Chapter Notes - Comparing Quantities

# Comparing Quantities Class 7 Notes Maths Chapter 7

 Table of contents Introduction Percentages Converting Ratios to Percent Converting Fractional Numbers into Percentage Converting Decimals into Percentages Conversion of Percentages into Fractions or Decimals Profit or Loss as a Percentage Charge given on Borrowed Money or Simple Interest

## Introduction

Imagine you and friend Sid got your report cards, and you scored 320 out of 400 whereas Sid scored 300 out of 360. You immediately say that, you have scored better than Sid, because 320 > 300. But, is that right? Report cards also have percentages written on them, and it shows that your percentage was 80% but Sid's percentage was 83.3%. So, this shows that, instead Sid has scored better.

Question for Chapter Notes - Comparing Quantities
Try yourself:
The concept of percentage is used to compare quantities by expressing them as a proportion of the maximum possible value. Which of the following situations illustrates the use of percentage for comparison?

## Percentages

Percentage is a method used to compare quantities. Percentages are numerators of fractions with the denominator 100. Percent is represented by the symbol - %.

 Percent is derived from the Latin word ‘per centum’ meaning ‘per hundred'

Example: What percent of ₹ 4500 is ₹ 9000?
Sol: Let us assume the percentage be Q,

Then, Q% of ₹ 4500 is ₹ 9000 => (Q/100) × 4500 = ₹ 9000

(4500Q/100) = ₹ 9000

45Q = ₹ 9000

Therefore, Q = ₹ 9000/45 = ₹ 200

Example: 30% of ₹ 360 = ________.
Sol: 30% of ₹ 360 = ₹ 108.

It can be written as = (30/100) × 360

= 10800/100

= ₹ 108

Example: 120 % of 50 km = ________.
Sol: 120 % of 50 km = 60 km.

It can be written as = (120/100) × 50

= 6000/100

= 60 km

Example: In a class of 50 students, 8 % were absent on one day. Find the number of students present on that day.
Sol: In a class of 50 students, 8 % were absent on one day. The number of students present on that day was 46.

From the question it is given that, number of students in the class = 50

Percentage of students who were absent on one day = 8%

Then, percentage of students who were present on one day = 100% – 8%

= 92%

So, 92% of 50

= (92/100) × 50

= 4600/100

= 46 students

## Converting Ratios to Percent

Ratios help us to compare quantities and determine the relation between them. We write ratios in the form of fractions and then compare them by converting them into like fractions. If these like fractions are equal, then we say that the given ratios are equivalent.
Example: The ratio of Fatima’s income to her savings is 4: 1. The percentage of money saved by her is :

(a) 20% (b) 25% (c) 40% (d) 80%

Sol: (a) 20%
Let’s assume the ratio of Fatima’s income to her savings be 4x: 1x.
Then, the percentage of money saved by her is = (her savings/(income + savings)) × 100
= ( (1x / 4 x) + x ) × 100 = 1/5 x 100 = 20 %

Example: Reena’s mother said, to make idlis, you must take two parts rice and one part urad dal. What percentage of such a mixture would be rice and what percentage would be urad dal?
Sol:
In terms of ratio we would write this as Rice: Urad dal = 2: 1.

Now, 2 + 1=3 is the total of all parts. This means 2/3 part is rice and 1/3 part is urad dal.

Then, percentage of rice would be (2/3) x 100 = (200/3) = 66.67%

Then, percentage of urad dal would be (1/3) x 100 = (100/3) = 33.33%

## Converting Fractional Numbers into Percentage

To compare fractional numbers, we need a common denominator. To convert a fraction into a percentage, multiply it by a hundred and then place the % symbol.

Percentages related to proper fractions are less than 100, whereas percentages related to improper fractions are more than 100.

Example: Out of 25 children in a class, 15 are girls. What is the percentage of girls?
Sol: Out of 25 children, there are 15 girls.
Therefore, the percentage of girls = 15/25 ×100 = 60.
There are 60% girls in the class.

Example: Convert 5/4 to percent.
Sol: We have, 5/4 x 100 = 125 %

## Converting Decimals into Percentages

To convert a decimal into a percentage:

Step 1: Convert the decimal into a fraction.

Step 2: Multiply the fraction by 100.

