Question 13 Marks
In a factory male workers are paid one and a half times more than their female counterparts for each hour of work. In a particular week a husband and wife team worked for a total of $60$ hours with the husband working twice as much as his wife. The total amount earned by both is $Rs. 960.$ If the husband's earning is $4$ times that of his wife, find the number of hours each worked.
AnswerLet the wife work for $x\ hrs.$
Since the husband works twice as much as his wife,
Therefore husband works for $2x\ hrs.$
Total numbers of hours worked $= 60$
$⇒ x + 2x = 60$
$⇒ 3x = 60$
$⇒ x = 20$
$⇒ 2x = 40$
Therefore, wife works for $20\ hrs$ and husband works for $40\ hrs.$
View full question & answer→Question 23 Marks
What number decreased by $18\%$ of itself gives $1599$?
AnswerLet the number be $x$.
Therefore,
$x-\frac{18}{100} \times x=1599 $
$\Rightarrow \frac{100 x-18 x}{100}=1599 $
$\Rightarrow 82 x=1599 \times 100 $
$\Rightarrow x=\frac{1599 \times 100}{82} $
$\Rightarrow x=1950$
Hence, $1950$ is the number.
View full question & answer→Question 33 Marks
What number decreased by $12\%$ of itself gives $1584$ ?
AnswerLet the number be $x$.
Therefore,
$x-\frac{12}{100} \times x=1584 $
$\Rightarrow \frac{100 x-12 x}{100}=1584 $
$\Rightarrow 88 x=1584 \times 100 $
$\Rightarrow x=\frac{1584 \times 100}{88} $
$\Rightarrow x=1800$
Hence, $1800$ is the number.
View full question & answer→Question 43 Marks
What number increased by $15\%$ of itself gives $2921$?
AnswerLet the number be $x$.
Therefore,
$x+\frac{15}{100} \times x=291 $
$\Rightarrow \frac{100 x+15 x}{100}=2921 $
$\Rightarrow 115 x =2921 \times 100 $
$\Rightarrow x =\frac{2921 \times 100}{115} $
$\Rightarrow x =2540$
Hence, $2540$ is the number.
View full question & answer→Question 53 Marks
What number increased by $8\%$ of itself gives $1620$?
AnswerLet the number be $x$.
Therefore,
$x+\frac{8}{100} x \times=1620 $
$\Rightarrow \frac{100 x+8 x}{100}=1620 $
$\Rightarrow 108 x =1620 \times 100 $
$\Rightarrow x =\frac{1620 \times 100}{108} $
$\Rightarrow x =1500$
Hence, $1500$ is the number.
View full question & answer→Question 63 Marks
The breadth of a rectangular room is $2\ m$ less than its length. If the perimeter of the room is $14\ m,$ find it's dimensions.
AnswerLet the length of the rectangle be $x$
breadth $= x - 2$
perimeter $= 2($length $+$ breadth$)$
$\therefore 2(x + x - 2) = 14$
$\Rightarrow 2x - 2 = 7$
$\Rightarrow x = 4.5$
$\therefore $ breadth
$= 4.5 - 2$
$= 2.5.$
View full question & answer→Question 73 Marks
In an isosceles triangle, each of the two equal sides is $4 \ cm$ more than its base. If the perimeter of the triangle is $29\ cm,$ find the sides of the triangle.
AnswerLet the base of the isosceles triangle be $x \ cm$ long.
Then, the equal sides are $(x + 4)\ cm.$
Given, the perimeter of the triangle is $29\ cm$
$\Rightarrow 2(x + 4) + x = 29\ cm$
$3x + 8 = 29\ cm$
$\Rightarrow 3x = 21$
$\Rightarrow x = 7\ cm$
Thus, the base of the isisceles triangle is $7\ cm$ long.
Then, the equal sides are $11\ cm.$
View full question & answer→Question 83 Marks
The length of a rectangle is $3 \ cm$ more than its breadth. If the perimeter of the rectangle is $18\ cm,$ find the length and breadth of the rectangle.
AnswerLet the length of the rectangular be $(x + 3)\ cm$ and breadth be $x \ cm.$
Given, the perimeter of a rectangular field is $18\ m.$
$\Rightarrow 2(x + x + 3) = 18\ m$
$\Rightarrow 2(2x + 3) = 18\ m$
$\Rightarrow 2x + 3 = 9$
$\Rightarrow 2x = 6$
$\Rightarrow x = 3\ cm$
Thus, the length of the rectangle is $6\ cm$ and breadth is $3\ cm.$
View full question & answer→Question 93 Marks
What number decreased by $12\%$ of itself gives $1584$?
