Linear Equations - Exercise 14(C)

If 7x - 12 = 13 - 5x - 5, then the value of x is

1. $\dfrac{3}{4}$

2. $\dfrac{5}{3}$

3. $\dfrac{6}{5}$

4. $\dfrac{7}{4}$

Answer

Given:

7x - 12 = 13 - 5x - 5

⇒ 7x - 12 = 8 - 5x

⇒ 7x + 5x = 8 + 12

⇒ 12x = 20

⇒ x = $\dfrac{20}{12}$

⇒ x = $\dfrac{5}{3}$

Hence, option 2 is the correct option.

If 5(n - 3) = 4(n - 2), then the value of n is

1. 5
2. 6
3. 7
4. 8

Answer

Given:

5(n - 3) = 4(n - 2)

⇒ 5n - 15 = 4n - 8

⇒ 5n - 4n = 15 - 8

⇒ n = 7

Hence, option 3 is the correct option.

If 7(5 - 6z) = 11(5 - 4z), then the value of z is

1. 10

2. $\dfrac{12}{5}$

3. 12

4. $\dfrac{18}{7}$

Answer

Given:

7(5 - 6z) = 11(5 - 4z)

⇒ 35 - 42z = 55 - 44z

Transpose +35 to RHS and -44z to LHS

⇒ 44z - 42z = 55 - 35

⇒ 2z = 20

⇒ z = $\dfrac{20}{2}$

⇒ z = 10

Hence, option 1 is the correct option.

If $2x + \dfrac{3}{5} = \dfrac{2}{5}x + 3$, then the value of x is

1. 1

2. $\dfrac{3}{2}$

3. 2

4. $\dfrac{5}{2}$

Answer

We have:

$\phantom{=} 2x + \dfrac{3}{5} = \dfrac{2}{5}x + 3 \\[1em] \Rightarrow 2x - \dfrac{2}{5}x = 3 - \dfrac{3}{5} \\[1em] \Rightarrow \dfrac{10x - 2x}{5} = \dfrac{15 - 3}{5} \\[1em] \Rightarrow \dfrac{8x}{5} = \dfrac{12}{5} \\[1em] \text{[Since the denominators are same, we cancel them]} \\[1em] \Rightarrow 8x = 12 \\[1em] \Rightarrow x = \dfrac{12}{8} \\[1em] \Rightarrow x = \dfrac{3}{2}$

Hence, option 2 is the correct option.

If $a - 3 = \dfrac{a}{2} + \dfrac{a}{3}$, then the value of $a$ is

1. 12
2. 15
3. 18
4. 24

Answer

We have:

$\phantom{=} a - 3 = \dfrac{a}{2} + \dfrac{a}{3} \\[1em] \Rightarrow a - 3 = \dfrac{3a + 2a}{6} \\[1em] \Rightarrow 6(a - 3) = 5a \\[1em] \Rightarrow 6a - 18 = 5a \\[1em] \text{[Transpose -18 to RHS and +5a to LHS]} \\[1em] \Rightarrow 6a - 5a = 18 \\[1em] \Rightarrow a = 18$

Hence, option 3 is the correct option.

If $\dfrac{k-2}{k+3} = \dfrac{4}{9}$, then the value of k is

1. 5
2. 6
3. 8
4. 9

Answer

We have:

$\phantom{=} \dfrac{k - 2}{k + 3} = \dfrac{4}{9} \\[1em] \text{By cross multiplication:} \\[1em] \Rightarrow 9(k - 2) = 4(k + 3) \\[1em] \Rightarrow 9k - 18 = 4k + 12 \\[1em] \text{Transpose -18 to RHS and +4k to LHS} \\[1em] \Rightarrow 9k - 4k = 12 + 18 \\[1em] \Rightarrow 5k = 30 \\[1em] \Rightarrow k = \dfrac{30}{5} \\[1em] \Rightarrow k = 6$

Hence, option 2 is the correct option.

