5 Percentage Increase Problems

 5 Percentage Increase Problems

• Percentage Increase:
  New Value = Original Value × (1 + Increase%/100)
  Increase = Original Value × (Increase%/100)
  Percentage Increase = [(New Value - Original Value) / Original Value] × 100
• Percentage Decrease:
  New Value = Original Value × (1 - Decrease%/100)
  Decrease = Original Value × (Decrease%/100)
  Percentage Decrease = [(Original Value - New Value) / Original Value] × 100

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Percentage Increase Problems

Problem 1: Price Increase

Problem: The price of a book is $50. If the price increases by 20%, find the new price.

Solution:

1. Identify the given values:
   • Original Price = $50
   • Percentage Increase = 20%
2. Calculate New Price:
   • New Price = Original Price × (1 + Increase%/100)
   • New Price = 50 × (1 + 20/100)
   • New Price = 50 × 1.2 = $60

Answer:

• New Price = $60

Explanation:

• A percentage increase adds a fraction of the original value to itself.
• The multiplier (1 + Increase%/100) accounts for the original value plus the increase.

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Problem 2: Population Growth

Problem: A town’s population is 10,000. It increases by 5% annually. Find the population after 1 year.

Solution:

1. Identify the given values:
   • Original Population = 10,000
   • Percentage Increase = 5%
2. Calculate New Population:
   • New Population = Original Population × (1 + Increase%/100)
   • New Population = 10,000 × (1 + 5/100)
   • New Population = 10,000 × 1.05 = 10,500

Answer:

• New Population = 10,500

Explanation:

• The population grows by 5% of its original size, calculated using the percentage increase formula.
• The result is the new population after one year.

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Problem 3: Salary Increase

Problem: An employee’s salary is $2000 per month. If it increases by 15%, find the new salary.

Solution:

1. Identify the given values:
   • Original Salary = $2000
   • Percentage Increase = 15%
2. Calculate New Salary:
   • New Salary = Original Salary × (1 + Increase%/100)
   • New Salary = 2000 × (1 + 15/100)
   • New Salary = 2000 × 1.15 = $2300

Answer:

• New Salary = $2300

Explanation:

• The salary increase is calculated by multiplying the original salary by (1 + Increase%/100).
• This gives the new salary after the raise.

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Problem 4: Find Original Value

Problem: After a 25% increase, the price of a gadget is $125. Find the original price.

Solution:

1. Identify the given values:
   • New Price = $125
   • Percentage Increase = 25%
2. Relate New Price to Original Price:
   • New Price = Original Price × (1 + Increase%/100)
   • 125 = Original Price × (1 + 25/100)
   • 125 = Original Price × 1.25
3. Solve for Original Price:
   • Original Price = 125 / 1.25
   • Original Price = $100

Answer:

• Original Price = $100

Explanation:

• The new price is 125% of the original price. Divide by 1.25 to find the original price.
• This reverses the percentage increase.

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Problem 5: Calculate Percentage Increase

Problem: The price of a car increased from $20,000 to $22,000. Find the percentage increase.

Solution:

1. Identify the given values:
   • Original Price = $20,000
   • New Price = $22,000
2. Calculate Increase:
   • Increase = New Price - Original Price
   • Increase = 22,000 - 20,000 = $2000
3. Calculate Percentage Increase:
   • Percentage Increase = (Increase / Original Price) × 100
   • Percentage Increase = (2000 / 20,000) × 100 = 10%

Answer:

• Percentage Increase = 10%

Explanation:

• The percentage increase is the increase divided by the original value, expressed as a percentage.
• This shows the relative change in price.

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Problem 6: Successive Increases

Problem: A product’s price is $100. It increases by 10% in the first year and 20% in the second year. Find the final price.

Solution:

1. Identify the given values:
   • Original Price = $100
   • First Increase = 10%
   • Second Increase = 20%
2. Calculate Price after First Increase:
   • Price after first increase = 100 × (1 + 10/100)
   • = 100 × 1.1 = $110
3. Calculate Price after Second Increase:
   • Final Price = 110 × (1 + 20/100)
   • = 110 × 1.2 = $132

Alternative Method:

• Final Price = Original Price × (1 + First Increase%/100) × (1 + Second Increase%/100)
• = 100 × 1.1 × 1.2 = $132

Answer:

• Final Price = $132

Explanation:

• Successive percentage increases are applied sequentially, multiplying by each increase factor.
• The alternative method combines both increases in one step.

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Problem 7: Increase to Match a Target

Problem: A shopkeeper wants to increase the price of an item from $80 to $100. Find the percentage increase required.

Solution:

1. Identify the given values:
   • Original Price = $80
   • New Price = $100
2. Calculate Increase:
   • Increase = New Price - Original Price
   • Increase = 100 - 80 = $20
3. Calculate Percentage Increase:
   • Percentage Increase = (Increase / Original Price) × 100
   • Percentage Increase = (20 / 80) × 100 = 25%

Answer:

• Percentage Increase = 25%

Explanation:

• The percentage increase is calculated to determine how much the original price must rise to reach the target.
• The formula gives the relative increase needed.

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Problem 8: Increase in Area

Problem: The side of a square is increased by 10%. Find the percentage increase in its area.

Solution:

1. Identify the given values:
   • Percentage Increase in Side = 10%
2. Relate Side Increase to Area:
   • Let original side = s
   • Original Area = s²
   • New Side = s × (1 + 10/100) = s × 1.1
   • New Area = (1.1s)² = 1.21s²
3. Calculate Percentage Increase in Area:
   • Increase in Area = New Area - Original Area = 1.21s² - s² = 0.21s²
   • Percentage Increase = (Increase / Original Area) × 100
   • = (0.21s² / s²) × 100 = 21%

Answer:

• Percentage Increase in Area = 21%

Explanation:

• The area of a square is proportional to the square of its side. A 10% increase in side results in a (1.1)² = 1.21 factor increase in area.
• The percentage increase is calculated relative to the original area.

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Problem 9: Increase with Fractional Percentage

Problem: A stock price of $200 increases by 7.5%. Find the new stock price.

Solution:

1. Identify the given values:
   • Original Price = $200
   • Percentage Increase = 7.5%
2. Calculate New Price:
   • New Price = Original Price × (1 + Increase%/100)
   • New Price = 200 × (1 + 7.5/100)
   • New Price = 200 × 1.075 = $215

Answer:

• New Stock Price = $215

Explanation:

• Fractional percentages are handled the same way as whole percentages in the formula.
• The new price reflects the original price plus the increase.

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Problem 10: Repeated Increase

Problem: A value of 100 increases by 5% each year for 2 years. Find the final value.

Solution:

1. Identify the given values:
   • Original Value = 100
   • Percentage Increase per Year = 5%
   • Time = 2 years
2. Calculate Value after First Year:
   • Value after Year 1 = 100 × (1 + 5/100)
   • = 100 × 1.05 = 105
3. Calculate Value after Second Year:
   • Final Value = 105 × (1 + 5/100)
   • = 105 × 1.05 = 110.25

Alternative Method:

• Final Value = Original Value × (1 + Increase%/100)^n
• = 100 × (1.05)^2
• = 100 × 1.1025 = 110.25

Answer:

• Final Value = 110.25

Explanation:

• Repeated percentage increases are compounded, similar to compound interest.
• The formula (1 + Increase%/100)^n accounts for multiple increases.