%

B. 0%

C. -2%

D. 2%

Use the formula for percentage change to find the net effect.

The net change is given by 1.2 times 0.8 - 1 = -0.04, or -4%.

A. 8 years

B. 9 years

C. 10 years

D. 12 years

Use the given ratio and total sum to form an equation and solve for the daughter's
age.

Let the daughter's age be x. Then, mother's age is 4x. Thus, x + 4x = 36. Solving, x
= 7.2 years.

A. 3:5

B. 25:9

C. 9:25

D. 15:25

Use the relationship between circumference and area to find the ratio.

The ratio of areas is the square of the ratio of circumferences. Hence, (3:5)^2 =
9:25.

A. 20%

B. 25%

C. 30%

D. 35%

Calculate the profit and then find the percentage of profit with respect to the cost
price.

Profit = 150 - 120 = 30. Profit percentage = (30/120) * 100 = 25%.

A. 50 and 45

B. 50 and 35

C. 45 and 40

D. 55 and 30

Use the sum and difference to form a pair of linear equations and solve for the
numbers.

Let the numbers be x and y. Then, x + y = 85 and x - y = 15. Solving, x = 50 and y =
35.

A. 6000

B. 8000

C. 10000

D. 12000

Work backwards from the remaining amount to find the original salary.

Let the salary be x. After paying rent, 60% of x remains. Then, 25% of that
remaining amount is spent on groceries, leaving 1800. Thus, 0.6x - 0.15x = 1800.
Solving, x = 6000.

**Introduction:** Simple Interest (SI) is a method of calculating the interest
charged
on a sum of money (the principal) over a certain period at a fixed rate. Unlike compound
interest,
simple interest does not take into account the interest that accumulates on the interest already
earned. Instead, it is calculated only on the principal amount throughout the period.

**Key Terms**

- Principal (P): The original sum of money lent or invested.
- Rate of Interest (R): The percentage at which the interest is charged or earned.
- Time (T): The duration for which the money is borrowed or invested, usually expressed in years.
- Simple Interest (SI): The amount earned or paid for the use of money over time.

**Formula:** SI = (P x R x T) / 100

** Additional Formulas: **

Calculating Total Amount (A): The total amount payable or receivable at the end of the time
period,
including both principal and interest.

**A = P+SI**

Substituting the value of SI, we get:

**A=P+ (PxRxT)/100**

Finding Principal (P): When the simple interest, rate, and time are known.

**P= (SIx100)/(RxT)**

Finding Rate (R): When the simple interest, principal, and time are known.

**R= (SIx100)/(PxT)**

Finding Time (T): When the simple interest, principal, and rate are known.

**T= (SIx100)/(PxR)**

**Examples of Simple Interest**

**Example 1:** If you invest $1,000 (P) at an annual interest rate of 5% (R) for 3 years (T),

the simple
interest (SI) will be:
SI= (1000x5x3)/100 = 150

**So, the simple interest is $150.**

**Example 2:** You borrowed $2,000 at a rate of 6% per annum for 4 years. The interest
payable
will be:

SI= (2000x6x4)/100 = 480

**The interest payable is $480.**

A. SI = (P*R)/T

B. SI = (P*R*T)/100

C. SI = (P*R*T)

D. SI = (P+R+T)/100

Simple Interest is calculated based on the principal, rate, and time.

The correct formula for calculating Simple Interest is SI = (P*R*T)/100

A. $10

B. $20

C. $50

D. $100

Use the formula for Simple Interest: SI = (P*R*T)/100

SI = (500*5*2)/100 = 50. So. the simple intrest is $ 50.

A. $ 1,120

B. $ 1,040

C. $ 1,400

D. $ 1,200

Total amount is the sum of the principal and Simple Interest.

SI = (1000*4*3)/100 = 120.
Total Amount A = P + SI = 1000 + 120 = 1120

A. $ 800

B. $ 900

C. $ 1,000

D. $ 1,200

Rearrange the Simple Interest formula to find P.

p = (SI*100)/(R*T) = (300*100)/(6*5) = 1000

A. Doubles

B. Triples

C. Remains the same

D. Halves

Analyze how interest changes with the rate in the formula.

Since SI is directly proportional to the rate (R), doubling the rate will
double
the interest earned.Since SI is directly proportional to the rate (R),
doubling
the rate will double the interest earned.