If $\dfrac{x+5}{6} - \dfrac{x+1}{9} = \dfrac{x+2}{4}$, then the value of x is

1. $\dfrac{5}{4}$

2. $\dfrac{6}{5}$

3. $\dfrac{7}{6}$

4. $\dfrac{8}{7}$

Answer

We have:

$\phantom{=} \dfrac{x+5}{6} - \dfrac{x+1}{9} = \dfrac{x+2}{4} \\[1em] \Rightarrow \dfrac{3(x + 5) - 2(x + 1)}{18} = \dfrac{x + 2}{4} \\[1em] \Rightarrow \dfrac{(3x + 15) - (2x + 2)}{18} = \dfrac{x + 2}{4} \\[1em] \Rightarrow \dfrac{3x + 15 - 2x - 2}{18} = \dfrac{x + 2}{4} \\[1em] \Rightarrow \dfrac{x + 13}{18} = \dfrac{x + 2}{4} \\[1em] \text{By cross multiplication:} \\[1em] \Rightarrow 4(x + 13) = 18(x + 2) \\[1em] \Rightarrow 4x + 52 = 18x + 36 \\[1em] \text{Transpose +4x to RHS and +36 to LHS} \\[1em] \Rightarrow 52 - 36 = 18x - 4x \\[1em] \Rightarrow 16 = 14x \\[1em] \Rightarrow x = \dfrac{16}{14} \\[1em] \Rightarrow x = \dfrac{8}{7}$

Hence, option 4 is the correct option.

A number is 6 more than another. If their sum is 32, then the smaller number is

1. 11
2. 12
3. 13
4. 14

Answer

Let smaller number be x.

Larger number is (x + 6).

According to the question their sum is 32:

∴ x + (x + 6) = 32

⇒ 2x + 6 = 32

⇒ 2x = 32 - 6

⇒ 2x = 26

⇒ x = $\dfrac{26}{2}$

⇒ x = 13

Hence, option 3 is the correct option.

A sum of ₹ 158 is to be divided between A, B and C, so that A gets ₹ 8 more than B and C gets ₹ 10 more than A. Then, A gets

1. ₹ 44
2. ₹ 48
3. ₹ 52
4. ₹ 60

Answer

Let B gets ₹ x.

Then A gets ₹ (x + 8)

C gets ₹ (x + 8 + 10) = ₹ (x + 18).

Total amount = ₹ 158

∴ x + (x + 8) + (x + 18) = 158

⇒ x + x + 8 + x + 18 = 158 $\quad$ [Removing brackets]

⇒ 3x + 26 = 158

⇒ 3x = 158 - 26 $\quad$ [Transposing +26 to RHS]

⇒ 3x = 132

⇒ x = $\dfrac{132}{3}$

⇒ x = 44

A gets ₹ (x + 8) = ₹ (44 + 8) = ₹ 52.

Hence, option 3 is the correct option.

Three times a number exceeds its one-third by 64. The number is

1. 24
2. 36
3. 42
4. 48

Answer

Let the number be x.

One-third of a number = $\dfrac{1}{3}x = \dfrac{x}{3}$

According to the question, three times a number exceeds its one-third by 64:

$\therefore 3x - \dfrac{x}{3} = 64 \\[1em] \Rightarrow \dfrac{9x - x}{3} = 64 \\[1em] \Rightarrow 8x = 64 \times 3 \\[1em] \Rightarrow 8x = 192 \\[1em] \Rightarrow x = \dfrac{192}{8} \\[1em] \Rightarrow x = 24$

Hence, option 1 is the correct option.

Two supplementary angles differ by 42°. One of these angles is

1. 57°
2. 69°
3. 75°
4. 81°

Answer

Let the required angles be x° and (x + 42)°.

Since the sum of supplementary angles is 180°, we have:

x + (x + 42) = 180

⇒ x + x + 42 = 180 $\quad$ [Removing brackets]

⇒ 2x + 42 = 180

⇒ 2x = 180 - 42 $\quad$ [Transposing +42 to RHS]

⇒ 2x = 138

⇒ x = 69°

Hence, option 2 is the correct option.

The sum of the digits of a two digit number is 9. On adding 27 to the number , its digits are reversed. The number is

1. 36
2. 45
3. 54
4. 72

Answer

Let the digit in the units place be x.

