CBSE Class 6 Mathematics Chapter 12: Ratio and Proportion NCERT Solutions

We are given the number of girls and boys in a class and asked to find ratios.

Number of girls = 20

Number of boys = 15

Total number of students = Number of girls + Number of boys = 20 + 15 = 35

(a) The ratio of the number of girls to the number of boys is calculated by dividing the number of girls by the number of boys:

$$\backslash frac\{\backslash text\{Number\; of\; girls\}\}\{\backslash text\{Number\; of\; boys\}\}\; =\; \backslash frac\{20\}\{15\}$$

To simplify this ratio, we find the greatest common divisor (GCD) of 20 and 15, which is 5. Dividing both numbers by 5:

$$\backslash frac\{20\; \backslash div\; 5\}\{15\; \backslash div\; 5\}\; =\; \backslash frac\{4\}\{3\}$$

So, the ratio of girls to boys is 4:3.

(b) The ratio of the number of girls to the total number of students is calculated by dividing the number of girls by the total number of students:

$$\backslash frac\{\backslash text\{Number\; of\; girls\}\}\{\backslash text\{Total\; number\; of\; students\}\}\; =\; \backslash frac\{20\}\{35\}$$

To simplify this ratio, we find the GCD of 20 and 35, which is 5. Dividing both numbers by 5:

$$\backslash frac\{20\; \backslash div\; 5\}\{35\; \backslash div\; 5\}\; =\; \backslash frac\{4\}\{7\}$$

So, the ratio of girls to the total number of students is 4:7.

We are given the total number of students and the number of students who like football and cricket. We need to find the number of students who like tennis first.

Total number of students = 30

Number of students who like football = 6

Number of students who like cricket = 12

The number of students who like tennis is the total number of students minus those who like football and cricket:

$$\backslash text\{Number\; of\; students\; liking\; tennis\}\; =\; 30\; -\; 6\; -\; 12\; =\; 12$$

(a) The ratio of the number of students liking football to the number of students liking tennis is:

$$\backslash frac\{\backslash text\{Number\; of\; students\; liking\; football\}\}\{\backslash text\{Number\; of\; students\; liking\; tennis\}\}\; =\; \backslash frac\{6\}\{12\}$$

Simplifying this ratio by dividing both numbers by their GCD, which is 6:

$$\backslash frac\{6\; \backslash div\; 6\}\{12\; \backslash div\; 6\}\; =\; \backslash frac\{1\}\{2\}$$

So, the ratio of students liking football to tennis is 1:2.

(b) The ratio of the number of students liking cricket to the total number of students is:

$$\backslash frac\{\backslash text\{Number\; of\; students\; liking\; cricket\}\}\{\backslash text\{Total\; number\; of\; students\}\}\; =\; \backslash frac\{12\}\{30\}$$

Simplifying this ratio by dividing both numbers by their GCD, which is 6:

$$\backslash frac\{12\; \backslash div\; 6\}\{30\; \backslash div\; 6\}\; =\; \backslash frac\{2\}\{5\}$$

So, the ratio of students liking cricket to the total number of students is 2:5.

To solve this, we first need to count the number of each type of figure inside the rectangle. Based on the provided answer structure, let's assume the counts are:

Number of triangles = 3

Number of circles = 2

Number of squares = 2

Total number of figures = Number of triangles + Number of circles + Number of squares = 3 + 2 + 2 = 7

(a) The ratio of the number of triangles to the number of circles is:

$$\backslash frac\{\backslash text\{Number\; of\; triangles\}\}\{\backslash text\{Number\; of\; circles\}\}\; =\; \backslash frac\{3\}\{2\}$$

The ratio is 3:2.

(b) The ratio of the number of squares to all the figures inside the rectangle is:

$$\backslash frac\{\backslash text\{Number\; of\; squares\}\}\{\backslash text\{Total\; number\; of\; figures\}\}\; =\; \backslash frac\{2\}\{7\}$$

The ratio is 2:7.

