Linear Equations in One Variable Test

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Linear Equations in One Variable Test - 33

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Question 1
1 / -0

A daily wage worker was paid Rs. $$1700$$ during a period of $$30$$ days. During his period he was absent for $$4$$ days and was fined Rs. $$15$$ per day for absence. He was paid the full salary only for $$18$$ days as he came late on the other days. Those who came late were given only half the salary for that day. What was the total salary paid per month to a worker who came on time every day and was never absent?

A
Rs. $$2400$$

B
Rs. $$3000$$

C
Rs. $$2700$$

D
Rs. $$2250$$

Solution

Let the salary of the worker per day be Rs. $$x$$.
Then, $$\displaystyle 18\times x+8\times \frac{x}{2}-4\times 15=1700 $$
$$\displaystyle \Rightarrow 18x+4x-60=1700$$
$$\Rightarrow 22x=1760$$
$$\displaystyle \rightarrow x=\frac{1760}{22}=80$$
$$\displaystyle \therefore $$ Total salary of a worker who came everyday on time for $$30$$ days $$= 30 \times$$ Rs. $$80 = $$ Rs. $$2400$$.
Question 2
1 / -0

3 years ago, the average age of a family of 5 members was 17 years. A baby having been born, the average age of the family is same today. The present age of the baby is

A
3 years

B
2 years

C
$$\displaystyle 1\frac{1}{2}$$ years

D
1 years

Solution

Total age of the family three years ago = 17 x 5 = 85 years.
Let the present age of the child be x years.
Present total age of the family = 85 + 5 x 3 + x = (100 + x) years.
Given $$\displaystyle \frac{100+x}{6}=17 $$
$$\displaystyle \Rightarrow 100+x=102\Rightarrow x=2 $$ years
Question 3
1 / -0

The ages of Rahul and Haroon are in the ratio $$5: 7$$. Four years later, the sum of their ages will be $$56$$ years. What are their present ages?

A
Rahul's age $$= 10$$ years and Haroon's age $$=17$$ years

B
Rahul's age $$= 25$$ years and Haroon's age $$= 42$$ years

C
Rahul's age $$= 28$$ years and Haroon's age $$= 50$$ years

D
Rahul's age $$= 20$$ years and Haroon's age $$= 28$$ years

Solution

Let the common ratio between Rahul's age and Haroon's age be $$x$$.
Therefore, age of Rahul and Haroon will be $$5x$$ and $$7x$$ years respectively.
After $$4$$ years, their ages will be $$(5x+4)$$ and $$(7x+4)$$ rexpectively.
According to the given condition, we have
$$5x+4+7x+4=56$$
$$\Rightarrow 12x+8=56$$
$$\Rightarrow x=4$$
Therefore, present age of rahul is $$5x=5\times 4=20$$ years and present age of Haroon is $$7x=7\times 4=28$$ years.
Question 4
1 / -0

If $$26 - (7m - 9) = -2 m$$, then $$m =$$

A
$$\displaystyle \frac {9}{17}$$

B
$$\displaystyle \frac {35}{9}$$

C
$$7$$

D
$$9$$

Solution

Given, $$26 - (7m - 9) = -2 m$$
$$\Rightarrow 26-7m+9=-2m$$
$$\Rightarrow 35=5m$$
Divide $$5$$ on both the sides, we get
$$ \dfrac {35}{5}=\dfrac {5m}{5} $$
$$\therefore m=7$$
Question 5
1 / -0

Solve $$\displaystyle \frac{x-2}{3} + \frac{3x-5}{5} = \frac{5x-7}{6} - \frac{1}{10}$$

A
$$x=4$$

B
$$x=8$$

C
$$x=5$$

D
$$x=12$$

Solution

Given, $$ \cfrac { x-2 }{ 3 } +\cfrac { 3x-5 }{ 5 } =\cfrac { 5x-7 }{ 6 } -\cfrac { 1 }{ 10 }$$
Multiplying both sides of the given equation by $$30$$ ....(LCM of $$3, 5, 6$$ and $$10$$), we get
$$10(x-2)+6(3x-5)=5(5x-7)-3\times 1$$
$$10x-20+18x-30=25x-35-3$$
$$28x-50=25x-38$$
Transposing the terms, we get
$$28x-25x=50-38$$
$$3x = 12$$
Dividing both sides by $$3$$, we get
$$x = 4$$
Question 6
1 / -0

Solve the linear equation:
$$3 (t - 3) = 5 (2t +1)$$

A
$$t = -2$$

B
$$t = 1$$

C
$$t = 6$$

D
$$t = -9$$

Solution

We have, $$3(t-3)=5(2t+1)$$
$$\Rightarrow 3t-9=10t + 5$$
$$\Rightarrow 3t-10t=5+9$$ [Transposing 10t to LHS and -9 to RHS]
$$\Rightarrow-7t = 14$$
$$\Rightarrow \displaystyle \frac{7t}{7} = - \frac{14}{7}$$
$$\Rightarrow t = - 2$$
Question 7
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The organisers of an essay competition decide that a winner in the competition gets a prize of Rs. $$100$$ and a participant who does not win gets a prize of Rs. $$25$$. The total prize money distributed is Rs. $$3000$$. Find the number of winners, if the total number of participants is $$63$$.

