A boat goes $30 ext{ km}$ upstream and $44 ext{ km}$ downstream in $10 ext{ hours}$. In $13 ext{ hours}$, it can go

Question

A boat goes $30	ext{ km}$ upstream and $44	ext{ km}$ downstream in $10	ext{ hours}$. In $13	ext{ hours}$, it can go $40	ext{ km}$ upstream and $55	ext{ km}$ downstream. Formulate the pair of linear equations and find the speed of the stream and that of the boat in still water.

Accepted Answer

Step-by-step Solution: Step 1: Define variables Let the speed of the boat in still water = $x	ext{ km/h}$ Let the speed of the stream = $y	ext{ km/h}$ 
Then, Speed of the boat upstream = $(x - y)	ext{ km/h}$ Speed of the boat downstream = $(x + y)	ext{ km/h}$ --- Step 2: Formulate equations using $	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}}$ Case 1: Time taken for $30	ext{ km}$ upstream + Time taken for $44	ext{ km}$ downstream = $10	ext{ hours}$ $$\frac{30}{x - y} + \frac{44}{x + y} = 10 \quad 	ext{--- (Equation 1)}$$ Case 2: Time taken for $40	ext{ km}$ upstream + Time taken for $55	ext{ km}$ downstream = $13	ext{ hours}$ $$\frac{40}{x - y} + \frac{55}{x + y} = 13 \quad 	ext{--- (Equation 2)}$$ --- Step 3: Reduce to linear equations
Let $\frac{1}{x - y} = u$ and $\frac{1}{x + y} = v$. 
Substituting $u$ and $v$ into Equations 1 and 2: $$30u + 44v = 10 \quad \implies \quad 15u + 22v = 5 \quad 	ext{--- (Equation 3)}$$ $$40u + 55v = 13 \quad 	ext{--- (Equation 4)}$$ --- Step 4: Solve the reduced linear equations
Multiply Equation 3 by $8$ and Equation 4 by $3$: $$120u + 176v = 40 \quad 	ext{--- (Equation 5)}$$ $$120u + 165v = 39 \quad 	ext{--- (Equation 6)}$$ 
Subtracting Equation 6 from Equation 5: $$(120u - 120u) + (176v - 165v) = 40 - 39$$ $$11v = 1 \implies v = \frac{1}{11}$$ 
Substitute $v = \frac{1}{11}$ in Equation 3: $$15u + 22\left(\frac{1}{11}ight) = 5$$ $$15u + 2 = 5 \implies 15u = 3 \implies u = \frac{3}{15} = \frac{1}{5}$$ --- Step 5: Find $x$ and $y$
Since $u = \frac{1}{x - y} = \frac{1}{5}$: $$x - y = 5 \quad 	ext{--- (Equation 7)}$$ 
Since $v = \frac{1}{x + y} = \frac{1}{11}$: $$x + y = 11 \quad 	ext{--- (Equation 8)}$$ 
Adding Equation 7 and Equation 8: $$(x - y) + (x + y) = 5 + 11$$ $$2x = 16 \implies x = 8$$ 
Substituting $x = 8$ in Equation 8: $$8 + y = 11 \implies y = 3$$ --- Final Answer: Speed of the boat in still water = $8	ext{ km/h}$ Speed of the stream = $3	ext{ km/h}$

Pair of Linear Equations in Two Variables

❓ Sibling Questions in this Chapter