CBSE Class 7 Mathematics Chapter 8: Comparing Quantities NCERT Solutions

To calculate a ratio between two quantities, it is essential that both quantities are expressed in the same unit. We will convert them to the smaller unit where applicable.

1. Ratio of ₹5 to 50 paise: First, convert ₹5 into paise. Since ₹1 = 100 paise, ₹5 = 5 \times 100 paise = 500 paise. Now, the ratio is 500 paise to 50 paise. The ratio can be written as a fraction: $\backslash frac\{500\; \backslash text\{\; paise\}\}\{50\; \backslash text\{\; paise\}\}$. Simplifying this fraction, we get $\backslash frac\{50\}\{5\}\; =\; \backslash frac\{10\}\{1\}$. Therefore, the ratio is 10:1.
2. Ratio of 15 kg to 210 g: Convert 15 kg into grams. Since 1 kg = 1000 g, 15 kg = 15 \times 1000 g = 15000 g. Now, the ratio is 15000 g to 210 g. The ratio as a fraction is $\backslash frac\{15000\; \backslash text\{\; g\}\}\{210\; \backslash text\{\; g\}\}$. To simplify, we can divide both numerator and denominator by 10: $\backslash frac\{1500\}\{21\}$. Both numbers are divisible by 3: $\backslash frac\{500\}\{7\}$. Therefore, the ratio is 500:7.

1. Ratio of 9 m to 27 cm: Convert 9 meters into centimeters. Since 1 m = 100 cm, 9 m = 9 \times 100 cm = 900 cm. Now, the ratio is 900 cm to 27 cm. The ratio as a fraction is $\backslash frac\{900\; \backslash text\{\; cm\}\}\{27\; \backslash text\{\; cm\}\}$. Both numbers are divisible by 9: $\backslash frac\{100\}\{3\}$. Therefore, the ratio is 100:3.
2. Ratio of 30 days to 36 hours: Convert 30 days into hours. Since 1 day = 24 hours, 30 days = 30 \times 24 hours = 720 hours. Now, the ratio is 720 hours to 36 hours. The ratio as a fraction is $\backslash frac\{720\; \backslash text\{\; hours\}\}\{36\; \backslash text\{\; hours\}\}$. We know that 720 is 20 times 36 (since 72 / 36 = 2). So, $\backslash frac\{720\}\{36\}\; =\; 20$. Therefore, the ratio is 20:1.

We are given that 6 students require 3 computers. We need to find out how many computers are needed for 24 students.

First, let's find out how many computers are needed for 1 student. If 6 students need 3 computers, then 1 student needs $\backslash frac\{3\}\{6\}$ computers.

Simplifying this, 1 student needs $\backslash frac\{1\}\{2\}$ (or 0.5) computers.

Now, to find the number of computers needed for 24 students, we multiply the number of computers per student by 24:

Number of computers = $\backslash frac\{1\}\{2\}\; \backslash times\; 24$

Number of computers = 12

Thus, 12 computers will be needed for 24 students.

To determine how many people are there per square kilometer (population density), we need to divide the total population of a state by its total area.

1. Population density in Rajasthan and U.P.: The formula for population density is: Population Density = $\backslash frac\{\backslash text\{Population\}\}\{\backslash text\{Area\}\}$. For Rajasthan: Population = 570 lakh Area = 3 lakh km² Population Density of Rajasthan = $\backslash frac\{570\; \backslash text\{\; lakh\}\}\{3\; \backslash text\{\; lakh\; km\}^2\}$ = 190 people per km². For U.P.: Population = 1660 lakh Area = 2 lakh km² Population Density of U.P. = $\backslash frac\{1660\; \backslash text\{\; lakh\}\}\{2\; \backslash text\{\; lakh\; km\}^2\}$ = 830 people per km². So, Rajasthan has 190 people per km², and U.P. has 830 people per km².
2. Which state is less populated? Comparing the population densities calculated above: Rajasthan: 190 people per km² U.P.: 830 people per km² Since 190 is less than 830, Rajasthan has a lower population density. Therefore, Rajasthan is less populated in terms of people per square kilometer.