[5 marks sum]

Question 15 Marks

If 2 is added to each of the two given numbers, then their ratio becomes 1 : 2. However, if 4 is subtracted from each of the given numbers, the ratio becomes 5 : 11. Find the numbers.

Answer

34, 70
[Hint. Let the given numbers be $x$ and $y$. Then, $\frac{x+2}{y+2}=\frac{1}{2}$ and $\frac{x-4}{y-4}=\frac{5}{11}$.]

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Question 25 Marks

Of the two numbers, 4 times the smaller one is less than 3 times the larger one by 6. Also, the sum of the numbers is larger than 6 times their difference by 5. Find the numbers.

Answer

62, 45

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Question 35 Marks

Find two numbers such that the sum of thrice the first and the second is 142 and four times the first exceeds the second by 138.

Answer

40, 22

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Question 45 Marks

A motorboat takes 6 hours to cover 100 km downstream and 30 km upstream. If the motorboat goes 75 km downstream and returns back to its starting point in 8 hours, find the speed of the motorboat in still water and the rate of the stream.

Answer

20 kmph, 5 kmph

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Question 55 Marks

The area of a rectangle gets reduced by 8 m², if its length is reduced by 5 m and breadth increased by 3 m. If we increase the length by 3 m and breadth by 2 m, the area is increased by 74 m². Find the length and breadth of the rectangle.

Answer

Length = 19 m, Breadth = 10 m

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Question 65 Marks

The sum of two numbers is 51. If the larger is doubled and the smaller is tripled, the difference is 12. Find the numbers.

Answer

33, 18

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Question 75 Marks

The length of a room exceeds its breadth by 3 metres. If the length is increased by 3 m and breadth is decreased by 2 metres, the area remains the same. Find the length and breadth of the room.

Answer

Length = 15 m, Breadth = 12 m

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Question 85 Marks

The present age of a man is 3 years more than three times the age of his son. Three years hence, the man's age will be 10 years more than twice the age of his son. Determine their present ages.

Answer

Father's age 33 years, Son's age 10 years

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Question 95 Marks

A and B each have a certain number of marbles. A says to B, "If you give 30 to me, I will have twice as many as left with you." B replies, "If you give me 10, I will have thrice as many as left with you." How many marbles does each have?

Answer

A = 34 marbles, B = 62 marbles

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Question 105 Marks

5 years ago, A was thrice as old as B and 10 years later, A shall be twice as old as B. What are the present ages of A and B?

Answer

50 years, 20 years

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Question 115 Marks

One year ago a man was four times as old as his son. After 6 years, his age exceeds twice his son's age by 9 years. Find their present ages.

Answer

33 years, 9 years

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Question 125 Marks

The sum of two numbers exceeds thrice the smaller by 2. If the difference between them is 19, find the numbers.

Answer

36, 17

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Question 135 Marks

The denominator of a fraction is greater than its numerator by 9 . If 7 is subtracted from both, its numerator and denominator, the fraction becomes $\frac{2}{3}$. Find the original fraction.

Answer

$\frac{25}{34}$
[Hint. Let the required fraction be ( $x / y$ ). Then, $y=x+9$ and $\frac{x-7}{y-7}=\frac{2}{3}$,]

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Question 145 Marks

The difference between two numbers is 12 and the difference between their squares is 456. Find the numbers.

Answer

25, 13
[Hint. $x-y=12$ and $\left.x^2-y^2=456 \Rightarrow x+y=\frac{x^2-y^2}{x-y}=\frac{456}{12}=38.\right]$

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Question 155 Marks

Tanusha went to a bank to withdraw money. She asked the cashier to give her ₹100 and ₹ 500 notes only. The cashier agreed. Tanusha got x, 100-rupee notes and y, 500-rupee notes.
Based on the above information, answer the following questions.
Q.1. If Tanusha withdrew ₹15,000, then the above information can be represented by the linear equation:
(a) x + 5y = 150 (b) 5x + y = 150 (с) x + 5y + 150 = 0 (d) x + y = 150
Q.2. If she got 54 notes in all, then the above information can be represented by the linear equation:
(a) 100x + 500y = 54 (b) x + y = 54 (c) 500x + 100y = 54 (d) 100x + y = 54
Q.3. If Tanusha withdraws ₹ 16,000, then which of the following combination of notes she may get?
(a) ₹500 notes - 30, ₹100 notes - 20
(b) ₹500 notes - 25, ₹100 notes - 25
(c) ₹500 notes - 20, ₹100 notes - 30
(d) ₹500 notes - 30, ₹100 notes -10
Q.4. If she gets twenty 500-rupee notes and twenty five 100-rupee notes, then the amount she withdraws is:
(a) ₹10000 (b) ₹11000 (c) ₹12000 (d) ₹12500
Q.5. Can Tanusha withdraw ₹10050, under the given conditions?
(a) yes (b) no (c) can't say anything (d) none of these

Answer

1. (a) 2. (b) 3. (d) 4. (d) 5. (b)

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Question 165 Marks

There are two mobile phone companies - P and Q, that offer different plans. Company P charges a monthly fee of ₹40 plus ₹0.5 per minute talk time. Company Q charges a monthly service fee of ₹30 plus 1 per minute talk time.