AnswerLet the number be $x$.
Therefore,
$x-\frac{12}{100} \times x=1584 $
$\Rightarrow \frac{100 x-12 x}{100}=1584 $
$\Rightarrow 88 x=1584 \times 100 $
$\Rightarrow x=\frac{1584 \times 100}{88} $
$\Rightarrow x=1800$
Hence, $1800$ is the number.
View full question & answer→Question 103 Marks
What number increased by $15\%$ of itself gives $2921$?
AnswerLet the number be $x$.
As per the given condition,
$x +15 \%$ of $x =2921 $
$\Rightarrow x+\frac{15}{100} x \times 2921 $
$\Rightarrow x+\frac{15 x}{100}=2921 $
$\Rightarrow \frac{115 x}{100}=2921 $
$\Rightarrow 115 x =292100 $
$\Rightarrow x =2540$
Hence, the number is $2540 .$
View full question & answer→Question 113 Marks
The sum of three consecutive natural numbers is $216.$ Find the numbers
AnswerLet the three consecutive natural numbers be $x, x + 1$ and $x + 2.$
Then, sum $= x + x + 1 + x + 2 = 216$
$\Rightarrow 3x + 3 = 216$
$\Rightarrow 3x = 213$
$x = 71$
$\Rightarrow x + 1 = 72, x + 2 = 73$
Thus, the $3$ natural numbers are $: 71, 72$ and $73.$
View full question & answer→Question 123 Marks
The angles of a triangle are$ : 2(x + 6)^\circ , 3(x - 1)^\circ $ and $6(x + 1)^\circ .$ Find $x,$ and show that the triangle is isosceles.
AnswerWe know that the sum of all angles of a triangle are $180^\circ $
Thus, $2(x + 6)^\circ + 3(x - 1)^\circ + 6(x + 1)^\circ = 180^\circ $
$\Rightarrow $ Collecting like terms, we get:
$(2x + 3x + 6x)^\circ + (12 + 6 - 3)^\circ = 180^\circ $
$\Rightarrow 11x^\circ = 180^\circ - 15^\circ = 165^\circ $
$\Rightarrow x = 15^\circ .$
View full question & answer→Question 133 Marks
Find the value of $x$ for which the sum of the expression $15(x + 1), 10(x + 2) $and $6(x + 3)$ is $270.$
AnswerGiven$: 15(x + 1) + 10(x + 2) +6(x + 3) = 270.$
$\Rightarrow $ Collecting like terms, we get:
$(15x + 10x + 6x) + (15 + 20 + 18) - (270) = 0$
$\Rightarrow 31x = 217$
$\Rightarrow x = 7.$
View full question & answer→Question 143 Marks
Find the value of $x$ which makes the expressions $10(3x + 12)$ and $3(9x + 50)$ equal to each other.
AnswerWe have to find the value of $x$ which makes the two expressions equal:
$10(3x + 12) = 3(9x + 50)$
$\Rightarrow 30x - 27x = 150 - 120$
$\Rightarrow 3x = 30$
$\Rightarrow x = 10.$
View full question & answer→Question 153 Marks
Two numbers are in the ratio $2:3.$ If their sum is $150,$ find the numbers
AnswerLet the two numbers be $x$ and $y$.
Then Given, $x+y=150$ and $x: y=2: 3$
Then, the number $x=\frac{2}{2+3}$ of $150=\frac{2 \times 150}{5}=60$
And $y=\frac{3}{2+3}$ of $150=\frac{3 \times 150}{5}=90$.
View full question & answer→Question 163 Marks
The measures of angles of a quadrilateral in degrees are $x^\circ , (3x-40)^\circ , 2x^o$ and $(4x+20)^o.$ Find the measures of the angles.
AnswerSum of angles of a quadrilateral $= 360^\circ$
$\Rightarrow x^\circ + (3x - 40)^\circ + 2x^\circ + (4x + 30)^\circ = 360^\circ$
$\Rightarrow 10x^\circ - 10^\circ = 360^\circ$
$\Rightarrow x = 37^\circ$
Therefore, the measures of the angles are $37^\circ , (3\times 37- 40)^\circ ,$
$2 \times 37^\circ$ and $(4 \times 37 + 20^\circ )$
i.e. $3^\circ 7, 71^\circ , 74^\circ a$ and $168^\circ .$
View full question & answer→Question 173 Marks
The measures of angles of a triangle are $(9x - 5)^\circ , (7x + 5)^\circ $ and $20x^\circ .$ Find the value of $x.$ Also, show that the triangle is isosceles.