Since the sum of the digits is 9, the digit in the tens place must be (9 - x).

A two-digit number is written as: 10 x (Tens digit) + (Units digit)

Original Number = 10(9 - x) + x

= 90 - 10x + x

= 90 - 9x

When digits are reversed, x becomes the tens digit and (9 - x) becomes the units digit.

Reversed Number = 10(x) + (9 - x)

= 10x + 9 - x

= 9x + 9

The problem states that adding 27 to the original number results in the reversed number.

Original Number + 27 = Reversed Number

(90 - 9x) + 27 = 9x + 9

⇒ 117 - 9x = 9x + 9

Transpose -9x to RHS and +9 to LHS

⇒ 117 - 9 = 9x + 9x

⇒ 108 = 18x

⇒ x = $\dfrac{108}{18}$

⇒ x = 6

Units digit = x = 6

Tens digit = (9 - x) = (9 - 6) = 3

∴ The original number is 36.

Hence, option 1 is the correct option.

One fourth of a number exceeds one sixth of the number by $\dfrac{1}{2}$. The number is

1. 4
2. 6
3. 8
4. 12

Answer

Let the number be x.

One fourth of a number = $\dfrac{1}{4}x = \dfrac{x}{4}$

One sixth of the number = $\dfrac{1}{6}x = \dfrac{x}{6}$

According to the question, one fourth of a number exceeds one sixth of the number by $\dfrac{1}{2}$:

$\therefore \dfrac{x}{4} - \dfrac{x}{6} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{3x - 2x}{12} = \dfrac{1}{2} \\[1em] \Rightarrow x = \dfrac{1}{2} \times 12 \\[1em] \Rightarrow x = \dfrac{1}{1} \times 6 \\[1em] \Rightarrow x = 6$

Hence, option 2 is the correct option.

What number must be added to 54 so that the resulting number is 7 times the number added?

1. 6
2. 7
3. 8
4. 9

Answer

Let the number be x.

According to the question, the resulting number is 7 times the number added:

∴ 54 + x = 7x

⇒ 54 = 7x - x

⇒ 54 = 6x

⇒ x = $\dfrac{54}{6}$

⇒ x = 9

Hence, option 4 is the correct option.

The ages of A and B are in the ratio 2 : 5. After 6 years, their ages will be in the ratio 1 : 2. The present age of B is

1. 20 years
2. 24 years
3. 30 years
4. 36 years

Answer

The ages of A and B are in the ratio 2 : 5.

Let the present age of A be 2x years.

Let the present age of B be 5x years.

After 6 years:

Age of A = (2x + 6) years.

Age of B = (5x + 6) years.

The problem states that after 6 years, their ages will be in the ratio 1 : 2.

$\therefore \dfrac{2x + 6}{5x + 6} = \dfrac{1}{2} \\[1em] \text{By cross multiplication:} \\[1em] \Rightarrow 2(2x + 6) = 1(5x + 6) \\[1em] \Rightarrow 4x + 12 = 5x + 6 \\[1em] \text{Transpose +12 to RHS and +5x to LHS} \\[1em] \Rightarrow 4x - 5x = 6 - 12 \\[1em] \Rightarrow -x = -6 \\[1em] \Rightarrow x = 6$

Present age of A = 2x = 2(6) = 12 years.

Present age of B = 5x = 5(6) = 30 years.

∴ The present age of B is 30 years.

Hence, option 3 is the correct option.

Fill in the blanks :

(i) An equation involving one variable with highest power ..............., is called a linear equation in that variable.

(ii) The solution of an equation is also called ............... of the given equation.

(iii) The process in which we drop a term from one side of an equation and put it on the other side with sign changed is called ............... .

(iv) A linear equation can have at the most ............... solution(s).

(v) If thrice a number increased by 8 gives 23, then the number is ............... .

(vi) If x - 7 = 4, then 3x - 15 is equal to ............... .

Answer

(i) An equation involving one variable with highest power 1, is called a linear equation in that variable.

(ii) The solution of an equation is also called root of the given equation.

