[3 marks sum]

Question 13 Marks

Find a number such that when $5$ is subtracted from $5$ times the number, the result is $4$ more than twice the number.

Answer

Let the number be $x.$
According to the question,
$5 x-5=2 x+4 $
or $5 x-2 x=4+5$
or $3 x=9 $
or $x=\frac{9}{3} $
or $x=3$
Thus, the number is $3.$

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Question 23 Marks

The difference between the squares of two consecutive numbers is $31.$ Find the numbers.

Answer

Let the numbers be $x$ and $x + 1 .$
According to the question,
$(x+1)^2-x^2=31 $
or $x^2+2 x+1-x^2=31 $
or $2 x=31-1 $
or $x=\frac{30}{2}$
or $x=15$
Thus, the numbers are $15$ and $16 .$

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Question 33 Marks

Find a number whose double is $45$ greater than its half.

Answer

Let the number be $x .$
According to the question,
$2x = \frac{1}{2}x + 45$
or $2x - \frac{1}{2}x = 45$
or $\frac{4x - x}{2} = 45$
or $3x = 90 ($After cross multiplication$)$
$\text{ or }x = \frac{90}{3}$
or $x = 30$
Thus, the number is $30$.

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Question 43 Marks

A man sold an article for $₹. 396$ and gained $10\%$ on it. Find the cost price of the article

Answer

$\text{S.P}$. of article $= ₹ 396$
Gain $= 10\%$
Let cost price $= ₹ x$
$\therefore \text { S.P. }=\frac{x \times(100+10)}{100}=\frac{110}{100} x $
$\therefore \frac{110}{100} x=396 $
$ \Rightarrow x=\frac{396 \times 100}{110}=360$
Cost price of an article $= ₹ 360$

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Question 53 Marks

Solve: $\frac{2}{5 \mathrm{x}}-\frac{5}{3 \mathrm{x}}=\frac{1}{15}$

Answer

$\frac{2}{5 x}-\frac{5}{3 x}=\frac{1}{15}$
$\Rightarrow \frac{2 \times 3}{5 \mathrm{x} \times 3}-\frac{5 \times 5}{3 \mathrm{x} \times 5}=\frac{1}{15}$
$\Rightarrow \frac{6-25}{15 x}=\frac{1}{15}$
$\Rightarrow \frac{-19}{15 x}=\frac{1}{15}$
$\Rightarrow \frac{-19}{\mathrm{x}}=\frac{15}{15}$
$\Rightarrow-19=\mathrm{x}$
$\therefore x=-19$

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Question 63 Marks

Solve the following equation and also verify your solution: $(x + 2)(x + 3) + (x − 3)(x − 2) − 2x(x + 1) = 0$

Answer

$(x + 2)(x + 3) + (x - 3)(x - 2) - 2x(x + 1) = 0$
or $x^2 + 5x + 6 + x^2 - 5x + 6 - 2 x^2 - 2x = 0$
or $12 - 2x = 0$
or $x = \frac{12}{2} = 6$
Verification:
$\text{ L . H . S . }= (6 + 2)(6 + 3) + (6 - 3)(6 - 2) - 2 \times 6(6 + 1)$
$ = 72 + 12 - 84 = 0 =\text{ R . H . S .}$

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Question 73 Marks

If Megha’s age is increased by three times her age, the result is $60$ years. Find her age

Answer

Let Megha’s age $= x$ yearsThree times Megha’s age $= 3x$ years
According to the statement :
$x + 3x = 60$
$=> 4x = 60$
$=> x = 15$
Megha’s age $= 15$ years

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Question 83 Marks

Fifteen less than $4$ times a number is $9$. Find the number.

Answer

Let the required number be $x4$ times the number $= 4x$
$15$ less than $4$ times the number $= 4x-15$
According to the statement :
$4x – 15 = 9$
$\Rightarrow 4x = 9 + 15$
$\Rightarrow 4x = 24$
$\Rightarrow x = 6$

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Question 93 Marks

The sum of three consecutive odd numbers is $57$. Find the numbers.

Answer

Let the three consecutive odd numbers be $x, x+2, x+4$. According to the statement :
$x + x + 2 + x + 4 = 57$
$⇒ x + x + x = 57 – 2 – 4$
$⇒ 3x = 51$
$⇒ x = 17$
Three consecutive odd numbers are $17, 19, 21$

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Question 103 Marks

The length of a rectangle is twice its width. If its perimeter is $54\ cm$; find its length.