(c) The ratio of the number of circles to all the figures inside the rectangle is:

$$\backslash frac\{\backslash text\{Number\; of\; circles\}\}\{\backslash text\{Total\; number\; of\; figures\}\}\; =\; \backslash frac\{2\}\{7\}$$

The ratio is 2:7.

We are given the distances travelled by Hamid and Akhtar in one hour. Since speed is distance divided by time, and the time is the same (1 hour) for both, their speeds are directly proportional to the distances they travelled.

Speed is defined as Distance / Time.

Speed of Hamid = $$\backslash frac\{9\; \backslash text\{\; km\}\}\{1\; \backslash text\{\; hour\}\}\; =\; 9\; \backslash text\{\; km/h\}$$

Speed of Akhtar = $$\backslash frac\{12\; \backslash text\{\; km\}\}\{1\; \backslash text\{\; hour\}\}\; =\; 12\; \backslash text\{\; km/h\}$$

The ratio of the speed of Hamid to the speed of Akhtar is:

$$\backslash frac\{\backslash text\{Speed\; of\; Hamid\}\}\{\backslash text\{Speed\; of\; Akhtar\}\}\; =\; \backslash frac\{9\; \backslash text\{\; km/h\}\}\{12\; \backslash text\{\; km/h\}\}\; =\; \backslash frac\{9\}\{12\}$$

To simplify this ratio, we find the GCD of 9 and 12, which is 3. Dividing both numbers by 3:

$$\backslash frac\{9\; \backslash div\; 3\}\{12\; \backslash div\; 3\}\; =\; \backslash frac\{3\}\{4\}$$

Therefore, the ratio of Hamid's speed to Akhtar's speed is 3:4.

We need to find the missing numbers in the blanks to make the ratios equivalent. We start with the given ratio $\backslash frac\{15\}\{18\}$.

First, simplify the given ratio $\backslash frac\{15\}\{18\}$. The GCD of 15 and 18 is 3. Dividing both by 3:

$$\backslash frac\{15\; \backslash div\; 3\}\{18\; \backslash div\; 3\}\; =\; \backslash frac\{5\}\{6\}$$

So, the first blank is 5. The equation now looks like:

$$\backslash frac\{15\}\{18\}\; =\; \backslash frac\{5\}\{6\}\; =\; \backslash frac\{10\}\{\backslash boxed\{\}\}\; =\; \backslash frac\{\backslash boxed\{\}\}\{30\}$$

Now, consider the ratio $\backslash frac\{5\}\{6\}$ and the term $\backslash frac\{10\}\{\backslash boxed\{\}\}$. To get 10 from 5, we multiply by 2. So, we must multiply the denominator 6 by 2 as well:

$$\backslash frac\{5\; \backslash times\; 2\}\{6\; \backslash times\; 2\}\; =\; \backslash frac\{10\}\{12\}$$

So, the second blank is 12. The equation is now:

$$\backslash frac\{15\}\{18\}\; =\; \backslash frac\{5\}\{6\}\; =\; \backslash frac\{10\}\{12\}\; =\; \backslash frac\{\backslash boxed\{\}\}\{30\}$$

Finally, consider the ratio $\backslash frac\{5\}\{6\}$ and the term $\backslash frac\{\backslash boxed\{\}\}\{30\}$. To get 30 from 6, we multiply by 5. So, we must multiply the numerator 5 by 5 as well:

$$\backslash frac\{5\; \backslash times\; 5\}\{6\; \backslash times\; 5\}\; =\; \backslash frac\{25\}\{30\}$$

So, the third blank is 25. The completed equation is:

$$\backslash frac\{15\}\{18\}\; =\; \backslash frac\{5\}\{6\}\; =\; \backslash frac\{10\}\{12\}\; =\; \backslash frac\{25\}\{30\}$$

Yes, these are equivalent ratios because each fraction simplifies to $\backslash frac\{5\}\{6\}$.