(iii) The process in which we drop a term from one side of an equation and put it on the other side with sign changed is called transposition.

(iv) A linear equation can have at the most 1 solution(s).

(v) If thrice a number increased by 8 gives 23, then the number is 5.

(vi) If x - 7 = 4, then 3x - 15 is equal to 18.

Explanation

(i) A linear equation is defined by the fact that the variable has an exponent of 1. If the power were 2, it would be a quadratic equation.

(ii) In algebra, the specific value of the variable that makes the equation true (LHS = RHS) is formally called the "root" or the solution.

(iii) Transposition is the mathematical term for moving terms across the equal sign (=) while reversing their operation (plus becomes minus, multiply becomes divide).

(iv) A linear equation in one variable can have at the most one solution.

(v)

Let the number be x.

According to the condition:

3x + 8 = 23

⇒ 3x = 23 - 8 $\quad$ [Transposing +8 to RHS]

⇒ 3x = 15

⇒ x = $\dfrac{15}{3}$

⇒ x = 5

∴ The required number is 5.

(vi)

Given equation:

x - 7 = 4

First, find the value of x:

x - 7 = 4

⇒ x = 4 + 7 $\quad$ [Transposing -7 to RHS]

⇒ x = 11

Now, substitute x = 11 into the expression 3x - 15:

3x - 15 = 3(11) - 15

= 33 - 15

= 18

∴ 3x - 15 = 18

Write true (T) or false (F) :

(i) The smaller of the two numbers whose sum is 26 and the difference is 6 is 10.

(ii) The solutions of an equation in variable z are the values of z for which L.H.S. = R.H.S.

(iii) Sum of two consecutive even natural numbers is 24. This statement can be represented by the equation x + (2x + 2) = 24.

(iv) 3x + 4 > 12 is a linear equation in one variable.

(v) x = 4 is a solution of the equation $\dfrac{5x}{2} - 1 = 4x - 7$.

Answer

(i) True
Reason —

Let the numbers be x and y.

Sum = x + y = 26

Difference = x - y = 6.

Adding the equations:

x + y + x - y = 26 + 6

⇒ 2x = 32

⇒ x = $\dfrac{32}{2}$

⇒ x = 16.

Substituting the value of x in (x + y = 26) we get:

x + y = 26

⇒ 16 + y = 26

⇒ y = 26 - 16 $\quad$ [Transposing +16 to RHS]

⇒ y = 10.

The smaller number is 10.

(ii) True
Reason — The statement is the fundamental definition of a solution. A value is only a "root" or "solution" if it makes the Left-Hand Side (L.H.S.) equal to the Right-Hand Side (R.H.S.).

(iii) False
Reason —

Consecutive even numbers are represented as x and x + 2. The equation should be:

x + (x + 2) = 24.

The expression 2x + 2 represents twice the first number plus two, which is not the definition of the next consecutive even number.

(iv) False
Reason — The symbol > makes this an inequality, not an equation. A linear equation must have an equal sign (=).

(v) True
Reason —

Given equation:

$\dfrac{5x}{2} - 1 = 4x - 7$

Substitute x = 4 into both sides:

$\therefore \dfrac{5x}{2} - 1 = 4x - 7 \\[1em] \Rightarrow \dfrac{5(4)}{2} - 1 = 4(4) - 7 \\[1em] \Rightarrow \dfrac{20}{2} - 1 = 16 - 7 \\[1em] \Rightarrow \dfrac{20 - 2}{2} = 9 \\[1em] \Rightarrow \dfrac{18}{2} = 9 \\[1em] \Rightarrow 9 = 9$

Since, L.H.S. = R.H.S., x = 4 is the correct solution.

Sanjay runs a computer repair service company. For an HP laptop, he charges ₹ 320 for diagnosis and ₹ 90 per hour for repairs. For a Dell laptop, he charges ₹ 480 for diagnosis and ₹ 70 per hour for repairs.

(1) Sanjay repairs an HP laptop of Mr Lal. He charges ₹ 950 for the work. How long did the repair work take place?

