Comparing Quantities is a chapter in 8th-grade mathematics that focuses on understanding and applying different methods to compare numerical values. It involves various concepts to express one quantity as a fraction of another, and vice versa.
Key Concepts:
- Ratio: A comparison of two quantities of the same kind in terms of division.
- Proportion: An equation stating that two ratios are equal.
- Percentage: A way of expressing a number as a fraction of 100.
- Profit and Loss: Calculation of profit or loss made in a business transaction.
- Simple Interest: Interest calculated on the principal amount.
- Compound Interest: Interest calculated on both the principal amount and the accumulated interest.
- Discount: Reduction in the marked price of an item.
- Tax: A compulsory levy imposed by the government on individuals and businesses.
Applications:
These concepts are widely used in real-life situations, such as:
- Calculating discounts and sales tax
- Determining profit or loss in business
- Understanding interest rates for loans and investments
- Analyzing data and statistics
By studying this chapter, students develop a strong foundation in mathematical reasoning and problem-solving skills.
Exercise 7.1
1. Find the ratio of the following:
(a) speed of a cycle 15 km per hour to the speed of scooter 30 km per hour.
(b) 5 m to 10 km
(c) 50 paise to ₹ 5
Ans :
(a)
Ratio = Speed of cycle / Speed of scooter
- Ratio = 15 km/hour / 30 km/hour
- Ratio = 1/2
So, the ratio is 1:2.
(b)
Let’s convert km to m.
- 1 km = 1000 m
- So, 10 km = 10 * 1000 m = 10000 m
- Ratio = 5 m / 10000 m
- Ratio = 1/2000
So, the ratio is 1:2000.
(c)
Let’s convert rupees to paise.
- 1 rupee = 100 paise
- So, ₹ 5 = 5 * 100 paise = 500 paise
- Ratio = 50 paise / 500 paise
- Ratio = 1/10
So, the ratio is 1:10.
2. Convert the following ratios to percentages:
(a) 3 : 4
(b) 2 : 3
Ans :
(a) 3 : 4
- Multiply the ratio by 100.
- Percentage = (3/4) * 100 = 75%
(b) 2 : 3
- Percentage = (2/3) * 100 = 66.67% (approximately)
Therefore, the percentages are:
- (a) 75%
- (b) 66.67%
3. 72% of 25 students are good in mathematics. How many are not good in mathematics?
Ans :
Step 1: Calculate the number of students good in mathematics:
- 72% of 25 students
- = (72/100) * 25
- = 18 students
Step 2: Calculate the number of students not good in mathematics:
- Total students – Students good in mathematics = 25 – 18 = 7 students
So, 7 students are not good in mathematics.
4. A football team won 10 matches out of the total number of matches they played. If their win percentage was 40, then how many matches did they play in all?
Ans :
Total number of matches played = x
We know:
- The team won 10 matches.
- Their win percentage is 40%.
Setting up the equation:
- Win percentage = (Number of matches won / Total number of matches) * 100
- 40 = (10 / x) * 100
Solving for x:
- x = (10 * 100) / 40
- x = 25
The football team played a total of 25 matches.
5. If Chameli had ₹ 600 left after spending 75% of her money, how much did she have in the beginning?
Ans :
Chameli spent 75% of her money.
She is left with Rs. 600.
Solution:
If Chameli spent 75% of her money, she is left with 100% – 75% = 25% of her money.
This 25% of her money is equal to Rs. 600.
So, 25% of her total money = Rs. 600.
To find 100% (her total money), we can set up a proportion:
25/100 = 600/Total money
Let’s denote Total money as X.
25/100 = 600/X
Cross-multiplying:
25X = 600 * 100
X = (600 * 100) / 25
X = 2400
Therefore, Chameli had Rs. 2400 in the beginning.
6. If 60% of people in a city like a cricket, 30% like football and the remaining like other games, then what per cent of the people like other games? If the total number of people are 50 lakh, find the exact number who like each type of game.
Ans :
Percentage of People Who Like Other Games
- Total percentage of people who like cricket and football = 60% + 30% = 90%
- Percentage of people who like other games = 100% – 90% = 10%
Number of People Who Like Each Type of Game
- Total number of people = 50 lakh = 50,00,000
- Number of people who like cricket = 60% of 50,00,000 = (60/100) * 50,00,000 = 30,00,000
- Number of people who like football
- = 30% of 50,00,000 = (30/100) * 50,00,000 = 15,00,000
- Number of people who like other games
- = 10% of 50,00,000 = (10/100) * 50,00,000 = 5,00,000
Therefore,
- 10% of the people like other games.
- 30,00,000 people like cricket.
- 15,00,000 people like football.
- 5,00,000 people like other games.
