NCERT Exemplar Solutions Class 10 Mathematics Solutions for Pair of Linear Equations in Two Variables - Exercise 3.4 in Chapter 3 - Pair of Linear Equations in Two Variables
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A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.
Let speed of boat in still water = x km/hr
and the speed of the stream = y km/hr
Speed of motor boat upstream = (x – y) km/hr
Speed of motor boat downstream = (x + y) km/hr
Case I: Time taken by motor boat in 30 km upstream = 30/(x-y) hr
Time taken by motor boat in 28 km downstream = 28/(x+y) hr
\therefore \frac{30}{(x-y)}+\frac{28}{(x+y)}=7\\ \Rightarrow 2\left[\frac{15}{(x-y)}+\frac{14}{(x+y)}\right]=7\\ \Rightarrow \frac{15}{x-y}+\frac{14}{x+y}=\frac{7}{2}
Case II: Time taken by motor boat in 21 km upstream = 21/(x-y) hr
Time taken by motor boat to return 21 km downstream = 21/(x+y) hr
\therefore=\frac{21}{x-y}+\frac{21}{x+y}=5\\ \Rightarrow 21\left[\frac{1}{x-y}+\frac{1}{x+y}\right]=5\\ \Rightarrow \frac{1}{x-y}+\frac{1}{x+y}=\frac{5}{21} \ldots from \ (i)\\ \frac{15}{x-y}+\frac{14}{x+y}=\frac{7}{2} \text { [From (i)] }
As equations (both) are symmetric to (x – y) and (x + y) so we can eliminate either (x–y) or (x + y).
Multiplying (ii) by 14, we get
\frac{14}{(x-y)}+\frac{14}{(x+y)}=\frac{70}{21} \ \ ...(iii)\\ \frac{15}{(x-y)}+\frac{14}{x+y}=\frac{7}{2} \ \ \text { [From (i) }
\frac{14}{(x-y)}-\frac{15}{(x-y)}=\frac{10}{3}-\frac{7}{2}[\text { Subtracting } \text { (i) from (iii)] }\\ \Rightarrow \frac{14-15}{(x-y)}=\frac{20-7 \times 3}{3 \times 2}\\ \Rightarrow \frac{-1}{(x-y)}=\frac{-1}{6} \ldots(\mathrm{iv})\\ \Rightarrow(x-y)=6
Now, substituting x – y = 6 in (ii), we have
\frac{1}{(x-y)}+\frac{1}{(x+y)}=\frac{5}{21} \\ \Rightarrow \frac{1}{6}+\frac{1}{(x+y)}=\frac{5}{21} \\ \Rightarrow \frac{1}{(x+y)}=\frac{5}{21}-\frac{1}{6} \\ \Rightarrow \frac{1}{(x-y)}=\frac{2 \times 5-7 \times 1}{3 \times 7 \times 2} \\ \Rightarrow \frac{1}{(x+y)}=\frac{3}{42} \\ \Rightarrow \frac{1}{(x+y)}=\frac{1}{14}
⇒ x + y= 14 ....(v)
x - y = 6 (From (iv))
2x = 20
⇒ x = 10 km/hr
Now, x + y = 14 (From (v))
⇒ 10 + y = 14
⇒ y = 4 km/hr
Hence, the speed of motorboat and stream are 10km/hr and 4 km/hr
respectively.
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