General Idea:
The relationship between speed, distance and time is given below:
[tex] Speed \; = \frac{Distance}{Time} \\ \\ Distance \; =Speed \times Time\\ \\ Time =\frac{Distance}{Speed} [/tex]
Upstream Speed = Speed of Boat - Speed of Current
Downstream Speed = Speed of Boat + Speed of Current
Speed against the wind= Speed of Plane - Speed of Wind
Speed with the wind = Speed of Plane + Speed of Wind
Applying the concept:
(i) We need to create a table filling the information given about speed, distance and time.
(ii) We need to assign variable for the unknown in the table
(iii) Then create an equation from the table.
(iv) Solve the equation. Solving means getting find the value of variable which makes the equation TRUE.
The table for all the problems and respective equations are attached.
Solution for Problem 1:
[tex] \frac{2}{x-6}=\frac{14}{x+6} \\ Cross \; Multiply \\\\ 2(x+6)=14(x-6) \\ Distribute\; in\; left\; and\; right\; side\; of\; the\; equation\\ \\ 2x+12=14x-84\\ Subtract \; 14x\; and\; 12\; on \; both\; sides\\ \\ 2x+12-84-14x=14x-84-12-14x\\ Combine \; Like \; Terms\; in\; right\; and\; left\; side\; of \; equation\\ \\ -12x=-96\\ Divide\; -12\; on\; both\; sides\\ \\ \frac{-12x}{-12} =\frac{-96}{-12} \\ Simplify\; fraction\; on\; both\; sides\\ \\ x=8 [/tex]
[tex] Time \; to\; travel\; upstream =\frac{Upstream \; Distance}{Upstream\; Speed} =\frac{16}{8-6} =16/2=8 \; hours\\ \\ Time \; to\; travel\; Downstream =\frac{Downstream \; Distance}{Downstream\; Speed} =\frac{16}{8+6} =16/14=1 \frac{1}{7}\; hours [/tex]
Solution to Problem 2:
[tex] \frac{140}{x}= \frac{500}{x+180} \\ Cross \; Multiply\\ \\ 140(x+180)=500x\\ Distribute \; 140 \; in \; left \; side \; of \; equation\\ \\ 140x+25200=500x\\ Subtract \; 140x \; on \; both \; sides\\ \\ 140x+25200-140x=500x-140x\\ Combine \; Like \; Terms\\ \\ 25200=360x\\ 360x=25200\\ Divide \; 360 \; on \; both \; sides\\ \\ \frac{360x}{360} =\frac{25200}{360} \\ Simplify \; fraction \; on \; both \; sides\\ \\ x=70 [/tex]
[tex] Speed \; of \; CAR=70 \; mph\\ \\ Speed \; of \; PLANE=70+180=250 \; mph [/tex]
Solution to Problem 3:
[tex] 8x+8(x+15)=920\\ Distribute \; 8\; in\; the\; left\; side\; of\; the\; equation\\ \\ 8x+8x+120=920\\ Combine\; Like\; Terms\; in\; left\; side\\ \\ 16x+120=920\\ Subtract\; 120\; on\; both\; sides\\ \\ 16x+120-120=920-120\\ Combine\; Like\; Terms\\ \\ 16x=800\\ Divide\; 16\; on\; both\; sides\\ \\ \frac{16x}{16}=\frac{800}{16} \\ Simplify \; fraction\; on\; both\; sides\\ \\ x=50 [/tex]
[tex] Speed\; of\; Train\; A=x=50\; mph\\ \\ Speed\; of\; Train\; B=x+15=65\; mph [/tex]
Solution to Problem 4:
[tex] \frac{440}{x-10} =\frac{495}{x+10} \\ Cross \; Multiply\\ \\ 440(x+10)=495(x-10)\\ Distribute\; 440 \; in \; the\; left\; side \; and \; 495\; in\; the\; right\; side\\ \\ 440x+4400=495x-4950\\ Subtract\; 4400\; and\; 495\; on\; both\; sides\\ \\ 440x+4400-4400-495x=495x-4950-495x-4400\\ Combine\; Like\; Terms\; on\; both\; sides\\ \\ -55x=-9350\\ Divide\; -55\; on\; both\; sides\\ \\ \frac{-55x}{-55}=\frac{-9350}{-55}\\ Simplifying\; fraction\; on\; both\; sides\\ \\ x=170 [/tex]
[tex] The \; Speed \; of\; Plane\; in\; Still \; Air=170\; mph [/tex]