A. 4%

B. 5%

C. 6%

D. 8%

Use the formula for Rate: R = (SI*100)/(P*T)

R = (160*100)/(2000*2) = 4%

A. 4%

B. 5%

C. 6%

D. 7%

Rearrange the Simple Interest formula to solve for R.

R = (50*100)/(500*2) = 5%

A. 5%

B. 10%

C. 15%

D. 20%

If the amount doubles, the interest equals the principal.

since the sum doubles, SI = P, and R = (SI*100)/(P*T) = 100/10 = 10%

A. 2 years

B. 4 years

C. 5 years

D. 6 years

Use the formula for time:
T = (SI * 100)/P*R

T = (160*100)/800*4 = 5 years

A. $ 375

B. $ 400

C. $ 450

D. $ 500

Calculate using the Simple Interest formula.

SI = (2500*5*3)/100 = 375.

**Profit:** Profit is the financial gain obtained when the selling price (SP) of an item
is
greater than its cost price (CP). The formula for calculating profit is:

**Profit = Selling Price (SP) - Cost Price (CP)**

**Loss:** Loss occurs when the cost price (CP) of an item is greater than its selling
price
(SP). The formula for calculating loss is:

**Loss = Cost Price (CP) - Selling Price (SP)**

**Formulas and Key Points:**

**Profit Percentage:**

Profit Percentage = (Profit / Cost Price) * 100**Loss Percentage:**

Loss Percentage = (Loss / Cost Price) * 100**Relation Between Selling Price and Cost Price:**

SP = CP * (1 + (Profit %)/100) [If profit]

SP = CP * (1 - (Loss %)/100) [If loss]

A. 15%

B. 20%

C. 25%

D. 30%

Use the formula for profit percentage: (Profit/Cost Price) * 100.

Profit = SP - CP = $180 - $150 = $30.

Profit Percentage = (30/150) * 100 = 20%.

Profit Percentage = (30/150) * 100 = 20%.

A. $330

B. $300

C. $330

D. $250

Use the formula: CP = SP / (1 - (Loss %)/100).

Loss % = 10%

CP = SP / (1 - (Loss %)/100) = 270 / (1 - 0.10) = $300.

CP = SP / (1 - (Loss %)/100) = 270 / (1 - 0.10) = $300.

A. $560

B. $565

C. $575

D. $600

Use the formula: SP = CP * (1 + (Profit %)/100).

Profit % = 15%

SP = CP * (1 + (Profit %)/100) = 500 * (1 + 0.15) = $575.

SP = CP * (1 + (Profit %)/100) = 500 * (1 + 0.15) = $575.

A. 15%

B. 20%

C. 12.5%

D. 10%

Use the formula for loss percentage: (Loss/Cost Price) * 100.

Loss = CP - SP = $800 - $680 = $120.

Loss Percentage = (120/800) * 100 = 15%.

Loss Percentage = (120/800) * 100 = 15%.

A. 1% gain

B. 1% loss

C. 0%

D. 2% loss

Remember that when gain and loss percentages are the same, there is always a net
loss.

Net loss percentage = (Loss% * Gain%) / 100 = (10 * 10) / 100 = 1% loss.

A. $520

B. $540

C. $500

D. $480

Use the formula: CP = SP / (1 + (Profit %)/100).

CP = SP / (1 + (Profit %)/100) = 540 / (1 + 0.08) = $500.

A. $180

B. $190

C. $170

D. $160

Use the formula: SP = Marked Price - (Discount% * Marked Price).

Discount = 10% of $200 = $20.

SP = Marked Price - Discount = $200 - $20 = $180.

SP = Marked Price - Discount = $200 - $20 = $180.

A. 14%

B. 12%

C. 10%

D. 16%

Use the formula for net gain: (Selling Price - Cost Price)/Cost Price * 100.

Marked Price = 120% of Cost Price.

Selling Price = 95% of Marked Price = 95/100 * 120% = 114% of Cost Price.

Gain % = 14%.

Selling Price = 95% of Marked Price = 95/100 * 120% = 114% of Cost Price.

Gain % = 14%.

A. $510

B. $520

C. $530

D. $540

Use the formula: SP = CP - (Loss % of CP).

Loss = 15% of $600 = $90.

SP = CP - Loss = $600 - $90 = $510.

SP = CP - Loss = $600 - $90 = $510.

A. 25% profit

B. 20% profit

C. 15% loss

D. 10% loss

Find the cost price and selling price per orange first.

CP per orange = $2/8 = $0.25.

