GSEB Class 8 Maths Solutions Chapter 8 Comparing Quantities Exercise 8.1

Question 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 Rs 5
Answer:
When working with ratios, the amounts must be in the same units. If they are not, we first change them to be identical.

(a) Speed of cycle = \( 15 \text{ km per hour} \)
Speed of scooter = \( 30 \text{ km per hour} \)
Ratio = \( \frac{\text{Speed of cycle}}{\text{Speed of scooter}} = \frac{15 \text{ km/hr}}{30 \text{ km/hr}} = \frac{15}{30} = \frac{1}{2} \) (or \( 1 : 2 \))

(b) Ratio = \( \frac{5 \text{m}}{10 \times 1000 \text{m}} \) (We change 10 km into meters)
\( = \frac{5}{10 \times 1000} = \frac{1}{2000} \) (or \( 1 : 2000 \))

(c) Ratio = \( \frac{50 \text{ paise}}{\text{Rs } 5} = \frac{50 \text{ paise}}{500 \text{ paise}} \) (We change Rs 5 to paise)
\( = \frac{50}{500} = \frac{1}{10} \) (or \( 1 : 10 \))
In simple words: To compare amounts using a ratio, always make sure their measurement units are the same. If they are different, change one to match the other before dividing.

Exam Tip: Remember to always express ratios in their simplest form. Also, verify that both quantities are in the same units before calculating the ratio.

Question 2. Convert the following ratios to percentages?
(a) 3:4
(b) 2:3
Answer:
(a) \( 3:4 = \frac{3}{4} \)
\( \implies \frac{3}{4} = \frac{3}{4} \times 100\% = (3 \times 25)\% = 75\% \)

(b) \( 2:3 = \frac{2}{3} \)
\( \implies \frac{2}{3} = \frac{2}{3} \times 100\% = \frac{200}{3}\% = 66\frac{2}{3}\% \)
In simple words: To change a ratio into a percentage, write it as a fraction and then multiply that fraction by 100%. Don't forget to add the percent symbol at the end.

Exam Tip: Ratios are often given in the format 'a:b'. Convert them into the fraction 'a/b' before multiplying by 100% to find the percentage equivalent.

Question 3. 72% of 25 students are good in Mathematics. How many are not good in Mathematics?
Answer:
72% of 25 students perform well in Mathematics.
\( \implies \) So, \( (100 - 72)\% \) of 25 students do not perform well in Mathematics.
\( \implies \) This means \( 28\% \) of 25 students are not good in Mathematics.
\( \implies \frac{28}{100} \times 25 = 7 \) students are not good in Mathematics.
In simple words: If most students are good at math, you can find the percentage of students who are not good by subtracting from 100%. Then, calculate that percentage of the total number of students.

Exam Tip: When dealing with percentages of a group, remember that "the remaining percentage" is found by subtracting the given percentage from 100%.

Question 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?
Answer:
The team won 10 matches.
The team achieved victory in \( 40\% \) of the total number of matches played.
\( \implies 40\% \) of [Total number of matches] = 10
\( \implies \frac{40}{100} \times \) [Total number of matches] = 10
\( \implies \) Total number of matches = \( \frac{10 \times 100}{40} = 25 \)
Therefore, the team played a total of 25 matches.
In simple words: If you know how many games a team won and what percentage that represents of all their games, you can calculate the total number of games they played. Just set up an equation where the percentage of the total equals the number of games won.

Exam Tip: When a percentage of a total is known, use the formula: Part = (Percentage/100) * Whole. Rearrange it to find the 'Whole' if the 'Part' and 'Percentage' are given.

Question 5. If Chameli had Rs 600 left after spending 75% of her money how much did she have in the beginning?
Answer:
Chameli used \( 75\% \) of her money for spending.
\( \implies \) So, she has \( (100 - 75)\% \) or \( 25\% \) of her money remaining.
She now possesses Rs 600.
\( \implies 25\% \) of her total money equals Rs 600.
\( \implies \) Total money = \( \frac{600 \times 100}{25} \)
\( = 600 \times 4 = \text{Rs } 2400 \)
Hence, she originally had Rs 2400.
In simple words: If a part of her money is left, first find what percentage that leftover part represents. Then, use that percentage and the amount of money to figure out how much money she had at the start.

Exam Tip: If a person spends a certain percentage, the remaining percentage is 100% minus the spent percentage. Use this remaining percentage to find the original amount.

Question 6. If 60% people in a city like 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?
Answer:
Number of people who prefer cricket = \( 60\% \)
Number of people who prefer football = \( 30\% \)
\( \implies \) People who prefer other games = \( [100 - (60 + 30)]\% \)
\( = [100 - 90]\% = 10\% \)
Currently, the total population is 50,00,000.
\( \implies 60\% \) of 50,00,000 = \( \frac{60}{100} \times 5000000 \)
\( = 6 \times 5000000 = 30,00,000 \)
\( 30\% \) of 50,00,000 = \( \frac{30}{100} \times 5000000 \)
\( = 3 \times 5000000 = 15,00,000 \)
\( 10\% \) of 50,00,000 = \( \frac{10}{100} \times 5000000 \)
\( = 1 \times 5000000 = 5,00,000 \)
Thus, the numbers are:
Cricket = 30,00,000
Football = 15,00,000
Other games = 5,00,000
In simple words: First, add up the percentages for the known preferences and subtract from 100% to find the percentage for "other games." Then, calculate the actual number of people for each preference by finding that percentage of the city's total population.

Exam Tip: Always check that the sum of all percentages for different categories equals 100%. When calculating actual numbers from percentages, multiply the percentage (as a decimal or fraction) by the total quantity.