Linear Equations in One Variable Test

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Linear Equations in One Variable Test - 14

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Weekly Quiz Competition

Question 1
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The perimeter of a rectangle is 28 cm. If the length and breadth of the rectangle are increased by 4 cm each, then the ratio of the breadth and the length becomes 5 : 6. What is the area of the rectangle?

A
24 cm²

B
36 cm²

C
45 cm²

D
48 cm²

Solution

We know that perimeter of a rectangle = 2 × (Length + Breadth).

∴ 2 × (Length + Breadth) = 28 cm

⇒ Length + Breadth

⇒ Length + Breadth = 14 cm

Let the length of the rectangle be l cm.

∴ Breadth = (14 − l) cm

Therefore, according to the given information, we get:

∴ Length, l = 8 cm

Breadth, (14 − l) = (14 − 8) cm = 6 cm

Thus, the area of the rectangle is (8 cm × 6 cm) = 48 cm².

Hence, the correct answer is option D.
Question 2
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Use the following information to answer the next question.

A motorboat covers a certain distance downstream in a river in 4 hours. It covers the same distance upstream in 5 hours. The speed of the motorboat in still water is 27 km/h.

What is the speed of the stream?

A
1.5 km/h

B
2 km/h

C
2.5 km/h

D
3 km/h

Solution

Let x be the speed of the stream.

∴ Speed of motorboat upstream = (27 − x) km/h

Speed of motorboat downstream = (27 + x) km/h

Distance covered upstream in 5 hours = 5 (27 − x) km

Distance covered downstream in 4 hours = 4 (27 + x) km

∴ 5 (27 − x) = 4 (27 + x)

⇒ 135 − 5x = 108 + 4x

⇒135 − 108 = 5x + 4x

⇒ 9x = 27

∴ x = 3

Thus, speed of the stream is 3 km/h.

The correct answer is D.
Question 3
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Use the following information to answer the next question.

The sum of the digits of a two digit number is 9. If the digits are interchanged, then the resulting number is 9 less than the original number.

What is the original number?

A
63

B
54

C
45

D
36

Solution

Let the digit in the unit's place be a.

Therefore, the digit in the ten's place is (9 − a).

∴ Original number = 10 (9 − a) + a

The number obtained by interchanging its digits is 10a + (9 − a).

Thus, the required number is 10(9 − a) + a = 10(9 − 4) + 4 = 54.

The correct answer is B.
Question 4
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Use the following information to answer the next question.

The present age of Rahim and Ram are in the ratio 11: 6. Six years ago, Rahim was twice as old as Ram.

What is the present age of Ram?

A
34 years

B
36 years

C
46 years

D
48 years

Solution

The present age of Rahim and Ram are in the ratio 11: 6.

Then, let the present age of Rahim and Ram be 11x and 6x respectively.

Six years ago, Rahim’s age was (11x − 6) and Ram’s age was (6x − 6).

∴ 11x − 6 = 2 (6x − 6)

⇒ 11x − 6 = 12x − 12

⇒ 12 − 6 = 12x − 11x

⇒ x = 6

∴ Ram’s present age = 6x = 6 × 6 = 36

Thus, Ram’s present age is 36 years.

The correct answer is B.
Question 5
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Use the following information to answer the next question.

The numerator of a fraction is greater than its denominator by 4. If the numerator is decreased by 6 and the denominator is increased by 3, then the fraction becomes.

What is the fraction?

A

B

C

D

Solution

Let x be the denominator of the fraction.

Therefore, the numerator of the fraction is (x + 4).

∴ Fraction =

Now, if the numerator is decreased by 6 and the denominator is increased by 3, then the fraction becomes.

∴ Denominator, x = 17

Numerator, x + 4 = 17 + 4 = 21

Thus, the required fraction is.

The correct answer is B.
Question 6
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Use the following information to answer the next question.

The present ages of Soni and Mani are in the ratio 6:7. Five years ago, the ratio of their ages was 11:13.

What is the mean of the ages of Soni and Mani?

A
60 years

B
65 years

C
73 years

D
78 years

Solution

The present ages of Soni and Mani are in the ratio 6: 7.

Then, let their ages be 6x and 7x respectively.

Five years ago, the ratio of their ages was 11:13.