1. 6 hours
2. 7 hours
3. 8 hours
4. 9 hours

(2) Sanjay repairs a Dell laptop of Mr Gupta. The total charges were ₹ 830. How long did the repair work take place?

1. 5 hours
2. 6 hours
3. 7 hours
4. 8 hours

(3) He repairs an HP laptop and a Dell laptop of Mr Subhash. He found that the repair work for both the laptops took equal time and also the service charges of the two laptops were the same. How long did the repair work take place for each laptop ?

1. 6 hours
2. 7 hours
3. 8 hours
4. 9 hours

(4) What were the total service charges that Mr. Subhash had to pay for both the laptops ?

1. ₹ 1040
2. ₹ 1260
3. ₹ 2040
4. ₹ 2520

Answer

Given:

Sanjay's Charge Rates:

HP: ₹ 320 (fixed) + ₹ 90 per hour

Dell: ₹ 480 (fixed) + ₹ 70 per hour

(1)

Let the repair time be x hours.

∴ 320 + 90x = 950

⇒ 90x = 950 - 320 $\quad$ [Transposing +320 to RHS]

⇒ 90x = 630

⇒ x = 7 hours.

Hence, option 2 is the correct option.

(2)

Let the repair time be y hours.

∴ 480 + 70y = 830

⇒ 70y = 830 - 480 $\quad$ [Transposing +480 to RHS]

⇒ 70y = 350

⇒ y = 5 hours.

Hence, option 1 is the correct option.

(3)

Let the time for each be t hours. Since the charges are equal:

HP Charge = Dell Charge

320 + 90t = 480 + 70t

⇒ 90t - 70t = 480 - 320 $\quad$ [Transposing +320 to RHS and +70t to LHS]

⇒ 20t = 160

⇒ t = $\dfrac{160}{20}$

⇒ t = 8 hours

Hence, option 3 is the correct option.

(4)

Let's calculate the cost for one laptop using t = 8:

HP: ₹ (320 + 90(8)) = ₹ (320 + 720) = ₹ 1040

Dell: ₹ (480 + 70(8)) = ₹ (480 + 560) = ₹ 1040

Total for both (HP + Dell) = ₹ 1040 + ₹ 1040 = ₹ 2080

The total service charges that Mr. Subhash had to pay for both the laptops is ₹ 2080. None of the options are correct.

The ABS school organised Science competition for seventh class students. In this exam, there were 40 questions, all MCQ types. Three marks were awarded for each correct answer and one mark was deducted for each question that was answered incorrectly or left unanswered.

(1) Aryan scored 76 marks. How many questions did he answer correctly ?

1. 23
2. 25
3. 7
4. 29

(2) Ramya scored 84 marks. If she attempted all the questions, how many of her answers were incorrect ?

1. 9
2. 10
3. 11
4. 12

(3) Hema's score was double the number of questions she attempted correctly. How many of her answers were correct ?

1. 10
2. 15
3. 20
4. 25

(4) Bilok's score was the highest in the class. His score was 12 marks less than the maximum marks. How many of his answers were correct ?

1. 35
2. 36
3. 37
4. 38

Answer

Given:

Total Questions: 40

Correct Answer: +3 marks

Incorrect/Unanswered: -1 mark

Let c = correct answers.

Then (40 - c) = incorrect/unanswered.

(1)

Aryan scores 76 marks:

∴ 3(c) - 1(40 - c) = 76

⇒ 3c - 40 + c = 76

⇒ 4c - 40 = 76

⇒ 4c = 76 + 40 $\quad$ [Transposing -40 to RHS]

⇒ 4c = 116

⇒ c = $\dfrac{116}{4}$

⇒ c = 29

Aryan answered 29 questions correctly.

Hence, option 4 is the correct option.

(2)

Ramya scored 84 marks:

∴ 3(c) - 1(40 - c) = 84

⇒ 3c - 40 + c = 84

⇒ 4c - 40 = 84

⇒ 4c = 84 + 40 $\quad$ [Transposing -40 to RHS]

⇒ 4c = 124

⇒ c = $\dfrac{124}{4}$

⇒ c = 31

Correct answers = 31

Incorrect answers = (40 - c) = (40 - 31) = 9.

