A motorboat can travel 30 km upstream and 28 km downstream in 7 h. It can travel 21 km upstream and return in 5 h. Find the speed of the boat in still water and the speed of the stream.

Let the speed of the motorboat in still water and the speed of the stream are u k m / h and v k m / h , respectively. Then a motorboat speed in downstream = u + v k m / h and a motorboat speed in upstream = u - v k m / h motorboat has taken time to travel 30 km upstream, t 1 = 30 u - v h and motorboat has taken time to travel 28 km downstream, t 2 = 28 u + v h By first condition, a motorboat can travel 30 km upstream and 28 km downstream in 7 h i.e., t 1 + t 2 = 7 h ⇒ 30 u - v + 28 u + v = 7 . . . ( i ) Now, motorboat has taken time to travel 21 km upstream and return i.e., t 3 = 21 u - v [for upstream] and t 4 = 21 u + v [for downstream] By second condition, t 4 + t 3 = 5 h ⇒ 21 u - v + 21 u + v = 5 . . . ( i i ) Let x = 1 u - v and y = 1 u + v Eqs. ( i ) and ( i i ) becomes 30 x + 28 y = 7 . . . ( i i i ) and 21 x + 21 y = 5 ⇒ x + y = 5 21 . . . ( i v ) Now, multiplying in Eq. ( i v ) by 28 and then subtracting from Eq. i i i , we get 30 x + 28 y = 7 28 x − + 28 y − = 140 21 − ¯ 2 x = 7 − 20 3 = 21 − 20 3 ⇒ 2 x = 1 3 ⇒ x = 1 6 On putting the value of x in Eq. i v , we get 1 6 + y = 5 21 ⇒ y = 5 21 − 1 6 = 10 − 7 42 = 3 42 ⇒ y = 1 14 ∴ x = 1 u - v = 1 6 ⇒ u - v = 6 . . . ( v ) and y = 1 u + v = 1 14 ⇒ u + v = 14 Now, adding Eqs. v and v i , we get 2 u = 20 ⇒ u = 10 On putting the value of u in Eq. v , we get 10 + v = 6 ⇒ v = - 4 Hence, the speed of the motorboat in still water is 10 k m / h and the speed of the stream 4 km/h.

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10th CBSE

IMPORTANT

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A person, rowing at the rate of $5$ km/h in still water, takes thrice as much time in going $40$ km upstream as in going $40$ km downstream. Find the speed of the stream.

Important Questions on Pair of Linear Equations in Two Variables

HARD

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 3 - Pair of Linear Equations in Two Variables>Exercise>Q 8

A motorboat can travel

30

km upstream and

28

km downstream in

7

h. It can travel

∴ v = \frac{10}{4} = 2 .5 k m / h

km upstream and return in

5

h. Find the speed of the boat in still water and the speed of the stream.

HARD

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 3 - Pair of Linear Equations in Two Variables>Exercise>Q 9

A two-digit number is obtained by either multiplying the sum of the digits by $8$ and then subtracting $5$ or by multiplying the difference of the digits by $16$ and then adding $3$ . Find the number.

HARD

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 3 - Pair of Linear Equations in Two Variables>Exercise>Q 10

A railway half ticket cost half the full fare but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from the stations

A

and

B

costs Rs.

2530

. Also, one reserved first class ticket and one reserved first class half ticket from stations

A

B

costs Rs.

3810

. Find the full first class fare from stations

A

and

B

and also the reservation charges for a ticket.

HARD

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 3 - Pair of Linear Equations in Two Variables>Exercise>Q 11

A shopkeeper sells a saree at $8 %$ profit and a sweater at $10 %$ discount, thereby, getting a sum Rs. $1008$ . If she had sold the saree at $10 %$ profit and the sweater at $8 %$ discount, she would have got Rs. $1028$ then find the cost of the saree and the list price (price before discounted) of the sweater.

HARD

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 3 - Pair of Linear Equations in Two Variables>Exercise>Q 12

Susan invented certain amount of money in two schemes $A$ and $B$ , which offer interest at the rate of $8 %$ per annum and $9 %$ per annum, respectively. She received Rs. $1860$ as annual interest. However, had she interchanged the amount of investments in the two schemes, she would have received Rs. $20$ more as annual interest. How much money did she invest in each scheme?

HARD

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 3 - Pair of Linear Equations in Two Variables>Exercise>Q 13

Vijay had some bananas and he divided them into two lots $A$ and $B$ . He sold the first lot at the rate of Rs. $2$ for $3$ bananas and the second lot at the rate of Rs. $1$ per banana and got a total of Rs. $400$ . If he had sold the first lot at the rate of Rs. $1$ per banana and the second lot at the rate of Rs. $4$ for $5$ bananas, his total collection would have been Rs. $460$ . Find the total number of bananas he had.

A person, rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km upstream as in going 40 km downstream. Find the speed of the stream.

Important Questions on Pair of Linear Equations in Two Variables

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A person, rowing at the rate of $5$ km/h in still water, takes thrice as much time in going $40$ km upstream as in going $40$ km downstream. Find the speed of the stream.