By cross multiplication, we get:

13 (6x − 5) = 11 (7x − 5)

⇒ 78x − 65 = 77x − 55

⇒ 78x − 77x = −55 + 65

⇒ x = 10

Therefore, present age of Soni, 6x = 6 × 10 = 60 years

Present age of Mani, 7x = 7 × 10 = 70 years

Thus, the mean of their ages is

The correct answer is B.

Question 7

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Use the following information to answer the next question.

Manish was given his weekly allowance on Monday. Out of this amount, he spent Rs 20 on Monday. He spent half the remaining amount on Tuesday. On Wednesday, he spent half the amount that he had. After this, he was left with Rs 45.

If Manish’s weekly allowance be taken as x, then which equation correctly represents the given situation?

Solution

Amount with Manish = x

Amount spent by him on Monday = Rs 20

∴ Amount left with him = x − Rs 20

Amount spent by him on Tuesday

∴Amount left with him

Amount spent by him on Wednesday

∴ Amount left with him

According to the given information, he is now left with Rs 45.

Thus, the equation correctly represents the given situation.

The correct answer is D.

Question 8
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Use the following information to answer the next question.

Three numbers x, y, and z are such that x and y are in the ratio 4:5 and z is 5 less than x. The sum of the three numbers is 60.

If the value of y is taken as 5p, then which situation correctly represents the given information?

A
4p + 5p + 5p + 5 = 60

B
4p + 5p + 5p − 5 = 60

C
4p + 5p + 4p − 5 = 60

D
4p + 5p + 4p + 5 = 60

Solution

It is given that the numbers x and y are in the ratio 4:5 and the value of y is 5p.

∴ x = 4p

It is also given that z is 5 less than x.

∴ z = x − 5 = 4p − 5

The sum of the three numbers is 60.

∴ x + y + z = 60

⇒ 4p + 5p + 4p − 5 = 60

Thus, the equation 4p + 5p + 4p − 5 = 60 correctly represents the given situation.

The correct answer is C.
Question 9
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Use the following information to answer the next question.

A square and a rectangle have equal perimeters. Each side of the square measures 45 cm, while the length of the rectangle is 10 cm more than its breadth.

If the length of the rectangle is assumed as x, then which equation correctly represents the given situation?

A
2 × (x + x − 10 cm) = 180 cm

B
2 × (x + x + 10 cm) = 180 cm

C
4 × (x + x − 10 cm) = 180 cm

D
4 × (x + x + 10 cm) = 180 cm

Solution

Length of each side of the square = 45 cm

∴ Perimeter of the square = 4 × 45 cm = 180 cm

Length of the rectangle = x

∴ Width of the rectangle = x − 10 cm

∴Perimeter of the rectangle = 2 × (x + x − 10) cm

It is given that the perimeter of the square equals that of the rectangle.

Thus, the equation that correctly represents the given situation is

2 × (x + x − 10 cm) = 180 cm.

The correct answer is A.
Question 10
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What is the value of y in the equation 2y + 5 = 19?

A

B
7

C

D
12

Solution

The given equation is 2y + 5 = 19.

Subtracting 5 from both the sides, we get:

2y + 5 − 5 = 19 − 5

⇒ 2y = 14

Dividing both sides by 2, we get:

Thus, the value of y in the given equation is 7.

The correct answer is B.
Question 11
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What is the value of x in the equation?

A
11

B
12

C
13

D
14

Solution

The given equation is

Subtracting 11 from both the sides, we get:

Now, multiplying both sides with 2, we get:

Dividing both sides by 7, we get:

Thus, the value of x in the given equation is 12.

The correct answer is B.
Question 12
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What is the value of y in the equation?

A
2.3

B
3.9

C
4.7

D
5.2

Solution

The given equation is

⇒ y − 3.5 = 1.2

Adding 3.5 on both sides, we get:

y − 3.5 + 3.5 = 1.2 + 3.5

⇒ y = 4.7

Thus, the value of y in the given equation is 4.7.

The correct answer is C.
Question 13
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What is the value of a in the equation?

A

B

C
2

D
5

Solution

The given equation is

Dividing both sides by 1.5, we get:

Thus, the value of a in the given equation is

The correct answer is A.