Solve the following three math problems: 1) The ages of Astha and Shiv are in the ratio 5:6. Five years from now their ages will be 6:7. Find their present ages, 2) The total surfa... Solve the following three math problems: 1) The ages of Astha and Shiv are in the ratio 5:6. Five years from now their ages will be 6:7. Find their present ages, 2) The total surface area of a rectangular block is 846 cm^2. Find the length, breadth, and height, if their dimensions are in the ratio 5:4:3 and 3) Draw a line passing through (1,6) and (3,2) on a graph, and extend the line both sides to meet the coordinate axes. What are the coordinates of the points where this line meets the x-axis and the y-axis? Read more

Steps to Solve

1. Solve problem i)

Let Astha's present age be $5x$ and Shiv's present age be $6x$. Five years from now, Astha's age will be $5x + 5$ and Shiv's age will be $6x + 5$. The ratio of their ages five years from now is 6:7. So, we have the equation:

$$ \frac{5x + 5}{6x + 5} = \frac{6}{7} $$

2. Solve the equation

Cross-multiply and solve for $x$:

$$ 7(5x + 5) = 6(6x + 5) $$ $$ 35x + 35 = 36x + 30 $$ $$ 35 = x + 30 $$ $$ x = 5 $$

3. Find the present ages

Astha's present age = $5x = 5 \times 5 = 25$ years Shiv's present age = $6x = 6 \times 5 = 30$ years

4. Solve problem ii)

Let the length, breadth, and height of the rectangular block be $5x$, $4x$, and $3x$ respectively.

5. Write the formula for the total surface area of a rectangular block

The total surface area of a rectangular block is given by $2(lb + bh + hl)$, where $l$ is the length, $b$ is the breadth, and $h$ is the height.

6. Set up the equation

Given the total surface area is 846 cm$^2$, we have:

$$ 2(5x \cdot 4x + 4x \cdot 3x + 3x \cdot 5x) = 846 $$ $$ 2(20x^2 + 12x^2 + 15x^2) = 846 $$ $$ 2(47x^2) = 846 $$ $$ 94x^2 = 846 $$

7. Solve the equation

$$ x^2 = \frac{846}{94} = 9 $$ $$ x = \sqrt{9} = 3 $$

8. Calculate the dimensions

Length = $5x = 5 \times 3 = 15$ cm Breadth = $4x = 4 \times 3 = 12$ cm Height = $3x = 3 \times 3 = 9$ cm

9. Solve problem iii)

First, we need to determine the equation of the line passing through the points (1,6) and (3,2). 10. Find the slope $m$

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 6}{3 - 1} = \frac{-4}{2} = -2$

11. Find the equation of the line Using the point-slope form, $y - y_1 = m(x - x_1)$, we have: $y - 6 = -2(x - 1)$ $y - 6 = -2x + 2$ $y = -2x + 8$

12. Find the $x$-intercept To find the $x$-intercept, set $y = 0$: $0 = -2x + 8$ $2x = 8$ $x = 4$ So, the $x$-intercept is $(4, 0)$.

13. Find the $y$-intercept To find the $y$-intercept, set $x = 0$: $y = -2(0) + 8$ $y = 8$ So, the $y$-intercept is $(0, 8)$.