## Prices Related to Buying and Selling

### Prices related to an item

- Prices related to an item are: (i) Selling price

(ii)Â Cost price **Selling price (SP)**Â is the price at which a product is sold out.**Cost price (CP)**is the buying price of an item.- Profit = Selling price – Cost price
- Loss = Cost price – Selling price
- If SP > CP , then it is profit.
- If SP = CP , then it is neither profit nor loss.
- If CP > SP , then it is loss.

### Finding the profit or loss percentage

- \(Profit\,Percentage=\frac{Profit}{Cost\,Price}\times 100\)
- \(Loss\,Percentage = \frac{Loss}{Cost Price}\times 100\)

## Percentages and Why Percentages

### Percentages

- Percentages are ratios expressed as a fractionÂ of 100.
- Percentages are represented by the symbol â€˜%â€™.
- Example: \(\frac{20}{100}=20\%\)and \(\frac{50}{100}=50\%\).

### Comparing percentages when denominator is not 100

- When a ratio is not expressed in fraction of 100, then convert the fraction to an equivalent fraction with denominator 100.
- Example: Consider a fraction \(\frac{3}{5}\).Multiply the numerator and denominator by 20.

\(\Rightarrow \frac{3\times 20}{5\times 20}=\frac{60}{100}=60\%\)

### Converting fractions/decimals to percentages

**Converting Decimals to Percentages**

Given decimal: 0.44

\(0.44=\frac{44}{100}=\frac{44}{100}\times 100\%=44\%\)**Converting Fractions to Percentages**

Given fraction: \(\frac{3}{5}\)\(\frac{3}{5}\times 100\%=3\times 20\%=60%\)

### Converting percentages to fractions/decimals

- \(0.25=\frac{25}{100}=\frac{1}{4}\)
- \(0.225=\frac{225}{1000}=\frac{9}{40}\)

### Estimation using percentages

- Estimation can be done using percentages.
- Example: What percentage of the given circle is shaded?

Solution: The given triangle consists of 8 regions, out of which 6 regions are shaded.

So, the percentage of shaded regions will be \(\frac{6}{8}\times 100=\frac{3}{4}\times100=75\%\).

### Interpreting percentage into usable data

- Percentages can be interpreted into useful data.
- Examples:

(i) 40% of Raghavâ€™s clothes are not washed.

â‡’ Raghavâ€™s 40 clothes out of 100 clothes are notÂ washed.

(ii) 30 % of students in class are infected by fever.

â‡’Â Out of 100 students in a class, 30 students areÂ infected by fever.

### Converting percentage to the form “how many”

- Example: 200 chocolates were distributed among two children: Joe and Tom. Joe got 60%Â and Tom got 40% ofÂ theÂ chocolates. How many chocolates will each get?

Solution: Total number of chocolates = 200

Joe got 60% of the chocolates = \(\frac{60}{100}\times 200=120\)Tom got 40% of the chocolates = \(\frac{40}{100}\times 200=80\) âˆ´ Joe and Tom will get 120 and 80 chocolates, respectively.

### Converting Ratios to percentages

- Ratios can be expressed as percentages to understand certain situations much better.
- Example: 200 chocolates were distributed among two children: James and Jacob. James got\(\frac{3}{5}\) and Jacob got \(\frac{2}{5}\) ofÂ theÂ chocolates. What is the percentage of chocolate that each got?

Solution: Total number of chocolates = 200

James got \(\frac{3}{5}\) of the chocolates = \(\frac{3}{5}\times100=60\%\) of the total chocolates.

Jacob got \(\frac{2}{5}\) of the chocolates = \(\frac{2}{5}\times100=40\%\) of the total chocolates.

## Introduction to Fractions and Ratios

### Comparing Quantities : Introduction

- To compare two quantities, the units must be the same.
- Examples:

(i) Joeâ€™s height is 150 cm and Tomâ€™s is 100 cm.

Ratio of Joe’s height to Tom’s heightÂ would beÂ**Joe’s height : Tom’s height**.

= 150 : 100 = 3: 2

(ii) Ratio of 3 km to 30 m isÂ**3 km : 30 m.**

= 3000 m : 30 m

= 300 : 1

### Ratios

- Ratio is a relation between two quantities showing the number of times one value contains or is contained within the other.
- Example: If there are four girls and seven boys in a class, then the ratio of number of girls to number of boys is
**4:7**.

### Equivalent Ratios

- By multiplyingÂ numerator and denominator of a rational number byÂ a non zero integer, we obtain another rational number equivalent to the given rational number. These are called
**equivalent fractions**. - Example:\(\frac{1}{3}=\frac{1}{3}\times \frac{2}{2}=\frac{2}{6}\) and \(\frac{1}{3}\) are equivalent fractions.

### Proportions

- If two ratios are equal, then they are said to be in proportion.
- Symbol “
**::**” or “**=**” is used to equate the two ratios. - Example: (i) Ratios
**2:3**and**6:9**are equal. They can be represented asÂ**2:3 :: 6:9**or**2:3 = 6:9**.

(ii) Ratios**1:2**and**x:4**are in proportion.

â‡’12=x4

â‡’1Ã—4=xÃ—2

â‡’2x=4

â‡’x=2

## Finding the Increase or Decrease in Percent

### Finding the percentage increase or decrease

- Example: Price of a book was changed from â‚¹20 to â‚¹25Â in a week. Calculate the percentage increased.

Solution: Change in price = â‚¹25 – â‚¹20 = â‚¹5

\(Percentage\,Increased =\frac{Change\,in\,Price}{Original\,Price}\) =\(\frac{5}{20}\times 100=25\%\)

## Simple and Compound Interest

### Sum / principal

- The money which has beenÂ borrowed is called
**sum or principal**. - This money can be used by the borrower for a particular time period before returning to the lender.
- Example: Loan that you take from a bank is the principal.

### Interest

**Interest**is the extra payment that a borrower should pay to the lender along with the principal.

### Amount

- A borrower should return the principal amount (he/she has borrowed) and the interest to the lender. This money is called amount.

â‡’ Amount = Principal + Interest.

### Simple Interest

- Simple interest(SI) is the interest charged on a borrowed money where the principal amount will be fixed for aÂ particular time period.
- \(Simple\,Interest=\frac{P\times R\times N}{100}\)P = Principal Amount, R = Interest rate

N = Number of years - Example: Calculate the simple interest for 3 years when the principal amount is 200 and interest rate is 10% for 1 year.

Solution: Given: P = 200; RÂ = 10%; T = 3 yrs

Simple Interest = \(\frac{200\times10\times\times3}{100}=60\)Amount = P + SI = Rs. (200 + 60) = Rs. 260