Step 3: Put a percent sign next to the number. Otherwise, shift the decimal point two places to the right.

Example: 2.5 = ________%
Sol:2.5 = 250 %

2.5 = 2.5 × 100

= 250%

Example: Convert the given decimals to per cents:
Sol: (a) 0.75

0.75 = 0.75 × 100 % = 75/100 × 100 % = 75%
(b) 49/50
49/50 = 49/50 x 100 = 98%

(c) 0.05
0.05 = 0.05 x 100 = 5%

## Conversion of Percentages into Fractions or Decimals

A given percentage can be converted into fractions and decimals.

Examples:

 Percent 2% 45% Fraction 2/100 45/100 Decimal 0.02 0.45

(a) To convert a percentage into a fraction:
Converting percentage into fraction

Step 1: Convert the mixed fraction percent into a proper fraction

Step 2: Now multiply the 1/100 to remove the percent symbol

Step 3: Reduce to the simplified fraction

Example: 225% is equal to

(a) 9: 4 (b) 4: 9 (c) 3: 2 (d) 2 : 3

Sol: (a) 9: 4

225% = (225/100)

(Because to remove the %, we have to divide the given number by 100.)

= 9/4

(b) To convert a percentage into a decimal:
Converting percentages into decimal

Step 1: Remove the percent sign.

Step 2: Divide the number by 100, or move the decimal point two places to the left in the numerator.

Example: Convert 77.5% into decimal

Sol: Dividing 77.5% by 100, we get;

77.5% = 77.5/100 = 7.75/10 = 0.775

Thus, 0.775 is the decimal equivalent of 77.5%.

## Profit or Loss as a Percentage

1. Cost Price (CP): The buying price of an item is known as its cost price written in short as CP.
2. Selling Price (SP): The price at which we sell an item is known as the selling price or in short SP. Naturally, it is better if we sell the item at a higher price than our buying price.
3. Profit or Loss: We can decide whether the sale was profitable or not depending on the CP and SP.

Based on the values of CP and SP, we calculate our profit or loss.
• If CP < SP then we have gained some amount, that is, we made a profit,
Profit = SP – CP
• If CP = SP then we are in a no profit no loss situation
• If CP > SP then we have lost some amount,
Loss = CP – SP
The profit or loss we find can be converted to a percentage. It is always calculated on the CP.
Formulas for profit and loss

Note: If we are given any two of the three quantities related to price, that is, CP, SP, and Profit or Loss percent, we can find the third.

Question for Chapter Notes - Comparing Quantities
Try yourself:
A shirt is bought for $30 and sold for$45. What is the profit percentage?

Example: Mohini bought a cow for ₹ 9000 and sold it at a loss of ₹ 900. The selling price of the cow is ________.
Sol: Mohini bought a cow for ₹ 9000 and sold it at a loss of ₹ 900. The selling price of the cow is ₹ 8100

From the question it is given that,

The cost price of the cow (CP) = ₹ 9000

Loss = ₹ 900

We know that, Selling price (SP) = CP – loss

= 9000 – 900 = ₹ 8100

Example: Suhana sells a sofa set for Rs. 9600 making a profit of 20%. What is the CP of the sofa?
Sol: Let the CP of the sofa be Rs. x

Example: If the price of sugar is decreased by 20%, then the new price of 3kg sugar, originally costing ₹ 120 will be _____.
Sol: If the price of sugar is decreased by 20%, then the new price of 3kg sugar originally costing ₹ 120 will be ₹ 96

From the question it is given that, price of 3kg sugar originally costing ₹ 120

The price of the sugar is decreased by 20%

Then, the new price of sugar = 120 – 20% of the original price

= 120 – (20/100) × 120 = 120 – (2400/100)

= 120 – 24 = ₹ 96

Example: Aahuti purchased a house for ₹ 50,59,700 and spent ₹ 40300 on its repairs. At what price should she sell the house to make a profit of 5%?
Sol: Aahuti purchased a house for ₹ 50,59,700 and spent ₹ 40300 on its repairs. To make a profit of 5%, she should sell the house for ₹ 5355000.