Exercise 7.2
1. During a sale, a shop offered a discount of 10% on the marked prices of all the items. What would a customer have to pay for a pair of Jeans marked at ₹ 1450 and two shirts marked at ₹ 850 each?
Ans :
Step 1: Find the total marked price:
Cost of one shirt = Rs. 850
Cost of two shirts = Rs. 850 * 2 = Rs. 1700
Total marked price = Cost of jeans + Cost of two shirts = Rs. 1450 + Rs. 1700 = Rs. 3150
Step 2: Calculate the discount:
Discount = 10% of Rs. 3150 = (10/100) * 3150
= Rs. 315
Step 3: Calculate the final price:
Final price = Marked price – Discount = Rs. 3150 – Rs. 315 = Rs. 2835
The customer would have to pay Rs. 2835.
2. The price of a TV is ₹ 13,000. The sales tax charged on it is at the rate of 12%. Find the amount that Vinod will have to pay if he buys it.
Ans :
- Price of the TV = Rs. 13,000
- Sales tax rate = 12%
Solution
- Calculate the sales tax amount:
- Sales tax = (12/100) * 13000 = Rs. 1560
- Calculate the total amount to be paid:
- Total amount = Price of TV + Sales tax = Rs. 13000 + Rs. 1560 = Rs. 14560
Therefore, Vinod will have to pay Rs. 14,560 for the TV.
3. Arun bought a pair of skates at a sale where the discount is given was 20%. If the amount he pays is ₹ 1,600, find the marked price.
Ans :
- Arun paid Rs. 1600 for the skates after a 20% discount.
- We need to find the original marked price.
Solution
- If Arun paid 100% – 20% = 80% of the marked price.
- This 80% is equal to Rs. 1600.
So, 80% of the marked price = Rs. 1600
To find 100% (the original marked price), we can set up a proportion:
- 80/100 = 1600/Marked price
Let’s denote the marked price as X.
- 80/100 = 1600/X
Cross-multiplying:
- 80X = 1600 * 100
- X = (1600 * 100) / 80
- X = 2000
4. I purchased a hair-dryer for ₹ 5,400 including 8% VAT. Find the price before VAT was added.
Ans :
Total price (including VAT) = Rs. 5400
VAT percentage = 8%
Solution:
- Calculate the VAT amount:
- VAT amount = (8/100) * 5400 = Rs. 432
- Calculate the price before VAT:
- Price before VAT = Total price – VAT amount = Rs. 5400 – Rs. 432 = Rs. 5000
5. The population of a place increased to 54,000 in 2003 at a rate of 5% per annum.
(i) Find the population in 2001.
(ii) What would be its population in 2005?
Ans :
Formula for population growth:
- P = P₀(1 + R/100)^n Where:
- P = Population after n years
- P₀ = Initial population
- R = Rate of growth per year
- n = Number of years
i) Find the population in 2001
- P = 54000 (population in 2003)
- R = 5%
- n = -2 (since we’re going back in time)
54000 = P₀ * (1 + 5/100)^(-2) 54000 = P₀ * (1.05)^(-2) P₀ = 54000 / (1.05)^(-2) P₀ ≈ 48979.59
Therefore, the population in 2001 was approximately 48,980.
ii) Find the population in 2005
- P₀ = 54000 (population in 2003)
- R = 5%
- n = 2 (since we’re going forward in time)
P = 54000 * (1 + 5/100)^(2) P = 54000 * (1.05)^(2) P ≈ 59535
The population in 2005 would be approximately 59,535.
2. In a Laboratory, the count of bacteria in a certain experiment was increasing at the rate of 2.5% per hour. Find the bacteria at the end of 2 hours if the count was initially 5,06,000.
Ans :
Understanding the problem:
- Initial bacteria count = 506000
- Growth rate = 2.5% per hour
- Time = 2 hours
Solution:
This is a problem of compound growth, similar to compound interest.
Formula for compound growth:
- Amount = Principal * (1 + Rate/100)^Time
Here,
- Amount = Final bacteria count
- Principal = Initial bacteria count = 506000
- Rate = 2.5% per hour
- Time = 2 hours
Substituting the values in the formula:
- Final bacteria count = 506000 * (1 + 2.5/100)^2 = 506000 * (1.025)^2 = 506000 * 1.050625 = 531616.25
The bacteria count at the end of 2 hours is approximately 531,616.
3. A scooter was bought at ₹ 42,000. Its value depreciated at the rate of 8% per annum. Find its value after one year.
Ans :
Understanding the Problem:
- Original price of the scooter = Rs. 42,000
- Depreciation rate = 8% per annum
- We need to find the value after 1 year.
Solution:
- Calculate the depreciation amount:
- Depreciation = 8% of 42000 = (8/100) * 42000 = Rs. 3360
- Value after 1 year = Original price – Depreciation = Rs. 42000 – Rs. 3360 = Rs. 38640