AnswerWe know that, sum of the measure of the angles of a triangle is $180^{\circ}$.
$9 x -5+7 x +5+20 x =180^{\circ} $
$\Rightarrow 36 x =\frac{180}{36} $
$\Rightarrow x =5$
To show that the triangle is isosceles, we can show that at least teo angles of the triangle are equal.
$9 x-5=9(5)-5=40 $
$7 x+5=7(5)+5=40$
So, the triangle is an isosceles triangle.
View full question & answer→Question 183 Marks
Two angles are supplementary and their measures are $(7x+6)^\circ $ and $(2x-15)^\circ .$ Find the measures of the angles.
AnswerLet the supplementary angles be
$\angle 1+\angle 2=180^{\circ} $
$\Rightarrow(7 x+6)+(2 x-15)=180^{\circ} $
$\Rightarrow 9 x=180+15-6=189^{\circ} $
$\Rightarrow \frac{189}{9}=21^{\circ}$
Thus, the measure of the angles are: $(7 x+6) v$ and $(2 x-15)^{\circ}$.
$=153^{\circ}$ and $27^{\circ}$.
View full question & answer→Question 193 Marks
Divide $300$ into two parts so that half of the one part is less than the other by $48.$
AnswerLet the two parts be $x$ and $y$.
Then, $x+y=300$
Also, one part is less than the other by $48.$
$\Rightarrow \frac{1}{2}(300-x)-x=48 $
$\Rightarrow 300-3 x=96 $
$\Rightarrow x=\frac{204}{3}=68$
Then,
$y=300-68 $
$=232$
View full question & answer→Question 203 Marks
The sum of two numbers is $50,$ and their difference is $10.$ Find the numbers.
AnswerLet the two nos be $x$ and $y$
Then, as per the questions,
Their Sum $= x + y = 50$
Their difference$ = x + y = 10$
Then, adding the two equations, $2x = 60$
$x = 30, y = 20.$
View full question & answer→Question 213 Marks
Twice a number decreased by $15,$ equals $25.$ Find the number.
AnswerLet the number be $x.$
Then As per the ques,
$2x - 15 = 25$
$2x - 15 = 25$
$\Rightarrow 2x = 40$
$x = 20.$
View full question & answer→Question 223 Marks
Find the number which, when added to its half, gives $60.$
AnswerLet the number be $x$.
Then As per the ques,
$x+\frac{x}{2}=60$
$\Rightarrow \frac{3 x}{2}=60$
Cross multiplying
$x=\frac{120}{3}=40$
View full question & answer→Question 233 Marks
Solve the following equations: $a(2x - b) - b(3x - a)+a(x + 1) = b(x + 5)$
Answer$a(2 x-b)-b(3 x-a)+a(x+1)=b(x+5)$
On simplyfying, we get:
$2 a x-a b-3 b x+a b+a x+a=b x+5 b $
$\Rightarrow 3 a x-4 b x=5 b-a $
$\Rightarrow x(3 a-4 b)=5 b-a $
$\Rightarrow x=\frac{5 b-a}{(3 a-4 b)} .$
View full question & answer→Question 243 Marks
Solve the following equations $:8x + a(x - b) = 10(ax - b)$
Answer$8 x+a(x-b)=10(a x-b) $
$\Rightarrow 8 x+a x-a b-10 a x+10 b=0 $
$ \Rightarrow 8 x-9 a x=a b-10 b $
$ \Rightarrow x(8-9 a)=b(a-10)$
$ \Rightarrow x=\frac{b(a-10)}{(8-9 a)} .$
View full question & answer→Question 253 Marks
Solve the following equations$: a(x - b) -x (x - 2b) = x + 5(x - b)$
Answer$a(x - b) -x (x - 2b) = x + 5(x - b)$
On simplyfying, we get:
$\Rightarrow ax - ab - x + 2b = x + 5x - 5b$
$\Rightarrow -ab + 2b + 5b = 6x + x - ax$
$\Rightarrow (7 - a)b = (7 - a)x$
$\Rightarrow x = b.$
View full question & answer→Question 263 Marks
Solve the following equations: $a(x - b) -b (x - a) = a^2 - b^2$
Answer$a(x - b) -b (x - a) = a^2 - b^2$
