7.1 Percentage – Another Way of Comparing Quantities

Understanding Percentages

A percentage is a way of expressing a number as a fraction of 100. The term “per cent” comes from the Latin “per centum,” which means “by the hundred.” It is denoted by the symbol %.

Comparing Scores Using Percentages

Let’s consider the example of Anita and Rita’s test scores to understand the importance of percentages in comparing quantities:

• Anita’s Report: She scored 320 out of 400 marks.
• Rita’s Report: She scored 300 out of 360 marks.

Anita argued she did better because she got more marks in total. However, when comparing their percentages:

• Anita’s Percentage: 320400×100=80%400320​×100=80%
• Rita’s Percentage: 300360×100=83.3%360300​×100=83.3%

Rita’s percentage is higher, indicating she performed better when considering the maximum possible marks.

Meaning of Percentage

• Per cent means “per hundred” or “out of 100.”
• Example: 1%=1100=0.011%=1001​=0.01

Practical Example

Rina made a table top with 100 colored tiles of different colors. Here’s how we can calculate the percentages for each color:

Calculating Percentages

When the total number of items isn’t 100, convert the fraction to an equivalent fraction with a denominator of 100:

• Example: A necklace has 20 beads with 8 red and 12 blue beads.
  • Red beads: 820=8×520×5=40100=40%208​=20×58×5​=10040​=40%
  • Blue beads: 1220=12×520×5=60100=60%2012​=20×512×5​=10060​=60%

Finding Percentages for Different Data

1. Height of Children: For given heights of children, convert the number to fractions and then percentages:

   • 110 cm: 22100×100=22%10022​×100=22%
   • 120 cm: 25100×100=25%10025​×100=25%
   • 128 cm: 32100×100=32%10032​×100=32%
   • 130 cm: 21100×100=21%10021​×100=21%

2. Shoe Sizes: If a shop has shoe pairs of different sizes, the percentages can be calculated similarly.

Example with Different Total

For collections where the total isn’t 100, you need to adjust:

• Example: A collection of 10 chips:
  • Green: 410=4×1010×10=40100=40%104​=10×104×10​=10040​=40%
  • Blue: 310=3×1010×10=30100=30%103​=10×103×10​=10030​=30%
  • Red: 310=3×1010×10=30100=30%103​=10×103×10​=10030​=30%

Real-Life Examples

Percentages are useful for understanding increases or decreases in quantities, comparing fractions or decimals, and interpreting various data sets.

Converting Percentages to Fractions or Decimals:

• Example: 25% = 25100=0.2510025​=0.25

Using Percentages in Real Life:

• Example: If 25% of children like football in a group of 40, then:
  • $25\%$ of 40 = 25100×40=1010025​×40=10 children.

Understanding Increases and Decreases:

• Example: If a population increased from 550,000 to 605,000, the percentage increase is:
  • Increase: 605,000 – 550,000 = 55,000
  • Percentage Increase: 55,000550,000×100≈10%550,00055,000​×100≈10%

7.2 Finding the Increase or Decrease Percentage

Understanding Increase and Decrease Percentages

In many real-life situations, we need to find out how much a quantity has increased or decreased and express this change as a percentage. This helps in understanding the magnitude of the change relative to the original quantity.

Calculating Percentage Increase

To find the percentage increase:

1. Find the Increase: Subtract the original quantity from the new quantity.
2. Calculate the Percentage: Divide the increase by the original quantity and multiply by 100.

Example:

• The price of a shirt increased from $500 to $550.
  • Increase = $550 – $500 = $50
  • Percentage Increase = 50500×100=10%50050​×100=10%

Calculating Percentage Decrease

To find the percentage decrease:

1. Find the Decrease: Subtract the new quantity from the original quantity.
2. Calculate the Percentage: Divide the decrease by the original quantity and multiply by 100.

Example:

• The price of a jacket decreased from $700 to $630.
  • Decrease = $700 – $630 = $70
  • Percentage Decrease = 70700×100=10%70070​×100=10%

Application in Real Life

These calculations are useful in various scenarios, such as understanding changes in prices, population, salaries, and more.