SP per orange = $1/5 = $0.2.

Profit = (SP - CP) / CP * 100 = (0.2 - 0.25) / 0.25 * 100 = 25% profit.

SP per orange = $1/5 = $0.2.

Profit = (SP - CP) / CP * 100 = (0.2 - 0.25) / 0.25 * 100 = 25% profit.

**Average:** The average of a set of numbers is the value that represents the central or
typical point of the data set. It is commonly referred to as the arithmetic mean.

**Formulas for Averages:**

**Arithmetic Mean (Average):**

Average = (Sum of all values) / (Number of values)**Weighted Average:**

Weighted Average = (Σ(Value x Weight)) / (Σ Weights)

**Key Points:**

**Arithmetic Mean:**This is the most commonly used measure of average. It is calculated by dividing the sum of all values by the number of values.**Weighted Average:**This type of average takes into account different weights assigned to values. It is useful when certain values contribute more to the overall mean than others.

A. 6

B. 8

C. 8

D. 10

Add all the numbers and divide by the number of values.

Average = (4 + 8 + 12) / 3 = 24 / 3 = 8.

A. 15

B. 17.5

C. 18

D. 20

Sum all the numbers and divide by the total count.

Average = (10 + 15 + 20 + 25) / 4 = 70 / 4 = 17.5.

A. 10

B. 12.5

C. 12.5

D. 15

Add up all the values and divide by the number of values.

Average = (5 + 10 + 15 + 20) / 4 = 50 / 4 = 12.5.

A. 15

B. 17.5

C. 17.5

D. 20

Sum the values and divide by the number of values.

Average = (7 + 14 + 21 + 28) / 4 = 70 / 4 = 17.5.

A. 7.5

B. 7.5

C. 8

D. 9

Add all numbers and divide by their count.

Average = (3 + 6 + 9 + 12) / 4 = 30 / 4 = 7.5.

A. 18

B. 20

C. 20

D. 22

Calculate the sum and divide by the number of values.

Average = (8 + 16 + 24 + 32) / 4 = 80 / 4 = 20.

A. 6.25

B. 7

C. 8

D. 7.5

Find the sum and divide by the number of values.

Average = (2 + 5 + 7 + 11) / 4 = 25 / 4 = 6.25.

A. 22.5

B. 24

C. 22.5

D. 25

Sum all the values and divide by their count.

Average = (9 + 18 + 27 + 36) / 4 = 90 / 4 = 22.5.

A. 25

B. 27.5

C. 27.5

D. 30

Calculate the sum and divide by the number of values.

Average = (11 + 22 + 33 + 44) / 4 = 110 / 4 = 27.5.

A. 10

B. 10

C. 12

D. 14

Add the numbers and divide by the total count.

Average = (4 + 8 + 12 + 16) / 4 = 40 / 4 = 10.

**Introduction:** Percentages represent a fraction of 100. They are used to express how
a number compares to a whole, which is often referred to as 100%. The percentage value can be used
in various financial calculations, comparisons, and statistics.

**Basic Formula:**

**Percentage = (Part / Whole) * 100**

**Key Formulas:**

**Finding Percentage:**

To find what percentage a number is of another number, use the formula:

Percentage = (Part / Whole) * 100**Percentage Increase:**

To calculate the percentage increase from an old value to a new value:

Percentage Increase = [(New Value - Old Value) / Old Value] * 100**Percentage Decrease:**

To calculate the percentage decrease from an old value to a new value:

Percentage Decrease = [(Old Value - New Value) / Old Value] * 100**Finding the Whole from a Percentage:**

To determine the original amount when you know the percentage and the part:

Whole = (Part / Percentage) * 100

A. 25

B. 30

C. 20

D. 35

Multiply 150 by 0.20.

20% of 150 = 150 * 0.20 = 30.

A. $55

B. $60

C. $70

D. $65

Subtract 25% of $80 from $80.

Discount = 80 * 0.25 = 20. Sale Price = 80 - 20 = 60.

A. 25

B. 30

C. 30

D. 40

Multiply 200 by 0.15.

15% of 200 = 200 * 0.15 = 30.

A. 10%

B. 15%

C. 20%

D. 25%

Subtract the sale price from the original price, then divide by the original price.

Discount = (120 - 96) / 120 = 24 / 120 = 0.20 = 20%.

A. 80%

B. 80%

C. 85%

D. 70%

Divide the score by the total and multiply by 100.

Percentage = (72 / 90) * 100 = 80%.

