Chapter Notes: Comparing Quantities

7.1 Introduction to Percentages

Percentage is a way to express a number as a fraction of 100. The term is derived from the Latin phrase per centum, which means 'per hundred'. The symbol for percentage is %. For instance, 1% means 1 out of 100 or a fraction equal to 1/100.

7.1.1 Meaning of Percentage

To compare scores effectively, as illustrated in the example of Anita and Rita, we cannot solely rely on total scores. Instead, we use percentages:

• Anita: 320 out of 400, which gives her a percentage of 80%.
• Rita: 300 out of 360, leading to a percentage of approximately 83.33%.

Thus, even if Anita scored more marks, Rita performed better in relative terms as her percentage is higher.

Constructing Percentages from Groups

If we look at a collection of items, such as colored tiles or children's heights, each can be represented as a percentage of the whole:

• For tiles: 14 yellow out of 100 tiles is 14%, and so forth for other colors.
• For heights: If we want the percentage of children at a certain height, we can follow a similar process.

Formula for Percentage Calculation

When the total does not equal 100, you can convert the fraction to percentage using: [ ext{Percentage} = rac{ ext{Part}}{ ext{Total}} imes 100 ] For example, to find the percentage of red beads in a necklace of 20 beads, if 8 beads are red: [ ext{Percentage of red} = rac{8}{20} imes 100 = 40 ext{%} ]

7.1.2 Converting Fractional Numbers to Percentage

• To convert fractions to percentages, multiply by 100. For instance, to express 3/4 as a percentage: [ rac{3}{4} imes 100 = 75 ext{%} ]
• Example: For a class with 25 children, if 15 are girls, the percentage of girls is: [ rac{15}{25} imes 100 = 60 ext{%} ]

7.1.3 Converting Decimals to Percentage

Similar to fractions, convert decimals to percentages by multiplying by 100. For example:

• 0.75 = 0.75 x 100 = 75%
• 0.2 = 0.2 x 100 = 20%

7.1.4 Converting Percentages to Fractions or Decimals

• Percentages can also be converted back into decimal or fractional form. For example:
• 25% as a decimal is 0.25 and as a fraction is ( \frac{25}{100} ) simplified to ( \frac{1}{4} ).

7.1.5 Fun with Estimation

Estimating percentages visually can be done with shaded areas in diagrams. For example, if half of a rectangle is shaded, it represents 50%.

7.2 Uses of Percentages

7.2.1 Interpreting Percentages

Percentages give a simple way to interpret statements like '5% of income is saved' or '20% of dresses are blue'. This shows how many parts are being referred to out of 100.

7.2.2 Converting Percentages to “How Many”

To find “how many” based on a percentage, apply the percentage to the total amount. For example, if 25% of 40 children like football, to find the number who do:

• [ 0.25 imes 40 = 10 ] children.

7.2.3 Ratios to Percentage

To convert ratios into percentages, first determine the total parts, then divide the parts of interest by the total parts and multiply by 100. For example, in a 2:1 mixture of rice and urad dal, rice is 66.67% and urad dal 33.33%.

7.2.4 Increase or Decrease as Percent

To determine percentage increase or decrease, evaluate the change over the original value. Example: If wins increase from 4 to 6: [ ext{Percentage increase} = rac{6-4}{4} imes 100 = 50 ext{%} ]

7.3 Pricing Related to Buying and Selling

Cost Price (CP) and Selling Price (SP)

• The cost price is the price at which an item is bought, and the selling price is the price at which it is sold. Profit or losses can be calculated based on these prices.
• Profit is when SP > CP, while loss occurs when CP > SP.

Profit or Loss as Percentage

To find profit or loss percentage: [ ext{Profit ext{ or } Loss ext{ %}} = \frac{ ext{Profit ext{ or } Loss}}{ ext{CP}} imes 100 ]

7.4 Simple Interest

Simple Interest is calculated on the principal amount over time. The formula is: [ I = \frac{P \times R \times T}{100} ] Where I is the interest, P is the principal, R is the rate, and T is time in years.

• Example: If a principal of `5000 at 5% for 1 year yields: [ I = \frac{5000 \times 5 \times 1}{100} = 250 ] The total amount payable would then be the principal plus the interest.

To summarize, percentages are integral in comparing quantities, analyzing data, and making financial decisions. Their applications range from school marks to sales analysis, making them essential in day-to-day calculations.

1. Percentage is a fraction of 100 used for comparison.
2. To convert fractions to percentages, multiply by 100.
3. Decimally represented numbers can also be converted to percent.
4. The concept of profit or loss is often expressed as a percentage.
5. It's crucial to understand increase or decrease as a percentage for trends.
6. Percentages are essential for interpreting ratios and data in real life.
7. The formula for calculating simple interest is vital for finance.
8. Profit or loss percentage is calculated based on cost price.
9. Understanding total percentages helps in estimating and budget planning.