Answer

Let width of the rectangle $= x cm$ Length of the rectangle $= 2x \ cm$
Perimeter of the rectangle $= 2 [$Length $+$ Width$] = 2 [2x + x] = 2 \times 3 = 6x \ cm$
Given perimeter $= 54 \ cm$
$6x = 54$
$⇒ x = 9$
Length $= 2x = 2 \times 9 = 18 \ cm$

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Question 113 Marks

Solve the following equation : $3(x + 1) = 12 + 4(x - 1)$

Answer

$3(x + 1) = 12 + 4(x – 1)$
$3x + 3 = 12 + 4x – 4$
$3x – 4x = 12 – 4 – 3$
$-x = 5$
$x = -5$

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Question 123 Marks

Solve the following equation: $5(8x + 3) = 9(4x + 7)$

Answer

$ \Rightarrow 40 x+15=36 x+63$
$ \Rightarrow 40 x-36 x=63-15$
$ \Rightarrow 4 x=48$
$ \Rightarrow x=\frac{48}{4}$
$ \Rightarrow x=12$

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Question 133 Marks

Solve the following equation: $\frac{x-8}{5}=\frac{x-12}{9}$

Answer

$\Rightarrow 9(x-8)=5(x-12) \ldots$ $($by cross multiplying$)$
$\Rightarrow 9 x-72=5 x-60$
$\Rightarrow 9 x-5 x=-60+72$
$\Rightarrow 4 \mathrm{x}=12$
$\Rightarrow x=\frac{12}{4}$
$\Rightarrow x=3$

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Question 143 Marks

Solve the following equation: $\frac{x+2}{9}=\frac{x+4}{11}$

Answer

$ \Rightarrow 11(x+2)=9(x+4) ($by cross multiplying$)$
$ \Rightarrow 11 x+22=9 x+36$
$ \Rightarrow 11 x-9 x=36-22$
$ \Rightarrow 2 x=14$
$ \Rightarrow x=\frac{14}{2}$
$ \Rightarrow x=7$

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Question 153 Marks

Solve the following equation: $3(b - 4) = 2(4 - b)$

Answer

$ \Rightarrow 3 b-12=8-2 b$
$ \Rightarrow 3 b+2 b=8+12$
$ \Rightarrow 5 b=20$
$ \Rightarrow b=\frac{20}{5}$
$ \Rightarrow b=4$

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Question 163 Marks

Solve the following equation: $\frac{3 x+2}{x-6}=-7$

Answer

$3x + 2 = -7 (x – 6) ($by cross multiplying$)$
$3x + 2 = -7x + 42$
$3x + 7x = 42 – 2$
$10x = 40$
$x = 4$

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Question 173 Marks

Solve: $5 x-\left(4 x+\frac{5 x-4}{7}\right)=\frac{4 x-14}{3}$

Answer

$ 5 x-\left(4 x+\frac{5 x-4}{7}\right)=\frac{4 x-14}{3}$
$ 5 x-\left(\frac{28 x+5 x-4}{7}\right)=\frac{4 x-14}{3}$
$ \frac{35 x-33 x+4}{7}=(4 x-14)$
$ 6 x+12=28 x-98$
$ 22 x=98+12$
$ x=\frac{110}{22}=5$

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Question 183 Marks

Solve: $0.25+\frac{1.95}{x}=0.9$

Answer

$0.25+\frac{1.95}{x}=0.9$
$ 0.25 x+1.95=0.9 x$
$ \Rightarrow 0.9 x-0.25 x=1.95$
$ \Rightarrow 0.65 x=1.95$
$ \Rightarrow x=\frac{1.95}{0.65}=3$
Hence, $x=3$

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Question 193 Marks

Solve the following equation: $\frac{x-1}{7 x-14}=\frac{x-3}{7 x-26}$

Answer

$ \frac{x-1}{7 x-14}=\frac{x-3}{7 x-26}$
$ \Rightarrow(x-1)(7 x-26)=(7 x-14)(x-3)$
$ \Rightarrow 7 x^2-7 x-26 x+26=7 x^2-14 x-21 x+42$
$ \Rightarrow-33 x+26=-35 x+42$
$ \Rightarrow 35 x-33 x=42-26$
$ \Rightarrow 2 x=16$
$ \Rightarrow x=8$

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Question 203 Marks

Solve the following equation: $(x - 5)(x + 3) = (x - 7)(x + 4)$

Answer

$(x - 5) (x + 3) = (x - 7)(x + 4)$
$\Rightarrow x^2 + 3x - 5x - 15$
$= x^2 + 4x - 7x - 28$
$\Rightarrow -2x - 15 = - 3x - 28$
$\Rightarrow 3x - 2x = 15 - 28$
$\Rightarrow x = -13$

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Question 213 Marks

Solve the following equation: $\frac{3 x}{4}-\frac{1}{4}(x-20)=\frac{x}{4}+32$

Answer

$ \Rightarrow \frac{3 x}{4}-\frac{x}{4}+5=\frac{x}{4}+32$
$ \Rightarrow \frac{3 x}{4}-\frac{x}{4}-\frac{x}{4}=32-5$
$ \Rightarrow \frac{3 x-x-x}{4}=27$
$ \Rightarrow \frac{x}{4}=27$
$ \Rightarrow x=27 \times 4$
$ \Rightarrow x=108$

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MATHS [3 marks sum]

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