Chain-Rule - 03

Explanation:-

Cost of 8 fan's and 14 oven's is Rs. 36520
Cost of 4 fan's and 7 oven's is Rs. $$\frac{{{\text{36520}}}}{2}$$   = Rs.18260
Cost of 12 fan's and 21 oven's is = 18260 × 3 = Rs. 54780

Explanation:-

1 man = 2 boys
⇒ (12 men + 18 boys) ≡ (12 × 2 ×18) boys = 42 boys
Let required number of boys = x
⇒ (21 men + x boys) ≡ (21 × 2 × x) boys = (42 + x) boys
Less days, More boys (Indirect proportion)
More hours per day, Less boys (Indirect proportion)
More work, More boys (Direct proportion)
$$\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 50}}:60 \hfill \\ {\text{Hours per day 9}}:\frac{{15}}{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Work }}1:2 \hfill \\ \end{gathered} \right\}::42:\left( {42 + x} \right)$$
$$\eqalign{ & \therefore \,\left[ {50 \times 98 \times 1 \times \left( {42 + x} \right)} \right] = \left( {60 \times \frac{{15}}{2} \times 2 \times 42} \right) \cr & \Leftrightarrow \left( {42 + x} \right) = \frac{{37800}}{{450}} \cr & \Leftrightarrow 42 + x = 84 \cr & \Leftrightarrow x = 42 \cr}$$

Explanation:-

Let the remaining food last for x days
500 men had provision for = (27 - 3) = 24 days
(500 + 300) men had provision for x days
More men, Less days (Indirect proportion)
$$\eqalign{ & \therefore \,800:500::24:x \cr & \Leftrightarrow \left( {800 \times x} \right) = \left( {500 \times 24} \right) \cr & \Leftrightarrow x = \frac{{\left( {500 \times 24} \right)}}{{800}} \cr & \Leftrightarrow x = 15 \cr}$$

Explanation:-

Let yje remaining food last for x days
4000 soldiers had provision for 160 days
3200 soldiers had provision for x days
Less men, More days (Indirect proportion)
$$\eqalign{ & \therefore \,3200:4000::160:x \cr & \Leftrightarrow 3200x = 4000 \times 160 \cr & \Leftrightarrow x = \frac{{\left( {4000 \times 160} \right)}}{{3200}} \cr & \Leftrightarrow x = 200 \cr}$$

Explanation:-

Let the required quantity of coal be x metric tonnes
More engines, More coal (Direct proportion)
More hours per day, More coal (Direct proportion)
More rate, More coal (Direct proportion)
$$\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Engines 5}}:8 \hfill \\ {\text{Hours per day 9}}:10 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Rate }}\frac{1}{3}:\frac{1}{4} \hfill \\ \end{gathered} \right\}::6:x$$
$$\eqalign{ & \therefore \,\,\left( {5 \times 9 \times \frac{1}{3} \times x} \right) = \left( {8 \times 10 \times \frac{1}{4} \times 6} \right) \cr & \Leftrightarrow 15x = 120 \cr & \Leftrightarrow x = 8 \cr}$$

Explanation:-

Let the required time be x seconds
Then,
More metres, More time (direct proportion)
$$\eqalign{ & \therefore 0.128:25::1:x \cr & \Leftrightarrow x = \frac{{25}}{{0.128}} \cr & \Leftrightarrow x = \frac{{25 \times 1000}}{{128}} \cr & \Leftrightarrow x = 195.31 \cr}$$ `
∴ Required time =195 seconds (approx)

Explanation:-

Let the required price be Rs. x
More mangoes, More price (Direct proportion)
$$\eqalign{ & \therefore 357:\left( {49 \times 12} \right)::1517.25:x \cr & \Leftrightarrow 357x = \left( {49 \times 12 \times 1517.25} \right) \cr & \Leftrightarrow x = \frac{{\left( {49 \times 12 \times 1517.25} \right)}}{{357}} \cr & \Leftrightarrow x = 2499 \cr}$$
Hence, the approximate price is Rs. 2500

Explanation:-

$$\eqalign{ & {\text{Cost of }}x{\text{ metres}} \cr & = {\text{Rs}}{\text{. }}d \cr & {\text{Coast of 1 metre}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{d}{x}} \right) \cr & {\text{Cost of }}y{\text{ metres}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{d}{x} \times y} \right) \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{yd}}{x}} \right) \cr}$$

Explanation:-

Let the required number of bottles be x.
More weavers, More mats (Direct Proportion)
More days, More mats (Direct Proportion)
$$\left. \begin{gathered} {\text{Wavers}}\,\,\,\,4:8 \hfill \\ \,\,\,\,\,\,{\text{Days}}\,\,\,\,4:8 \hfill \\ \end{gathered} \right\}::4:x$$
$$\eqalign{ & \therefore 4 \times 4 \times x = 8 \times 8 \times 4 \cr & \Rightarrow x = \frac{{\left( {8 \times 8 \times 4} \right)}}{{4 \times 4}} \cr & \Rightarrow x = 7 \cr}$$

Explanation:-

Le the required time be x seconds.
More metres, More time (Direct Proportion)
$$\eqalign{ & \therefore 0.128:25::1:x\,\,\, \Leftrightarrow \,\,\,0.128x = 25 \times 1 \cr & x = \frac{{25}}{{0.128}} = \frac{{25 \times 1000}}{{128}} \cr & \Rightarrow x = 195.31 \cr}$$
∴ Required time = 195 sec (approximately)