Hence, option 1 is the correct option.

(3)

Hema's score was double the number of questions she attempted correctly:

Score = 2c

∴ 3c - (40 - c) = 2c

⇒ 3c - 40 + c = 2c

⇒ 4c - 40 = 2c

⇒ 4c - 2c = 40 $\quad$ [Transposing +2c to LHS and -40 to RHS]

⇒ 2c = 40

⇒ c = $\dfrac{40}{2}$

⇒ c = 20

∴ Hema answered 20 questions correctly.

Hence, option 3 is the correct option.

(4)

Bilok's score was 12 marks less than the maximum marks

Maximum possible marks = 40 x 3 = 120.

Bilok's score = 120 - 12 = 108.

∴ 3c - (40 - c) = 108

⇒ 3c - 40 + c = 108

⇒ 4c - 40 = 108

⇒ 4c = 108 + 40 $\quad$ [Transposing -40 to RHS]

⇒ 4c = 148

⇒ c = $\dfrac{148}{4}$

⇒ c = 37

∴ Bilok answered 37 questions correctly.

Hence, option 3 is the correct option.

Assertion: If 9 is the value of variable x in the equation $\dfrac{2x - 7}{2} = y$, then the value of y is 28.

Reason: The value which when substituted for the variable in an equation, makes LHS = RHS is called a solution of the given equation.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is false but Reason (R) is true.

Explanation

Assertion:

Given equation:

$\dfrac{2x - 7}{2} = y$

Substitute x = 9 into the equation:

$\phantom{=} \dfrac{2x - 7}{2} = y \\[1em] \Rightarrow \dfrac{2(9) - 7}{2} = y \\[1em] \Rightarrow \dfrac{18 - 7}{2} = y \\[1em] \Rightarrow \dfrac{11}{2} = y \\[1em] 5.5 = y$

Since y = 5.5 and not 28, the Assertion is False.

Reason:

The definition provided for a "solution" or "root" of an equation is mathematically perfect. Therefore, the Reason is True.

Hence, option 4 is the correct option.

Assertion: Any term of an equation may be transposed from one side of the equation to the other side of the equation by changing the sign of the term.

Reason: Transposition is a process in which we drop a term from one side of an equation and put it on the other side with sign changed.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation

Assertion:

The statement describes the standard rule of algebra. If we move +5 to the other side, it becomes -5. This is a true statement.

Reason provides the formal definition of the process mentioned in the assertion. It is also true and correct explanation of assertion.

Hence, option 1 is the correct option.

Refer to the statements below about expressing an equation in statement form:

Statement 1: x − 6 = 16 can be expressed as the difference between x and 6 is 16.

Statement 2: (2 + 3)t = 35 can be expressed as 3 times a number plus 2 gives 35.

Statement 3: p − 1 = 11 can be expressed as 1 subtracted from p is 11.

Which of these statement(s) is/are true?

1. Only statement 3
2. Both statements 1 and 3
3. Both statements 2 and 3
4. Statements 1, 2 and 3

Answer

Statement 1: x − 6 = 16

The expression x − 6 means 6 is subtracted from x, which represents the difference between x and 6. So, the equation can be read as "the difference between x and 6 is 16."

Hence, Statement 1 is True.

Statement 2: (2 + 3)t = 35

The given expression (2 + 3)t means (2 + 3) is multiplied by t, i.e., 5t = 35.

The statement "3 times a number plus 2 gives 35" would be written as 3t + 2 = 35, which is not the same as (2 + 3)t = 35.

Hence, Statement 2 is False.

Statement 3: p − 1 = 11

The expression p − 1 means 1 is subtracted from p. So, the equation can be read as "1 subtracted from p is 11."

Hence, Statement 3 is True.

Hence, option 2 is the correct option.

To solve the equation 2(x − 4) = 16, two students rewrite it as shown:

Kartik: x − 4 = 8

Sanjay: 2x − 8 = 16

Which student rewrites the equation correctly to solve it?