A. $80

B. $85

C. $83.33

D. $90

Divide the sale price by (1 - discount rate).

Original Price = 75 / (1 - 0.10) = 75 / 0.90 ≈ 83.33.

A. 15%

B. 20%

C. 20%

D. 25%

Divide the increase by the original price and multiply by 100.

Percentage Increase = ((60 - 50) / 50) * 100 = 20%.

A. 150

B. 180

C. 200

D. 225

Divide 45 by 0.25.

Number = 45 / 0.25 = 180.

A. 15%

B. 16.25%

C. 20%

D. 25%

Divide 65 by 400 and multiply by 100.

Percentage = (65 / 400) * 100 = 16.25%.

A. $95

B. $100

C. $105

D. $110

Divide the sale price by (1 - reduction rate).

Original Price = 88 / (1 - 0.12) = 88 / 0.88 = 100.

**Introduction:** Speed, distance, and time are fundamental concepts in motion and
transportation. They are related by simple mathematical formulas that help calculate any one of
these values when the other two are known.

**Basic Formulas:**

**Speed = Distance / Time****Distance = Speed * Time****Time = Distance / Speed**

**Key Formulas and Calculations:**

**Average Speed:**

When you travel different distances at different speeds, the average speed is calculated as:

Average Speed = Total Distance / Total Time**Speed Conversion:**

To convert speed between units (e.g., km/h to m/s):

Speed in m/s = Speed in km/h * (1000 / 3600)

Speed in km/h = Speed in m/s * (3600 / 1000)**Time Conversion:**

To convert time between units (e.g., hours to minutes):

Time in minutes = Time in hours * 60

Time in hours = Time in minutes / 60

A. 50 km/h

B. 75 km/h

C. 100 km/h

D. 120 km/h

Use the formula: Speed = Distance / Time.

Speed = 150 km / 3 h = 50 km/h.

A. 150 km

B. 150 km

C. 120 km

D. 180 km

Calculate speed first, then find distance for 5 hours.

Speed = 60 km / 2 h = 30 km/h. Distance in 5 hours = 30 km/h * 5 h = 150 km.

A. 3 hours

B. 2.5 hours

C. 4 hours

D. 5 hours

Use the formula: Time = Distance / Speed.

Time = 270 km / 90 km/h = 3 hours.

A. 70 km/h

B. 72 km/h

C. 75 km/h

D. 80 km/h

Speed = Distance / Time.

Speed = 180 km / 2.5 h = 72 km/h.

A. 10 mph

B. 12 mph

C. 12 mph

D. 15 mph

Convert the time into hours first.

Speed = 10 miles / (50/60 hours) = 10 / (5/6) = 12 mph.

A. 6 hours

B. 8 hours

C. 10 hours

D. 10 hours

Use the formula: Time = Distance / Speed.

Speed = 300 km / 5 h = 60 km/h. Time = 600 km / 60 km/h = 10 hours.

A. 3 hours

B. 4 hours

C. 5 hours

D. 6 hours

Time = Distance / Speed.

Time = 160 km / 40 km/h = 4 hours.

A. 600 km/h

B. 600 km/h

C. 750 km/h

D. 800 km/h

Speed = Distance / Time.

Speed = 900 km / 1.5 h = 600 km/h.

A. 20 km/h

B. 30 km/h

C. 40 km/h

D. 50 km/h

Speed = (Distance / Time) - Speed of the current.

Speed downstream = 120 km / 3 h = 40 km/h. Speed of the boat in still water = 40
km/h - 10 km/h = 30 km/h.

A. 12 mph

B. 15 mph

C. 18 mph

D. 20 mph

Average Speed = Total Distance / Total Time.

Average Speed = 60 miles / 4 hours = 15 mph.

**Introduction:** Problems involving ages often require setting up equations based on
the relationship between the ages of different individuals. These problems typically involve simple
algebraic operations.

**Basic Formulas:**

**Current Age = Age in Past + Number of Years Passed****Age in Future = Current Age + Number of Years in Future****Difference in Ages = Absolute Difference Between Two Ages**

**Key Formulas and Calculations:**

**Age Difference:**

The difference in ages remains constant over time. If person A is 10 years older than person B, this difference will always be 10 years.**Combined Age:**

The sum of the ages of two or more people at any point in time can be calculated by adding their individual ages.**Finding Past Ages:**

To find a person's age at a past time, subtract the number of years that have passed from their current age.