From the question it is given that,

CP of house purchased by Aahuti = ₹ 50,59,700

Amount spent to repair the house = ₹ 40,300

Total CP of house = ₹ 50,59,700 + 40,300 = ₹ 5100000

Profit % = (profit/CP) × 100 & profit = SP - CP

5 = ((SP – CP)/CP) × 100

5 = ((SP – 5100000)/5100000) × 100
(5 × 5100000)/100 = SP – 5100000  => SP =  ₹ 5355000

## Charge given on Borrowed Money or Simple Interest

Simple Interest is a method of calculating the amount of interest charged on a sum at a given rate and for a given period of time.

1. Principal: The money borrowed is known as sum borrowed or principal.
2. Interest: We have to pay some extra money (or charge) to the bank for the money being used by us for some time. This is known as interest.
3. Amount: We can find the amount we have to pay at the end of the year by adding the above two. That is.
Amount = Principal + Interest
Simple Interest Calculation

Example: Interest is typically expressed as a percentage for one year, denoted as per annum (p.a.). For instance, 10% p.a. implies that on every ₹100 borrowed, the borrower must pay ₹10 as interest for one year. This arrangement illustrates how the total amount owed is determined.

The general formula for simple interest over multiple years is derived by recognizing that the interest paid for one year on a principal amount (P) at an annual interest rate (R%) is given by

�⋅�1Therefore, the interest (I) paid for T years is expressed as �⋅�⋅�100. The total amount to be repaid at the end of T years is given by �=�+�
�=�⋅�⋅�100If you are provided with any two of the quantities (I, P, T, R), you can use these formulas to calculate the remaining quantity.

Example: Find simple interest on ₹ 12500 at 18% per annum for a period of 2 years and 4 months.
Sol: Interest on ₹ 12500 at 18% per annum for a period of 2 years and 4 months is ₹ 5250.

From the question it is given that, Principal= ₹ 12500
Time = 2 years 4 months = (2 + (4/12)) = (2 + (1/3)) = 7/3 year

Rate = 18%

Then, we know the formula of Simple interest SI = (P × R × T)/100

SI= (12500 × 18 × (7/3)) /100

SI = ₹ 5250

Example: The difference of interest for 2 years and 3 years on a sum of ₹ 2100 at 8% per annum is _________.
Sol: The difference of interest for 2 years and 3 years on a sum of ₹ 2100 at 8% per annum is ₹ 168.

From the question it is given that, P = ₹ 2100 ,Time = 2 years, Rate = 8%

Then, we know the formula of Simple interest SI = (P × R × T)/100

SI = (2100 × 2 × 8)/100 = 33600/100 = ₹ 336

Then, taking Time = 3 years, simple interest is

SI= (2100 × 3× 8)/100 = 50400/100 = ₹504

The difference of interest for 2 years and 3 years = 3 years – 2 years

= ₹ 504 – ₹ 336

= ₹ 168

The document Comparing Quantities Class 7 Notes Maths Chapter 7 is a part of the Class 7 Course Mathematics (Maths) Class 7.
All you need of Class 7 at this link: Class 7

## Mathematics (Maths) Class 7

76 videos|345 docs|39 tests

## FAQs on Comparing Quantities Class 7 Notes Maths Chapter 7

 1. How do you convert ratios into percentages?
Ans. To convert ratios into percentages, you need to multiply the ratio by 100 and add a percentage sign. For example, if the ratio is 2:3, you would calculate (2/3) x 100 = 66.67%.
 2. How can you convert fractional numbers into percentages?
Ans. To convert fractional numbers into percentages, you can simply multiply the fraction by 100. For example, if the fraction is 3/4, you would calculate (3/4) x 100 = 75%.
 3. What is the formula to convert decimals into percentages?
Ans. To convert decimals into percentages, you need to multiply the decimal by 100 and add a percentage sign. For example, if the decimal is 0.75, you would calculate 0.75 x 100 = 75%.
 4. How do you convert percentages into fractions or decimals?
Ans. To convert percentages into fractions, you can simply write the percentage as a fraction with a denominator of 100. To convert percentages into decimals, you divide the percentage by 100. For example, 25% can be written as 25/100 or 0.25.
 5. How can you calculate profit or loss as a percentage?
Ans. To calculate profit or loss as a percentage, you can use the formula: (Profit or Loss / Cost Price) x 100. If the result is positive, it represents profit, and if it is negative, it represents a loss.

## Mathematics (Maths) Class 7

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