Chain Rules

Let the required quantity be×kg.
More workers, More quantity (Direct proportion)
More days, More quantity (Direct proportion)
\[\left. \begin{gathered} {\text{Workers 5}}:8 \hfill \\ \,\,\,\,\,\,\,\,{\text{Days 3}}:5 \hfill \\ \end{gathered} \right\}::60:x\]
$$\therefore {\text{ }}5 \times 3 \times×= 8 \times 5 \times 60$$
$$ ⇒ x = \frac{{\left( {8 \times 5 \times 60} \right)}}{{\left( {5 \times 3} \right)}}$$
$$ ⇒ x = 160$$

Let the required number of days be x
Less people, More days (Indirect proportion)
Less quantity, Less days ( Direct proportion)
\[\left. \begin{gathered} \,\,\,\,\,\,\,{\text{People 35}}:50 \hfill \\ {\text{Quantity 350}}:50 \hfill \\ \end{gathered} \right\}::30:x\]
$$\therefore {\text{ }}35 \times 350 \times×= 50 \times 50 \times 30$$
$$ ⇒ x = \frac{{\left( {50 \times 50 \times 30} \right)}}{{\left( {35 \times 350} \right)}}$$
$$ ⇒ x = \frac{{300}}{{49}}$$
$$ ⇒ x = 6\frac{6}{{49}}$$

Let the required number of days be×
Less working hours, More days (Indirect Proportion)
More men, Less days (Indirect Proportion)
\[\left. \begin{gathered} {\text{Working hours 4}}:8 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Men 18}}:12 \hfill \\ \end{gathered} \right\}::30:x\]
$$\therefore {\text{ }}4 \times 18 \times×= 8 \times 12 \times 30$$
$$ ⇒ x = \frac{{\left( {8 \times 12 \times 30} \right)}}{{\left( {4 \times 18} \right)}}$$
$$ ⇒ x = 40$$

Let's working hours per day be x
More persons, Less working hours ( Indirect proportion)
Less days, More working hours (Indirect proportion)
\[\left. \begin{gathered} {\text{Persons 7}}:5 \hfill \\ {\text{Quantity 4}}:8 \hfill \\ \end{gathered} \right\}::7:x\]
$$\therefore {\text{ }}7 \times 4 \times×= 5 \times 8 \times 7$$
$$ ⇒ x = \frac{{\left( {5 \times 8 \times 7} \right)}}{{\left( {7 \times 4} \right)}}$$
$$ ⇒ x = 10$$

Let's working hours per day be x
More persons, Less working hours ( Indirect proportion)
Less days, More working hours (Indirect proportion)
\[\left. \begin{gathered} {\text{Persons 7}}:4 \hfill \\ {\text{Quantity 4}}:8 \hfill \\ \end{gathered} \right\}::7:x\]
$$\therefore {\text{ }}7 \times 4 \times×= 4 \times 8 \times 7$$
$$ ⇒ x = \frac{{\left( {4 \times 8 \times 7} \right)}}{{\left( {7 \times 4} \right)}}$$
$$ ⇒ x = 8$$

Initially, Let there be×men having food for y days
After 10 days,×men had food for days (y - 10)
Also, $$\left( {x - \frac{x}{5}} \right)$$   men had food for y days
$$\therefore \,x\left( {y - 10} \right) = \frac{{4x}}{5} \times y$$
$$ ⇒ 5xy - 50x = 4xy$$
$$ ⇒ xy - 50x = 0$$
$$ ⇒ x\left( {y - 50} \right) = 0$$
$$ ⇒ y - 50 = 0$$
$$ ⇒ y = 50$$

Let there be×men originally
So,×men had provisions 40 days whereas (x + 500) men consumed it in 35 days
More men, Less days (Indirect proportion)
$$\therefore \,\left( {x + 500} \right):x::40:35$$
$$ ⇒ 35 \times \left( {x + 500} \right) = 40x$$
$$ ⇒ 5x = 35 \times 500$$
$$ ⇒ x = \left( {\frac{{35 \times 500}}{5}} \right)$$
$$ ⇒ x = 3500$$

Let the team take×days to finish 360 pieces
Then, number of pieces made each day = $$\frac{{360}}{x}$$
More number of pieces per day, Less days (Indirect proportion)
$$\therefore \,\left( {\frac{{360}}{x} + 4} \right):\frac{{360}}{x}::x:\left( {x - 1} \right)$$
$$ ⇒ \left( {\frac{{360}}{x} + 4} \right) \left( {x - 1} \right) = \frac{{360}}{x} \times× $$
$$ ⇒ 360 - \frac{{360}}{x} + 4x - 4 = 360$$
$$ ⇒ 4x - \frac{{360}}{x} - 4 = 0$$
$$ ⇒×- \frac{{90}}{x} - 1 = 0$$
$$ ⇒ {x^2} -×- 90 = 0$$
$$ ⇒ \left( {x - 10} \right)\left( {x + 9} \right) = 0$$
$$ ⇒ x = 10$$

Ratio of time taken by a woman, a man and a boy
$$ = 8:6:12$$
$$ = 4:3:6$$
So, 4 women ≡ 3 men ≡ 6 boy
(12 mens + 12 womens + 12 boys)
$$ = \left[ {12 + \left( {\frac{3}{4} \times 12} \right) + \left( {\frac{3}{6} \times 12} \right)} \right]{\text{men}}$$
$${\text{ = }}\left( {12 + 9 + 6} \right){\text{men}}$$
$$ = 27{\text{ men}}$$
Let the required number of days be x
More men, Less days (Indirect proportion)
More working hours, Less days (Indirect proportion)
\[\left. \begin{gathered} {\text{Working hours 8}}:6 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Men 27}}:9 \hfill \\ \end{gathered} \right\}::6:x\]
$$\therefore \,27 \times 8 \times×= 9 \times 6 \times 6$$
$$ ⇒ x = \frac{{\left( {9 \times 6 \times 6} \right)}}{{\left( {27 \times 8} \right)}}$$
$$ ⇒ x = \frac{3}{2}$$
$$ ⇒ x = 1\frac{1}{2}$$

1 man ≡ 2 boys ⇔ (12 men + 18 boys)
          ≡ (12 × 2 ×18) boys = 42 boys
Let required number of boys = x
⇒ (21 men +×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)
\[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 50}}:60 \hfill \\ {\text{Hours per day 9}}:\frac{{15}}{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Work }}1:2 \hfill \\ \end{gathered} \right\}::42:\left( {42 + x} \right)\]
$$\therefore \left[ {50 \times 9 \times 1 \times \left( {42 + x} \right)} \right] = $$     $$\left( {60 \times \frac{{15}}{2} \times 2 \times 42} \right)$$
$$ ⇒ \left( {42 + x} \right) = \frac{{37800}}{{450}}$$
$$ ⇒ 42 +×= 84$$
$$ ⇒ x = 42$$