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

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

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

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

Converting percentages to fractions/decimals

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

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?Estimation using percentages
    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.

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Introduction to Fractions and Ratios

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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

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Ratios

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  • 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

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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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