Based on the above information, answer the following questions:
Q.1. The linear equation which expresses the plan of company P is:
(a) y = 0.5x + 40 (b) y = 40x + 0.5 (c) y = x + 40.5 (d) y = 40 - 0.5x
Q.2. The linear equation which expresses the plan of company Q is:
(a) y = 30x + 1 (b) y = x + 30 (c) y = x - 30 (d) y = 30x - 1
Q.3. How many minutes of talk time would yield equal expenditure from both the companies?
(a) 10 minutes (b) 15 minutes (c) 20 minutes. (d) 25 minutes
Q.4. Manisha took the plan of company P and used 400 minutes of talk time. She spent:
(a) ₹240 (b) ₹430 (c) ₹220 (d) ₹215
Q.5. If in a month, Anurag wants to use only 300 minutes of talk time, then which company's plan is better for him?
(a) Company P
(b) Company Q
(c) Both offers the same plan
(d) Can't be determined

Answer

1. (a) 2. (b) 3. (c) 4. (a) 5. (a)

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Question 175 Marks

Ritesh bought a new well furnished two bedroom flat in a society. The layout of the flat is shown in the figure alongside. The builder claims that the areas of the two bedrooms and the kitchen together is 95 square metres. All the dimensions in the figure are in meter (m). Study the above information and answer the following questions.

Q.1.Which of the following pair of linear equations represent the given situation?
(a) x + y = 19, 2x + y = 13 (b) x + y = 13, 2x + y = 19
(c) x - y = 19, 2x - y = 13 (d) x + y = 13, 2x - y = 19
Q.2. The perimeter of the outer boundary of the layout is:
(a) 54 m (b) 27 m (c) 50 m (d) 52 m
Q.3. Total area of bedroom 1 and kitchen is:
(a) $60 m^2$ (b) $70 m^2$ (c) $65 m^2$ (d) $95 m^2$
Q.4. The area of the living room is:
(a) $50 m^2$ (b) $60 m^2$ (c) $70 m^2$ (d) $75 m^2$
Q.5. The cost of laying tiles on the floor of the kitchen at the rate of ₹200 per sq mis:
(a) ₹7000 (b) ₹6000 (c) ₹5200 (d) ₹5000

Answer

1. (b) 2. (a) 3. (c) 4. (d) 5. (a)

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Question 185 Marks

Case Study III: Tanusha went to a bank to withdraw money. She asked the cashier to give her ₹100 and ₹ 500 notes only. The cashier agreed. Tanusha got x, 100-rupee notes and y, 500-rupee notes. Based on the above information, answer the following questions.
1. If Tanusha withdrew ₹ 15,000, then the above information can be represented by the linear equation:
(a) x + 5y = 150
(b) 5x + y = 150
(c) x + 5y + 150 = 0
(d) x + y = 150
2. If she got 54 notes in all, then the above information can be represented by the linear equation:
(a) 100x + 500y = 54
(b) x + y = 54
(c) 500x + 100y = 54
(d) 100x + y = 54
3. If Tanusha withdraws ₹ 16,000, then which of the following combination of notes she may get?
(a) ₹ 500 notes - 30, ₹ 100 notes - 20
(b) ₹ 500 notes - 25, ₹ 100 notes - 25
(c) ₹ 500 notes - 20, ₹ 100 notes - 30
(d) ₹ 500 notes - 30, ₹ 100 notes - 10
4. If she gets twenty 500-rupee notes and twenty five 100-rupee notes, then the amount she withdraws is:
(a) ₹ 10000 (b) ₹ 11000 (c) ₹ 12000 (d) ₹ 12500
5. Can Tanusha withdraw ₹ 10050, under the given conditions?
(a) yes
(b) no
(c) can't say anything
(d) none of these

Answer

Case Study III (1. a), (2. b), (3. d), (4. d), (5. b)

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Question 195 Marks

Case Study II : There are two mobile phone companies - P and Q, that offer different plans. Company P charges a monthly fee of ₹ 40 plus ₹ 0.5 per minute talk time. Company Q charges a monthly service fee of ₹ 30 plus ₹ 1 per minute talk time.
Based on the above information, answer the following questions:
1. The linear equation which expresses the plan of company P is:
(a) y = 0.5x + 40 (b) y = 40x + 0.5 (c) y = x + 40.5 (d) y = 40 - 0.5x
2. The linear equation which expresses the plan of company Q is:
(a) y = 30x + 1 (b) y = x + 30 (c) y = x - 30 (d) y = 30x - 1
3. How many minutes of talk time would yield equal expenditure from both the companies?
(a) 10 minutes (b) 15 minutes (c) 20 minutes (d) 25 minutes
4. Manisha took the plan of company P and used 400 minutes of talk time. She spent:
(a) ₹ 240 (b) ₹ 430 (c) ₹ 220 (d) ₹ 215
5. If in a month, Anurag wants to use only 300 minutes of talk time, then which company's plan is better for him?
(a) Company P (b) Company Q
(c) Both offers the same plan (d) Can't be determined

