Linear Equations in One Variable Test

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Linear Equations in One Variable Test - 41

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Weekly Quiz Competition

Question 1
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Wayne would like to buy a school jacket priced at $$\$81$$, but the price of the jacket is $$\$59$$ more than he has. In which of the following equations does $$x$$ represent the number of dollars Wayne has?

A
$$x + 81 = 59$$

B
$$x - 81 = 59$$

C
$$x - 59 = - 81$$

D
$$x - 81 = - 59$$

E
$$x - 59 = 81$$

Solution

The price of jacket is $$ $81 $$.
Let the number of dollars Wayne had be $$x$$.
Given that $$81-x = 59$$
Which implies $$ x-81 = -59$$
Question 2
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The acute angles of a right triangle have a ratio of $$2:3$$. What is the difference between the two angle measures?

A
$$42$$ degrees

B
$$54$$ degrees

C
$$18$$ degrees

D
$$72$$ degrees

Solution

Use the fact that the sum of the angles in any triangle $$= 180^o$$ degrees.

Let the measure of angle be $$2x$$ and $$3x$$

$$2x + 3x + 90^o = 180^o$$ (Combine like-terms).

$$\Rightarrow 5x + 90^o = 180^o$$ (Subtract $$90$$ from both sides).

$$\Rightarrow 5x = 90^o$$ (Divide both sides by $$5$$).

$$\Rightarrow x = 18^o$$

$$\Rightarrow 2x = 36^o$$

$$\Rightarrow 3x = 54^o$$

The measures of the acute angles in a right triangle whose ratio is $$2:3$$ are:

$$36$$ degrees and $$54$$ degrees.

Therefore, difference between the two angles $$=54^o-36^o=18^o$$

Hence, option C is correct.
Question 3
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What is the solution of $$\displaystyle \frac{x-5}{2} - \frac{x-3}{5} = \frac{1}{2}$$?

A
$$x = 5$$

B
$$x = 7$$

C
$$x = 8$$

D
$$x = 9$$

Solution

Given ,$$\dfrac{x-5}{2}-\dfrac{x-3}{5}=\dfrac{1}{2}$$

$$\dfrac{5x-25-2x+6}{5\times 2}=\dfrac{1}{2}$$

$$3x-19=5$$

$$3x=24$$

$$x=8$$
Question 4
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The acute angles of a right triangle have a ratio of $$12 : 3.$$ What is the difference between the two angle measures?

A
$$42^\circ$$

B
$$54^\circ$$

C
$$64^\circ$$

D
$$72^\circ$$

Solution

Let the measure of acute angles be $$12x$$ and $$3x$$ respectively.

Now, we will apply the angle sum property of a triangle,
$$12x+3x+90=180$$
$$\Rightarrow 15x+90^\circ=180^\circ$$
$$\Rightarrow 15x=90^\circ$$
$$\Rightarrow x=6^\circ$$

So,
$$\Rightarrow 12x=12\times6^\circ$$
$$=72^\circ$$
$$\Rightarrow 3x=3\times6^\circ$$
$$=18^\circ$$

Therefore, the acute angles are $$72^\circ$$ and $$18^\circ$$ degrees.
The difference between these angles is $$72^\circ-18^\circ=54^\circ.$$
Question 5
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The diagonal of a rectangle is thrice its smaller side. What is the ratio of its sides?

A
$$\sqrt {2} : 1$$

B
$$2\sqrt {2} : 1$$

C
$$3 : 2$$

D
$$\sqrt {3} : 1$$

Solution

Let, The smaller side of rectangle be $$x$$ m.
$$\therefore$$ the diagonal of rectangle is $$3x$$ m.
$$\therefore$$ By Pythagorous theorem,
$$(Diagonal)^2=(smaller \space side)^2+ (Greater \space side)^2$$
$$\implies (3x)^2= (x)^2+(Greater \space side)^2$$
$$\implies (Greater \space side)^2=9x^2-x^2=8x^2$$
$$\implies Greater side =\sqrt{8x^2}=2\sqrt2x$$
$$\therefore$$ Required ratio=$$2\sqrt2x:x=2\sqrt2:1$$
Question 6
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Two numbers are in the ratio $$3 : 8$$. If the sum of the numbers is $$165$$, then find the numbers.

