CBSE Class 7 Mathematics Chapter 8: Comparing Quantities NCERT Solutions

To find the ratio between two quantities, it is essential that both quantities are expressed in the same unit. We will convert the larger unit to the smaller unit for each part.

1. Ratio of ₹5 to 50 paise:

   First, convert ₹5 to paise. Since ₹1 = 100 paise, ₹5 = 5 × 100 = 500 paise.

   Now, find the ratio of 500 paise to 50 paise.

   Ratio = $\backslash frac\{500\; \backslash text\{\; paise\}\}\{50\; \backslash text\{\; paise\}\}$ = $\backslash frac\{50\}\{5\}$ = $\backslash frac\{10\}\{1\}$

   Thus, the ratio is 10:1.

2. Ratio of 15 kg to 210 g:

   Convert 15 kg to grams. Since 1 kg = 1000 g, 15 kg = 15 × 1000 = 15000 g.

   Now, find the ratio of 15000 g to 210 g.

   Ratio = $\backslash frac\{15000\; \backslash text\{\; g\}\}\{210\; \backslash text\{\; g\}\}$ = $\backslash frac\{1500\}\{21\}$

   Divide both numerator and denominator by their greatest common divisor, which is 3: $\backslash frac\{1500\; \backslash div\; 3\}\{21\; \backslash div\; 3\}$ = $\backslash frac\{500\}\{7\}$

   Thus, the ratio is 500:7.

3. Ratio of 9 m to 27 cm:

   Convert 9 meters to centimeters. Since 1 m = 100 cm, 9 m = 9 × 100 = 900 cm.

   Now, find the ratio of 900 cm to 27 cm.

   Ratio = $\backslash frac\{900\; \backslash text\{\; cm\}\}\{27\; \backslash text\{\; cm\}\}$

   Divide both numerator and denominator by their greatest common divisor, which is 9: $\backslash frac\{900\; \backslash div\; 9\}\{27\; \backslash div\; 9\}$ = $\backslash frac\{100\}\{3\}$

   Thus, the ratio is 100:3.

4. Ratio of 30 days to 36 hours:

   Convert 30 days to hours. Since 1 day = 24 hours, 30 days = 30 × 24 = 720 hours.

   Now, find the ratio of 720 hours to 36 hours.

   Ratio = $\backslash frac\{720\; \backslash text\{\; hours\}\}\{36\; \backslash text\{\; hours\}\}$

   Divide both numerator and denominator by their greatest common divisor, which is 36: $\backslash frac\{720\; \backslash div\; 36\}\{36\; \backslash div\; 36\}$ = $\backslash frac\{20\}\{1\}$

   Thus, 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 the number of computers needed for 1 student. This is a unit rate calculation.

Number of computers per student = $\backslash frac\{\backslash text\{Number\; of\; computers\}\}\{\backslash text\{Number\; of\; students\}\}$ = $\backslash frac\{3\}\{6\}$ = $\backslash frac\{1\}\{2\}$ computer per student.

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

Computers needed for 24 students = 24 students × $\backslash frac\{1\}\{2\}$ computer/student

Computers needed = $24\; \backslash times\; \backslash frac\{1\}\{2\}\; =\; 12$

Therefore, 12 computers will be needed for 24 students.

To determine the population density, we need to calculate the number of people per square kilometer for each state.

1. Population density calculation:

   The formula for population density is: Population Density = $\backslash frac\{\backslash text\{Total\; Population\}\}\{\backslash text\{Total\; Area\}\}$

   In Rajasthan:

   Population = 570 lakhs

   Area = 3 lakh km²

   Population Density of Rajasthan = $\backslash frac\{570\; \backslash text\{\; lakhs\}\}\{3\; \backslash text\{\; lakh\; km\}^2\}$ = 190 people per km².

   In U.P.:

   Population = 1660 lakhs

   Area = 2 lakh km²

   Population Density of U.P. = $\backslash frac\{1660\; \backslash text\{\; lakhs\}\}\{2\; \backslash text\{\; lakh\; km\}^2\}$ = 830 people per km².

2. Determining the less populated state:

   Comparing the population densities calculated above:

   Rajasthan has a population density of 190 people per km².

   U.P. has a population density of 830 people per km².

   Since 190 is less than 830, Rajasthan is less populated per square kilometer.