On simplyfying we get:
$\Rightarrow ax - ab - bx + ab = a^2 - b^2$
$\Rightarrow (a - b)x = a^2 - b^2$
$\Rightarrow x = (a + b).$
View full question & answer→Question 273 Marks
Solve the following equations for the unknown: $\frac{1}{5}=\frac{3 \sqrt{x}}{3}$
Answer$\frac{1}{5}=\frac{3 \sqrt{x}}{3} $
$\frac{1}{5}=\sqrt{x}$
Squaring both sides
$\Rightarrow\left(\frac{1}{5}\right)^2=x $
$\Rightarrow x=\frac{1}{25} .$
View full question & answer→Question 283 Marks
Solve the following equations for the unknown: $7-\frac{1}{\sqrt{y}}=0$
Answer$7-\frac{1}{\sqrt{y}}=0 $
$\Rightarrow 7=\frac{1}{\sqrt{y}}$
Squaring both sides
$\Rightarrow(7)^2=\frac{1}{y} $
$\Rightarrow 49=\frac{1}{y} $
$\Rightarrow y=\frac{1}{49} .$
View full question & answer→Question 293 Marks
Solve the following equations for the unknown: $\frac{3 x-5}{7 x-5}=\frac{1}{9}, x \neq \frac{5}{7}$
Answer$\frac{3 x-5}{7 x-5}=\frac{1}{9}, x \neq \frac{5}{7}$
$ \Rightarrow 9(3 x-5)=7 x-5$
$ \Rightarrow 27 x-45=7 x-5$
$\Rightarrow 20 x=40$
$\Rightarrow x=2 .$
View full question & answer→Question 303 Marks
Solve the following equations for the unknown: $\frac{2 x+3}{x+7}=\frac{5}{8}, x \neq-7$
Answer$\frac{2 x+3}{x+7}=\frac{5}{8}, x \neq-7$
$ \Rightarrow 8(2 x+3)=5(x+7)$
$ \Rightarrow 16 x+24=5 x+35$
$\Rightarrow 11 x=11$
$ \Rightarrow x=1 .$
View full question & answer→Question 313 Marks
Solve the following equations for the unknown: $\frac{5}{3 x-2}-\frac{1}{8}=0, x \neq 0, x \neq \frac{2}{3}$
Answer$\frac{5}{3 x-2}-\frac{1}{8}=0, x \neq 0, x \neq \frac{2}{3} $
$ \Rightarrow \frac{5}{3 x-2}=\frac{1}{8}$
$ \Rightarrow 40=3 x-2 $
$ \Rightarrow 3 x=42 $
$\Rightarrow x=14 .$
View full question & answer→Question 323 Marks
Solve the following equations for the unknown: $\frac{5}{x}-11=\frac{2}{x}+16, x \neq 0$
Answer$\frac{5}{x}-11=\frac{2}{x}+16, x \neq 0$
$\Rightarrow \frac{5}{x}-\frac{2}{x}=11+16 $
$ \Rightarrow \frac{5-2}{x}=27$
$\Rightarrow \frac{3}{x}=27$
$ \Rightarrow x=\frac{3}{27} $
$ \Rightarrow x=\frac{1}{9} .$
View full question & answer→Question 333 Marks
Solve the following equation for the unknown: $\frac{2 m}{3}-\frac{m}{2}=1$
Answer$\frac{2 m}{3}-\frac{m}{2}=1 $
$\therefore \frac{4 m-3 m}{6}=1 $
$ \therefore 4 m-3 m=6 $
$ \therefore m=6 .$
View full question & answer→Question 343 Marks
Solve the following equation for the unknown: $-\frac{3.4\ m }{2.7}=\frac{10.2}{9}$
Answer$-\frac{3.4\ m }{2.7}=\frac{10.2}{9}$
$ \therefore(-3.4\ m )(9)=(10.2)(2.7)$
$\therefore y=-\frac{10.2 \times 2.7}{3.4 \times 9}$
$\therefore y=-\frac{102 \times 27}{34 \times 9 \times 10} $
$ \therefore y=-\frac{3 \times 3}{1 \times 1 \times 10} $
$ \therefore y=-0.9$
View full question & answer→Question 353 Marks
Solve the following equation for the unknown: $\frac{1.5 y}{3}=\frac{7}{2}$
Answer$\frac{1.5 y}{3}=\frac{7}{2}$
$\therefore(1.5 y)(2)=(7)(3) $
$ \therefore y=\frac{7 \times 3}{1.5 \times 2}$
$ \therefore y=\frac{70 \times 3}{15 \times 2}$
$ \therefore y=7 .$
View full question & answer→Question 363 Marks
Solve the following equation for the unknown: $\frac{4 x}{27}=\frac{8}{9}$