A
$$45,120$$

B
$$65,120$$

C
$$35,130$$

D
None of these

Solution

let the no. be $$3x$$ and $$8x$$
As $$3x+8x=165$$
$$\Rightarrow 11x=165$$
$$\Rightarrow x=15$$
$$\therefore $$ No. are $$45$$ and $$120$$
Question 7
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If $$\dfrac{18\times 72\times x}{48\times 315} = \sqrt{81}$$, find the value of $$x$$.

A
$$115$$

B
$$150$$

C
$$15$$

D
$$105$$

Solution

$$\dfrac{18\times 72\times x}{48\times 315}=\sqrt{81}$$

$$\implies \dfrac{9\times 72 \times x}{24\times 315}=9$$

$$\implies \dfrac{9\times 3\times x}{315}=9$$

$$\implies x=\dfrac{9\times 315}{9 \times 3}$$

$$\implies x=105$$
Question 8
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A piece of string is $$40$$ centimeters long. It is cut into three pieces. The longest piece is $$3$$ times as long as the middle-sized and the shortest piece is $$23$$ centimeters shorter than the longest piece. Find the length of the shortest piece(in cm).

A
$$27$$

B
$$5$$

C
$$4$$

D
$$9$$

Solution

Let the length of largest piece be $$3x$$.

Thus length of middle sized part will be $$x$$.

According to the given condition, length of shortest part is $$3x – 23$$.

Or, $$ 3x + x + (3x – 23) = 40. $$
$$7x=63$$

Or, $$x = 9$$

Or, the shortest piece $$= 3(9) – 23 = 4$$
Question 9
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Kiran scored $$80$$ marks in his Science test which is $$\dfrac { 5 }{ 6 } $$ of the total marks. What are the total marks for the test?

A
$$96$$

B
$$100$$

C
$$104$$

D
$$108$$

Solution

Given : Kiran scored $$80$$ marks in his science test which is $$\dfrac{5}{6}$$ of total marks
Let the total marks be $$x$$
Then $$\dfrac{5}{6}$$ of total marks scored by Kiran in science $$=\dfrac{5}{6}\times x=\dfrac{5x}{6}$$
But Kiran scored $$80$$ marks
$$\therefore \dfrac{5x}{6}=80$$
Multiplying by $$6$$ on both the sides, we get
$$\dfrac{5x}{6}\times 6=80\times 6$$
$$5x=480$$
Dividing by $$5$$ on both sides, we get
$$\dfrac{5x}{5}=\dfrac{480}{5}$$
$$\Rightarrow x=96$$
Thus, the total marks is $$96$$.
Question 10
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The height of pole $$A$$ is $$\dfrac { 5 }{ 6 } $$ of pole $$B$$, and the height of pole $$B$$ is $$\dfrac { 1 }{ 2 } $$ of pole $$C$$. If the height of pole $$C$$ is $$72 m$$, how tall is pole $$A$$?

A
$$60 m$$

B
$$30 m$$

C
$$18 m$$

D
$$12 m$$

Solution

Given height of pole C is 72 m
And given height of pole B is $$\dfrac{1}{2}$$ of pole C
Then height of pole B $$=\dfrac{1}{2}\times 72=36$$ m
And given height of pole A is $$\dfrac{5}{6}$$ of pole B
Then height of pole A $$=\dfrac{5}{6}\times 36=30$$ m

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Linear Equations in One Variable Test - 41

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Linear Equations in One Variable Test - 41

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SHARING IS CARING

Question 1

$$x + 81 = 59$$

$$x - 81 = 59$$

$$x - 59 = - 81$$

$$x - 81 = - 59$$

$$x - 59 = 81$$

Solution

Question 2

$$42$$ degrees

$$54$$ degrees

$$18$$ degrees

$$72$$ degrees

Solution

Question 3

$$x = 5$$

$$x = 7$$

$$x = 8$$

$$x = 9$$

Solution

Question 4

$$42^\circ$$

$$54^\circ$$

$$64^\circ$$

$$72^\circ$$

Solution

Question 5

$$\sqrt {2} : 1$$

$$2\sqrt {2} : 1$$

$$3 : 2$$

$$\sqrt {3} : 1$$

Solution

Question 6

$$45,120$$

$$65,120$$

$$35,130$$

None of these

Solution

Question 7

$$115$$

$$150$$

$$15$$

$$105$$

Solution

Question 8

$$27$$

$$5$$

$$4$$

$$9$$

Solution

Question 9

$$96$$

$$100$$

$$104$$

$$108$$

Solution

Question 10

$$60 m$$

$$30 m$$

$$18 m$$

$$12 m$$

Solution

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