Practical Exercise

If the population of a town increased from 20,000 to 22,500:

1. Increase = 22,500 – 20,000 = 2,500
2. Percentage Increase = 2,50020,000×100=12.5%20,0002,500​×100=12.5%

If the population then decreased to 21,500:

1. Decrease = 22,500 – 21,500 = 1,000
2. Percentage Decrease = 1,00022,500×100≈4.44%22,5001,000​×100≈4.44%

7.3 Finding Discounts – Estimation in Percentages

Understanding Discounts

A discount is a reduction in the marked price of an item. Discounts are often given during sales to attract customers. Calculating discounts as a percentage helps in understanding how much you save.

Calculating Discount Percentage

To find the discount percentage:

1. Find the Discount: Subtract the sale price from the marked price.
2. Calculate the Percentage: Divide the discount by the marked price and multiply by 100.

Example:

• A pair of shoes marked at $800 is sold for $600.
  • Discount = $800 – $600 = $200
  • Discount Percentage = 200800×100=25%800200​×100=25%

Estimating Discounts

Estimation helps in quickly figuring out the approximate discount without precise calculation. It is useful for quick decision-making while shopping.

Example:

• A TV marked at $20,000 is sold at a discount of 15%.
  • Estimated Discount = $15\%$ of $20,000 = 0.15 times 20,000 = $3,000
  • Sale Price = $20,000 – $3,000 = $17,000

Practical Exercise

Consider a shop offering a 20% discount on all items. Calculate the sale price of:

1. A dress marked at $1,200:

   • Discount = $20\%$ of $1,200 = 0.20 times 1,200 = $240
   • Sale Price = $1,200 – $240 = $960

2. A laptop marked at $50,000:

   • Discount = $20\%$ of $50,000 = 0.20 times 50,000 = $10,000
   • Sale Price = $50,000 – $10,000 = $40,000

Applying Discounts to Bulk Purchases

When buying multiple items with the same discount percentage, calculate the total marked price first and then apply the discount percentage.

Example:

• 3 shirts each marked at $500 with a 10% discount:
  • Total Marked Price = 3 times 500 = $1,500
  • Total Discount = $10\%$ of $1,500 = 0.10 times 1,500 = $150
  • Total Sale Price = $1,500 – $150 = $1,350

7.4 Finding Original Prices and Quantities Using Percentage Increase/Decrease

Understanding Original Prices and Quantities

When given the new price or quantity after a percentage increase or decrease, we often need to find the original value. This is useful in scenarios like calculating the original price before a discount or the original population before a change.

Finding the Original Price/Quantity after a Percentage Increase

To find the original price/quantity before an increase:

1. Identify the New Value: The value after the increase.
2. Identify the Percentage Increase: The rate at which the value increased.
3. Calculate the Original Value:
   Original Value=New Value1+(Percentage Increase100)Original Value=1+(100Percentage Increase​)New Value​

Example:

• A bike now costs $660 after a 10% increase.
  • New Value = $660
  • Percentage Increase = 10%
  • Original Value = frac{660}{1 + 0.10} = frac{660}{1.10} = $600

Finding the Original Price/Quantity after a Percentage Decrease

To find the original price/quantity before a decrease:

1. Identify the New Value: The value after the decrease.
2. Identify the Percentage Decrease: The rate at which the value decreased.
3. Calculate the Original Value:
   Original Value=New Value1−(Percentage Decrease100)Original Value=1−(100Percentage Decrease​)New Value​

Example:

• A TV now costs $720 after a 20% decrease.
  • New Value = $720
  • Percentage Decrease = 20%
  • Original Value = frac{720}{1 – 0.20} = frac{720}{0.80} = $900

Application in Real Life

These calculations help in various scenarios, such as finding the original price of an item before a sale, determining the original population of a town before a decrease, or figuring out the initial salary before an increment.

Practical Exercise

1. Increase Example:

   • A laptop now costs $1,100 after a 10% increase. Find the original price.
     • New Value = $1,100
     • Percentage Increase = 10%
     • Original Value = frac{1,100}{1 + 0.10} = frac{1,100}{1.10} = $1,000

2. Decrease Example:

   • A book now costs $240 after a 20% decrease. Find the original price.
     • New Value = $240
     • Percentage Decrease = 20%
     • Original Value = frac{240}{1 – 0.20} = frac{240}{0.80} = $300.