Answer$\frac{4 x}{27}=\frac{8}{9} $
$ \therefore(4 x)(9)=(8)(27)$
$ \therefore x=\frac{8 \times 27}{4 \times 9} $
$\therefore x=\frac{2 \times 3}{1 \times 1}$
$ \therefore x=6$
View full question & answer→Question 373 Marks
Solve the following equations for the unknown: $(3x - 1)^2 + (4x + 1)^2 = (5x + 1)^2 + 5$
Answer$(3x - 1)^2 + (4x + 1)^2 = (5x + 1)^2 + 5$
Opening squares, we get
$[9x^2 + 1 + 2(3x)(1)] + [16x^2+ 1 + 2(4x)(1)] = [25x^2 + 1 2(5x)(1)] + 5$
$\Rightarrow 25x^2 + 2 + 8x + 6x = 25x^2 + 6 + 10x$
$\Rightarrow 4x = 4$
$\Rightarrow x = 1.$
View full question & answer→Question 383 Marks
Solve the following equations for the unknown$: 5x + 10 - 4x + 6 = 12x + 20 - 3x + 12$
Answer$5x + 10 - 4x + 6 = 12x + 20 - 3x + 12$
Collecting like terms,
$\Rightarrow x + 16 = 9x + 32$
$\Rightarrow 8x = -16$
$\Rightarrow x = -2.$
View full question & answer→Question 393 Marks
Solve the following equations for the unknown$: 15y - 20 = 2y + 6$
Answer$15y - 20 = 2y + 6$
Collecting like terms,
$13y = 26$
$\Rightarrow y = 2.$
View full question & answer→Question 403 Marks
Solve the following equations for the unknown: $2 x+\sqrt{2}=3 x-4-3 \sqrt{2}$
Answer$2 x+\sqrt{2}=3 x-4-3 \sqrt{2}$
Collecting like terms,
$\Rightarrow x=4 \sqrt{2}+4 $
$=4(\sqrt{2}+1)$
View full question & answer→Question 413 Marks
Solve the following equations for the unknown$: 2x - (3x - 4) = 3x - 4$
Answer$2x - (3x - 4) = 3x - 4$
Collecting like terms,
$\Rightarrow 2x - 3x - 3x = -8$
$\Rightarrow -4x = -8$
$\Rightarrow x = 2.$
View full question & answer→Question 423 Marks
Solve the following equations for the unknown$: 8x - 21 = 3x - 11$
Answer$8x - 21 = 3x - 11$
Collecting like terms,
$8x - 3x = 21 - 11$
$\Rightarrow 5x = 10$
$\Rightarrow x = 2.$
View full question & answer→Question 433 Marks
Solve the following equations for the unknown$: 3x + 8 = 35$
Answer$3x + 8 = 35$
Collecting like terms,
$3x = 27$
$\Rightarrow x = 9.$
View full question & answer→Question 443 Marks
In the following equations, verify if the given value is a solution of the equation$: 3x + 8 = x - 7; x = 3$
Answer$3x + 8 = x - 7; x = 3$
can be written as$ : 2x + 15 = 0$
Putting, $x = 3,$ we get $\text{L.H.S. = R.H.S.}$
Thus, x = 3 is a solution of the equation $3x + 8 = x - 7.$
View full question & answer→Question 453 Marks
Find the value of a when $x = 3$ is a solution set of $ax^2 + (a - 4)x + 1 = a.$
AnswerGiven $x = 3$ is a solution of $ax^2 + (a - 4)x + 1 = a$
$⇒ x = 3$ must satisfy the equation.
$⇒ 9a + (a - 4)3 + 1 - a = 0$
$⇒ 9a + 3a - 12 + 1 - a = 0$
$⇒ 11a - 11 = 0$
$⇒ a = 1.$
View full question & answer→Question 463 Marks
In the following equations, verify if the given value is a solution of the equation: $5x - 2 = 18; x = 4$
Answer$5x - 2 = 18; x = 4$
We know a value is a solution of the equation if it satisfies the equation.
i.e., $x_1$ is a solution if $(x_1) = 0$
Here, $x_1 = 4$.
Put in $5x - 2 = 18$
we get $\text{L.H.S. = R.H.S.}$
Thus, $x = 4$ is a solution of the equation $5x - 2 = 18$.
View full question & answer→