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.

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Finding the profit or loss percentage

• \(\begin{array}{l}Profit\,Percentage=\frac{Profit}{Cost\,Price}\times 100\end{array} \)
• \(\begin{array}{l}Loss\,Percentage = \frac{Loss}{Cost Price}\times 100\end{array} \)

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Percentages and Why Percentages

Percentages

• Percentages are ratios expressed as a fraction of 100.
• Percentages are represented by the symbol ‘%’.
• Example:
  \(\begin{array}{l}\frac{20}{100}=20\%\end{array} \)
  and
  \(\begin{array}{l}\frac{50}{100}=50\%\end{array} \)
  .

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
  \(\begin{array}{l}\frac{3}{5}\end{array} \)
  .Multiply the numerator and denominator by 20.
  
  \(\begin{array}{l}\Rightarrow \frac{3\times 20}{5\times 20}=\frac{60}{100}=60\%\end{array} \)

Converting fractions/decimals to percentages

• Converting Decimals to Percentages
  Given decimal: 0.44
  
  \(\begin{array}{l}0.44=\frac{44}{100}=\frac{44}{100}\times 100\%=44\%\end{array} \)
• Converting Fractions to Percentages
  Given fraction:
  \(\begin{array}{l}\frac{3}{5}\end{array} \)
  \(\begin{array}{l}\frac{3}{5}\times 100\%=3\times 20\%=60%\end{array} \)

To know more about Fraction to Percentage Conversion, visit here.

Converting percentages to fractions/decimals

• \(\begin{array}{l}0.25=\frac{25}{100}=\frac{1}{4}\end{array} \)
• \(\begin{array}{l}0.225=\frac{225}{1000}=\frac{9}{40}\end{array} \)

To know more about Percentage to Fraction Conversion, visit here.

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
  \(\begin{array}{l}\frac{6}{8}\times 100=\frac{3}{4}\times100=75\%\end{array} \)
  .

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 =
  \(\begin{array}{l}\frac{60}{100}\times 200=120\end{array} \)
  Tom got 40% of the chocolates =
  \(\begin{array}{l}\frac{40}{100}\times 200=80\end{array} \)
  ∴ 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
  \(\begin{array}{l}\frac{3}{5}\end{array} \)
  and Jacob got
  \(\begin{array}{l}\frac{2}{5}\end{array} \)
  of the chocolates. What is the percentage of chocolate that each got?
  Solution: Total number of chocolates = 200
  James got
  \(\begin{array}{l}\frac{3}{5}\end{array} \)
  of the chocolates =
  \(\begin{array}{l}\frac{3}{5}\times100=60\%\end{array} \)
  of the total chocolates.
  Jacob got
  \(\begin{array}{l}\frac{2}{5}\end{array} \)
  of the chocolates =
  \(\begin{array}{l}\frac{2}{5}\times100=40\%\end{array} \)
  of the total chocolates.

To know more about Ratios to Percentages Conversion, visit here.

Introduction to Fractions and Ratios

For more information on Introduction to Fractions and Ratios, watch the below video.

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

To know more on Comparing Quantities, visit here.

Ratios

For more information on Ratios, watch the below video.

• 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:
  \(\begin{array}{l}\frac{1}{3}=\frac{1}{3}\times \frac{2}{2}=\frac{2}{6}\end{array} \)
  and
  \(\begin{array}{l}\frac{1}{3}\end{array} \)
  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

To know more about Ratios and Proportion, visit here.

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

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.
• \(\begin{array}{l}Simple\,Interest=\frac{P\times R\times N}{100}\end{array} \)
  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 =
  \(\begin{array}{l}\frac{200\times10\times\times3}{100}=60\end{array} \)
  Amount = P + SI = Rs. (200 + 60) = Rs. 260