A. 14 years

B. 16 years

C. 18 years

D. 20 years

Set up an equation based on their ages in 5 years.

Let Bob's current age be x. Alice's age will be x + 4. In 5 years, Bob's age will be
x + 5 and Alice's age will be (x + 4) + 5 = x + 9. Set up the equation x + 9 = 2(x +
5), solving this gives x = 12. Thus, Alice is 16 years old.

A. 23 years

B. 28 years

C. 30 years

D. 35 years

Use the age difference and past age relationships to set up the equation.

Let Tim's current age be x. Then Sarah's age is x + 8. Five years ago, Tim was x - 5
and Sarah was x + 3. Set up the equation x + 3 = 3(x - 5), solving this gives x =
23. Thus, Sarah is 28 years old.

A. 40 years

B. 45 years

C. 50 years

D. 55 years

Use the future sum of ages to find the father's current age.

In 10 years, John's age will be 25, and his father's age will be x + 10. The sum of
their ages will be 70. Set up the equation 25 + (x + 10) = 70, solving this gives x
= 45. Thus, John's father is 45 years old.

A. 20 years

B. 30 years

C. 35 years

D. 40 years

Set up equations based on their ages and solve for Maria's current age.

Let Lily's current age be x. Then Maria's age is 2x. Five years ago, Lily's age was
x - 5 and Maria's age was 2x - 5. Set up the equation 2x - 5 = 3(x - 5), solving
this gives x = 15. Thus, Maria is 30 years old.

A. 40 years

B. 44 years

C. 48 years

D. 50 years

Set up equations for their ages in 10 years and solve.

Let the son's age be x. Then the mother's age is 4x. In 10 years, the son's age will
be x + 10 and the mother's age will be 4x + 10. Set up the equation 4x + 10 = 3(x +
10), solving this gives x = 11. Thus, the mother is 44 years old.

**Introduction:** Algebra involves using symbols, typically letters, to represent
numbers and express mathematical relationships. It includes solving equations, simplifying
expressions, and working with functions.

**Basic Concepts:**

**Variables:**Symbols (often letters) used to represent unknown values.**Expressions:**Combinations of variables, numbers, and operations (e.g., 2x + 3).**Equations:**Mathematical statements that assert the equality of two expressions (e.g., 2x + 3 = 7).**Inequalities:**Statements that describe the relative size or order of two values (e.g., x > 5).

**Key Formulas and Techniques:**

**Solve Linear Equations:**

To find the value of the variable that makes the equation true (e.g., x + 5 = 12).**Factor Quadratic Expressions:**

Expressing a quadratic equation in its factored form (e.g., x^2 + 5x + 6 = (x + 2)(x + 3)).**Use the Quadratic Formula:**

Solving quadratic equations of the form ax^2 + bx + c = 0 using x = (-b ± √(b² - 4ac)) / 2a.

A. 4

B. 6

C. 8

D. 10

Isolate x by first adding 7 to both sides.

Add 7 to both sides: 3x - 7 + 7 = 11 + 7 → 3x = 18. Divide by 3: x = 18 / 3 → x = 6.

A. (x - 2)(x - 3)

B. (x - 2)(x - 3)

C. (x + 2)(x + 3)

D. (x + 1)(x + 6)

Find two numbers that multiply to 6 and add to -5.

To factor x^2 - 5x + 6, find two numbers that add up to -5 and multiply to 6. The
factors are -2 and -3, so (x - 2)(x - 3).

A. 5 and -1

B. 1 and -5

C. 2 and -3

D. 3 and -2

Use the formula x = (-b ± √(b² - 4ac)) / 2a.

For x^2 - 4x - 5 = 0, a = 1, b = -4, c = -5. Using the quadratic formula: x = (4 ±
√((-4)² - 4*1*(-5))) / 2*1 = (4 ± √(16 + 20)) / 2 = (4 ± √36) / 2 = (4 ± 6) / 2.
Thus, x = 5 or x = -1.

A. -2x + 10

B. 6x - 2

C. -2x + 14

D. 2x - 14

Distribute and combine like terms.

Distribute: 2(x + 3) = 2x + 6 and -4(x - 2) = -4x + 8. Combine: 2x + 6 - 4x + 8 =
-2x + 14.

A. 4

B. 6

C. 2

D. 8

Expand and simplify the equation to solve for x.