Answer

Case Study II (1. a), (2. b), (3. c), (4. a), (5. a)

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Question 205 Marks

Case Study I: Ritesh bought a new well furnished two bedroom flat in a society. The layout of the flat is shown in the figure alongside. The builder claims that the areas of the two bedrooms and the kitchen together is 95 square metres. All the dimensions in the figure are in meter (m). Study the above information and answer the following questions.
1. Which of the following pair of linear equations represent the given situation?
(a) x + y = 19, 2x + y = 13
(b) x + y = 13, 2x + y = 19
(c) x - y = 19, 2x - y = 13
(d) x + y = 13, 2x - y = 19
2. The perimeter of the outer boundary of the layout is :
(a) 54 m (b) 27 m (c) 50 m (d) 52 m
3. Total area of bedroom 1 and kitchen is :
(a) $60 m^2$ (b) $70 m^2$ (c) $65 m^2$ (d) $95 m^2$
4. The area of the living room is :
(a) $50 m^2$ (b) $60 m^2$ (c) $70 m^2$ (d) $75 m^2$
5. The cost of laying tiles on the floor of the kitchen at the rate of ₹ 200 per sq m is :
(a) ₹ 7000 (b) ₹ 6000 (c) ₹ 5200 (d) ₹ 5000

Answer

Case Study I (1. b), (2. a), (3. c), (4. d), (5. a)

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Question 215 Marks

Answer

20 kmph, 5 kmph

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Question 225 Marks

Answer

Length = 19 m, Breadth = 10 m

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Question 235 Marks

The length of a room exceeds its breadth by 3 metres. If the length is increased by 3 m and breadth is decreased by 2 metres, the area remains the same. Find the length and breadth of the room.

Answer

Length = 15 m, Breadth = 12 m

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Question 245 Marks

The present age of a man is 3 years more than three times the age of his son. Three years hence, the man's age will be 10 years more than twice the age of his son. Determine their present ages.

Answer

Father's age 33 years, Son's age 10 years

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Question 255 Marks

Answer

A = 34 marbles, B = 62 marbles

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Question 265 Marks

There are two examination halls A and B. If 12 pupils are sent from A to B, the number of pupils in each room becomes the same. If 11 pupils are sent from room B to room A, then the number of pupils in A is double their number in B. Find the number of pupils in each room.

Answer

A = 81 pupils, B = 57 pupils

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Question 275 Marks

5 years ago, A was thrice as old as B and 10 years later, A shall be twice as old as B. What are the present ages of A and B?

Answer

50 years, 20 years

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Question 285 Marks

One year ago a man was four times as old as his son. After 6 years, his age exceeds twice his son's age by 9 years. Find their present ages.

Answer

33 years, 9 years

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Question 295 Marks

The denominator of a fraction is greater than its numerator by 9 . If 7 is subtracted from both, its numerator and denominator, the fraction becomes $\frac{2}{3}$. Find the original fraction.
[Hint. Let the required fraction be $(x / y)$. Then, $y=x+9$ and $\frac{x-7}{y-7}=\frac{2}{3}$.]

Answer

$\frac{25}{34}$

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Question 305 Marks

The difference between two numbers is 12 and the difference between their squares is 456. Find the numbers.
[Hint. $x-y=12$ and $x^2-y^2=456 \Rightarrow x+y=\frac{x^2-y^2}{x-y}=\frac{456}{12}=38$.]

Answer

25, 13

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Question 315 Marks

If 2 is added to each of the two given numbers, then their ratio becomes 1 : 2. However, if 4 is subtracted from each of the given numbers, the ratio becomes 5 : 11. Find the numbers.
[Hint. Let the given numbers be $x$ and $y$. Then, $\frac{x+2}{y+2}=\frac{1}{2}$ and $\frac{x-4}{y-4}=\frac{5}{11}$.]

Answer

34, 70

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Question 325 Marks

Of the two numbers, 4 times the smaller one is less than 3 times the larger one by 6. Also, the sum of the numbers is larger than 6 times their difference by 5. Find the numbers.

Answer

62, 45

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Question 335 Marks

Find two numbers such that the sum of twice the first and thrice the second is 103 and four times the first exceeds seven times the second by 11.

Answer

29, 15

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Question 345 Marks

The sum of two numbers is 51. If the larger is doubled and the smaller is tripled, the difference is 12. Find the numbers.

Answer

33, 18