1. Only Kartik
2. Only Sanjay
3. Both Kartik and Sanjay
4. Neither Kartik nor Sanjay

Answer

Given equation:

2(x − 4) = 16

Kartik's Method — Dividing both sides by 2:

$\phantom{=} 2(x - 4) = 16 \\[1em] \Rightarrow \dfrac{2(x - 4)}{2} = \dfrac{16}{2} \\[1em] \Rightarrow x - 4 = 8$

So, Kartik's rewriting is correct.

Sanjay's Method — Removing the bracket by multiplication:

$\phantom{=} 2(x - 4) = 16 \\[1em] \Rightarrow 2 \times x - 2 \times 4 = 16 \\[1em] \Rightarrow 2x - 8 = 16$

So, Sanjay's rewriting is correct.

Both methods are valid ways of simplifying the same equation and will lead to the same solution x = 12.

Hence, option 3 is the correct option.

A shopkeeper sells bananas in two types of boxes, one small and one large. A large box contains as many as 6 small boxes plus 2 loose bananas. Form an equation which gives the number of bananas in each small box, if the number of bananas in 1 large box is 50.

1. 3x + 1 = 50
2. x + 1 = 20
3. 6x + 2 = 50
4. 2x + 1 = 20

Answer

Let the number of bananas in each small box be x.

Then, the number of bananas in 6 small boxes = 6x.

According to the question, a large box contains as many as 6 small boxes plus 2 loose bananas, and the total number of bananas in 1 large box is 50.

∴ 6x + 2 = 50

Hence, option 3 is the correct option.

Study the following statements carefully and select the correct option.

P: If 3(x + 3) − 2(x − 1) = 5(x − 5), then x = 9.

Q: If $\dfrac{y}{2} + \dfrac{3}{2} = \dfrac{2y}{5} + 1$, then y = 5

1. Both P and Q are true
2. Both P and Q are false
3. P is true but Q is false
4. P is false but Q is true

Answer

Statement P:

Given equation:

3(x + 3) − 2(x − 1) = 5(x − 5)

⇒ 3x + 9 − 2x + 2 = 5x − 25 $\quad$ [Removing brackets]

⇒ x + 11 = 5x − 25

⇒ 11 + 25 = 5x − x $\quad$ [Transposing -25 to LHS and +x to RHS]

⇒ 36 = 4x

⇒ x = $\dfrac{36}{4}$

⇒ x = 9

Since the value of x is 9, Statement P is True.

Statement Q:

Given equation:

$\dfrac{y}{2} + \dfrac{3}{2} = \dfrac{2y}{5} + 1$

$\phantom{=} \dfrac{y}{2} + \dfrac{3}{2} = \dfrac{2y}{5} + 1 \\[1em] \Rightarrow \dfrac{y + 3}{2} = \dfrac{2y + 5}{5} \\[1em] \Rightarrow 5(y + 3) = 2(2y + 5) \quad\text{[By cross multiplication]} \\[1em] \Rightarrow 5y + 15 = 4y + 10 \quad\text{[Removing brackets]} \\[1em] \Rightarrow 5y - 4y = 10 - 15 \quad\text{[Transposing +15 to RHS and +4y to LHS]} \\[1em] \Rightarrow y = -5$

Since the value of y is −5 and not 5, Statement Q is False.

Hence, option 3 is the correct option.

Some people are sitting in three rooms. In first room, 9 people are more than the second room, and in the third room, 1 person is less than the second room. If there are 41 people in all the three rooms, how many people are there in second room?

1. 10
2. 11
3. 20
4. 21

Answer

Let the number of people in the second room be x.

Then, the number of people in the first room = (x + 9).

And, the number of people in the third room = (x − 1).

According to the question, the total number of people in all three rooms is 41:

∴ (x + 9) + x + (x − 1) = 41

⇒ x + 9 + x + x − 1 = 41 $\quad$[Removing brackets]

⇒ 3x + 8 = 41

⇒ 3x = 41 − 8 $\quad$[Transposing +8 to RHS]

⇒ 3x = 33

⇒ x = $\dfrac{33}{3}$

⇒ x = 11

∴ The number of people in the second room is 11.

Hence, option 2 is the correct option.