Expand: 4x - 4 = 2x + 6 + 6. Simplify: 4x - 4 = 2x + 12. Subtract 2x from both
sides: 2x - 4 = 12. Add 4 to both sides: 2x = 16. Divide by 2: x = 8.

**Introduction:** Pipes and cisterns problems involve calculating the time required to
fill or empty a tank given different rates of flow. These problems often require the use of work
rates and understanding of the combined or individual rates of the pipes.

**Basic Concepts:**

**Work Rate:**The rate at which a pipe fills or empties a tank, typically measured in units of volume per time (e.g., liters per hour).**Time:**The duration required to fill or empty a tank, calculated based on the work rates of the pipes.**Combined Rate:**The total rate at which multiple pipes fill or empty a tank when working together.

**Key Formulas:**

**Time = Volume / Rate:**

To find the time required to fill or empty a tank, divide the total volume by the rate of flow.**Combined Rate:**

When pipes work together to fill or empty a tank, the combined rate is the sum of their individual rates if they are filling or the difference if they are emptying.**Time Taken by Each Pipe:**

If Pipe A fills the tank in 'a' hours and Pipe B fills it in 'b' hours, then the time taken to fill the tank together is given by: Time = 1 / (1/a + 1/b).

A. 4.8 hours

B. 4.5 hours

C. 5 hours

D. 6 hours

Use the formula for combined rates of filling the tank.

Combined rate = 1/8 + 1/12 = 5/24.

Time taken = 1 / (5/24) = 24/5 = 4.8 hours.

Time taken = 1 / (5/24) = 24/5 = 4.8 hours.

A. 20 hours

B. 30 hours

C. 25 hours

D. 35 hours

Calculate the individual rates and use the combined rate formula to find Pipe B’s
time.

Let the time taken by Pipe B be x hours.

Combined rate of Pipe A and B = 1/10.

Rate of Pipe A = 1/15, Rate of Pipe B = 1/x.

1/15 + 1/x = 1/10.

Solving for x gives x = 30 hours.

Combined rate of Pipe A and B = 1/10.

Rate of Pipe A = 1/15, Rate of Pipe B = 1/x.

1/15 + 1/x = 1/10.

Solving for x gives x = 30 hours.

A. 24 hours

B. 36 hours

C. 30 hours

D. 48 hours

Use the combined rate of the pipes and the rate of Pipe A to find the rate of Pipe
B.

Combined rate = 1/12.

Rate of Pipe A = 1/18.

Rate of Pipe B = 1/12 - 1/18 = 1/36.

Time taken by Pipe B = 36 hours.

Rate of Pipe A = 1/18.

Rate of Pipe B = 1/12 - 1/18 = 1/36.

Time taken by Pipe B = 36 hours.

A. 12 hours

B. 20 hours

C. 15 hours

D. 18 hours

Calculate the net rate of filling and emptying the cistern.

Rate of Pipe A = 1/9.

Rate of Pipe B = -1/15.

Net rate = 1/9 - 1/15 = 1/45.

Time taken = 1 / (1/45) = 45 hours.

Rate of Pipe B = -1/15.

Net rate = 1/9 - 1/15 = 1/45.

Time taken = 1 / (1/45) = 45 hours.

A. 3 PM

B. 4 PM

C. 5 PM

D. 6 PM

Calculate the volume filled by each pipe and find the time at which the cistern is
full.

Rate of Pipe A = 1/6.

Rate of Pipe B = 1/8.

Time taken together = 1 / (1/6 + 1/8) = 24/7 hours.

Pipe A fills for 2 hours before Pipe B starts, so it fills 2/6 of the tank.

Remaining = 1 - 2/6 = 4/6.

Time to fill remaining with both pipes = (4/6) / (1/6 + 1/8) = 4/7 hours.

Total time = 2 hours + 4/7 hours = 2 hours 34 minutes. Cistern will be full at 12 PM + 2 hours 34 minutes = 2:34 PM, or approximately 5 PM.

Rate of Pipe B = 1/8.

Time taken together = 1 / (1/6 + 1/8) = 24/7 hours.

Pipe A fills for 2 hours before Pipe B starts, so it fills 2/6 of the tank.

Remaining = 1 - 2/6 = 4/6.

Time to fill remaining with both pipes = (4/6) / (1/6 + 1/8) = 4/7 hours.

Total time = 2 hours + 4/7 hours = 2 hours 34 minutes. Cistern will be full at 12 PM + 2 hours 34 minutes = 2:34 PM, or approximately 5 PM.