# Comparing Quantities Class 7 Notes: Chapter 8

## Prices Related to Buying and Selling

### Prices related to an item

• Prices related to an item are: (i) Selling price
(ii) Cost price
• Selling price(SP) 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 fraction of 100.
• Percentages are represented by the symbol ‘%’.
• 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 Raghav’s clothes are not washed.
⇒ Raghav’s 40 clothes out of 100 clothes are not washed.
(ii) 30 % of students in class are infected by fever.
⇒ Out of 100 students in a class, 30 students are infected by fever.

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

• Example: 200 chocolates were distributed among two children: Joe and Tom. Joe got 60% and Tom got 40% of the 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 = $\frac{40}{100}\times 200=80$ ∴ 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 $\frac{2}{5}$ of the 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) Joe’s height is 150 cm and Tom’s is 100 cm.
Ratio of Joe’s height to Tom’s height would be Joe’s height : Tom’s height.
= 150 : 100 = 3: 2
(ii) Ratio of 3 km to 30 m is 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 multiplying numerator and denominator of a rational number by 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 as 2:3 :: 6:9 or 2:3 = 6:9.
(ii) Ratios 1:2 and x:4 are in proportion.
⇒12=x4
⇒1×4=x×2
⇒2x=4
⇒x=2

## Finding the Increase or Decrease in Percent

### Finding the percentage increase or decrease

• Example: Price of a book was changed from ₹20 to ₹25 in a week. Calculate the percentage increased.
Solution: Change in price = ₹25 – ₹20 = ₹5
$Percentage\,Increased =\frac{Change\,in\,Price}{Original\,Price}$ =$\frac{5}{20}\times 100=25\%$

## Simple and Compound Interest

### Sum / principal

• The money which has been 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.
⇒ Amount = Principal + Interest.

### Simple Interest

• Simple interest(SI) is the interest charged on a borrowed money where the principal amount will be fixed for a 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; R = 10%; T = 3 yrs
Simple Interest = $\frac{200\times10\times\times3}{100}=60$

Amount = P + SI = Rs. (200 + 60) = Rs. 260

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