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

(a) To find the ratio of the number of girls to the number of boys, we divide the number of girls by the number of boys:

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

Now, we simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 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) To find the ratio of the number of girls to the total number of students, we first need to find the total number of students.

Total number of students = Number of girls + Number of boys

$$Total\; students\; =\; 20\; +\; 15\; =\; 35$$

Now, we find the ratio of girls to the total students:

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

We simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 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:

$$Number\; of\; students\; liking\; tennis\; =\; Total\; students\; -\; (Students\; liking\; football\; +\; Students\; liking\; cricket)$$

$$Number\; of\; students\; liking\; tennis\; =\; 30\; -\; (6\; +\; 12)\; =\; 30\; -\; 18\; =\; 12$$

(a) To find the ratio of the number of students liking football to the number of students liking tennis, we divide the number of students liking football by the number of students liking tennis:

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

Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor, 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 those liking tennis is 1:2.

(b) To find the ratio of the number of students liking cricket to the total number of students, we divide the number of students liking cricket by the total number of students:

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

Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor, 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 and the total number of figures inside the rectangle. Assuming the figure contains 3 triangles, 2 circles, and 2 squares.

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

$$Total\; figures\; =\; 3\; +\; 2\; +\; 2\; =\; 7$$

(a) The ratio of the number of triangles to the number of circles is found by dividing the count of triangles by the count of circles:

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

This ratio is 3:2.

(b) The ratio of the number of squares to all the figures inside the rectangle is found by dividing the count of squares by the total count of figures:

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

This ratio is 2:7.

(c) The ratio of the number of circles to all the figures inside the rectangle is found by dividing the count of circles by the total count of figures:

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

This ratio is 2:7.

We are given the distances travelled by Hamid and Akhtar in one hour. Since speed is defined as distance travelled per unit time, and the time is the same (1 hour) for both, the ratio of their speeds will be the same as the ratio of the distances they travelled.

Distance travelled by Hamid in 1 hour = 9 km

Distance travelled by Akhtar in 1 hour = 12 km

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

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

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

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

Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

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

So, the ratio of the speed of Hamid to the speed of Akhtar 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, let's simplify the given ratio $\backslash frac\{15\}\{18\}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 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:

Now, let's find the second blank. We have . To get 10 in the numerator from 5, we multiply by 2. So, we must also multiply the denominator by 2:

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

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

Finally, let's find the third blank. We have . To get 30 in the denominator from 6, we multiply by 5. So, we must also multiply the numerator by 5:

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

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 $\backslash frac\{5\}\{6\}$.