MP Board · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesA boat goes $30\text{ km}$ upstream and $44\text{ km}$ downstream in $10\text{ hours}$. In $13\text{ hours}$, it can go $40\text{ km}$ upstream and $55\text{ km}$ downstream. Formulate the pair of linear equations and find the speed of the stream and that of the boat in still water.

Step-by-step Solution:

Step 1: Define variables

• Let the speed of the boat in still water = $x\text{ km/h}$
• Let the speed of the stream = $y\text{ km/h}$ \nThen,
• Speed of the boat upstream = $(x - y)\text{ km/h}$
• Speed of the boat downstream = $(x + y)\text{ km/h}$

Step 2: Formulate equations using $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

• Case 1: Time taken for $30\text{ km}$ upstream + Time taken for $44\text{ km}$ downstream = $10\text{ hours}$ $$\frac{30}{x - y} + \frac{44}{x + y} = 10 \quad \text{--- (Equation 1)}$$

• Case 2: Time taken for $40\text{ km}$ upstream + Time taken for $55\text{ km}$ downstream = $13\text{ hours}$ $$\frac{40}{x - y} + \frac{55}{x + y} = 13 \quad \text{--- (Equation 2)}$$

Step 3: Reduce to linear equations\nLet $\frac{1}{x - y} = u$ and $\frac{1}{x + y} = v$. \nSubstituting $u$ and $v$ into Equations 1 and 2: $$30u + 44v = 10 \quad \implies \quad 15u + 22v = 5 \quad \text{--- (Equation 3)}$$ $$40u + 55v = 13 \quad \text{--- (Equation 4)}$$

Step 4: Solve the reduced linear equations\nMultiply Equation 3 by $8$ and Equation 4 by $3$: $$120u + 176v = 40 \quad \text{--- (Equation 5)}$$ $$120u + 165v = 39 \quad \text{--- (Equation 6)}$$ \nSubtracting Equation 6 from Equation 5: $$(120u - 120u) + (176v - 165v) = 40 - 39$$ $$11v = 1 \implies v = \frac{1}{11}$$ \nSubstitute $v = \frac{1}{11}$ in Equation 3: $$15u + 22\left(\frac{1}{11}\right) = 5$$ $$15u + 2 = 5 \implies 15u = 3 \implies u = \frac{3}{15} = \frac{1}{5}$$

Step 5: Find $x$ and $y$\nSince $u = \frac{1}{x - y} = \frac{1}{5}$: $$x - y = 5 \quad \text{--- (Equation 7)}$$ \nSince $v = \frac{1}{x + y} = \frac{1}{11}$: $$x + y = 11 \quad \text{--- (Equation 8)}$$ \nAdding Equation 7 and Equation 8: $$(x - y) + (x + y) = 5 + 11$$ $$2x = 16 \implies x = 8$$ \nSubstituting $x = 8$ in Equation 8: $$8 + y = 11 \implies y = 3$$

Final Answer:

• Speed of the boat in still water = $8\text{ km/h}$
• Speed of the stream = $3\text{ km/h}$