A
$$14$$

B
$$16$$

C
$$19$$

D
$$12$$

Solution

Let x be the number of the winners. So, the number of participants who does not win $$(63-x)$$.

Prize money given to $$x$$ winners = Rs. $$(100 \times x) =$$ Rs. $$100x$$
and prize money given to $$(63 - x)$$ participants = Rs. $$25 (63-x)$$

Given, total prize money distributed = Rs. $$3000$$
$$\therefore 100 x + 25(63-x) = 3000$$
$$\Rightarrow 4x + (63-x) = 120$$ ...[On dividing both sides by 25]

$$\Rightarrow 3x + 63 =120$$
$$\Rightarrow 3x = 120 -63$$ ...[Transposing 63 to RHS]

$$\Rightarrow 3x - 57 \Rightarrow x = \displaystyle \frac{57}{3} = 19$$

$$\therefore$$ The number of winners is $$19$$.
Question 8
1 / -0

If a scooterist drives at the rate of $$25\text{ km/h}$$ he reaches his destination $$7\text{ min}$$ late and if he drives at the rate of $$30\text{ km/h}$$ he reaches his destination $$5\text{ min}$$ earlier. How far is his destination?

A
$$20\text{ km}$$

B
$$25\text{ km}$$

C
$$30\text{ km}$$

D
$$32\text{ km}$$

Solution

Let the distance be $$x\text{ km}$$ far.

From the given information we can conclude that the difference between the time taken at the given two speeds is $$12 \text{ min}.$$

Now, by using the relation between distance, time, and speed,
$$\begin{aligned}{}\frac{x}{{25}} - \frac{x}{{30}}& = \frac{{12}}{{60}}\\\frac{{6x - 5x}}{{150}}& = \frac{1}{5}\\x &= \frac{{150}}{5}\\& = 30\text{ km}\end{aligned}$$

Hence, the desired destination is $$30 \text{ km}$$ far.
Question 9
1 / -0

A grand father is ten times older than his grand daughter. He is also $$54$$ years older than her. Find their present ages.

A
The present age of granddaughter is $$5$$ years and that of grandfather is $$50$$ years.

B
The present age of granddaughter is $$4$$ years and that of grandfather is $$40$$ years.

C
The present age of granddaughter is $$7$$ years and that of grandfather is $$70$$ years.

D
The present age of granddaughter is $$6$$ years and that of grandfather is $$60$$ years.

Solution

Let the present age of granddaughter be $$x$$ years.
So, the present age of grandfather is $$10x$$ years.
According to the given condition, we have
$$10x-x=54$$
$$\Rightarrow 9x=54$$
$$ \Rightarrow x=6$$ ....[On dividing both sides by $$9$$]
Hence, the present age of granddaughter is $$6$$ years and that of grandfather is $$60$$ years.
Question 10
1 / -0

The age of a girl in months is equal to the age of her grandmother in years. If the difference between their ages is $$66$$ years, find their ages.

A
The age of girl is $$5$$ years and that of grandmother is $$60$$ years

B
The age of girl is $$6$$ years and that of grandmother is $$72$$ years

C
The age of girl is $$7$$ years and that of grandmother is $$84$$ years

D
The age of girl is $$8$$ years and that of grandmother is $$96$$ years

Solution

Let the age of grandmother is $$12x$$ years.
Then, the age of girl $$=$$ $$\cfrac { 12x }{ 12 } =x$$ years
According to the given condition,
$$12x - x = 66$$
$$\Rightarrow 11x=66$$
$$\Rightarrow x=6$$
Therefore, the age of girl $$= 6$$ years and that of grandmother $$= 12 \times 6 = 72$$ years.

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Re-Attempt Weekly Quiz Competition

Linear Equations in One Variable Test - 33

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Linear Equations in One Variable Test - 33

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Question 1

Rs. $$2400$$

Rs. $$3000$$

Rs. $$2700$$

Rs. $$2250$$

Solution

Question 2

3 years

2 years

$$\displaystyle 1\frac{1}{2}$$ years

1 years

Solution

Question 3

Rahul's age $$= 10$$ years and Haroon's age $$=17$$ years

Rahul's age $$= 25$$ years and Haroon's age $$= 42$$ years

Rahul's age $$= 28$$ years and Haroon's age $$= 50$$ years

Rahul's age $$= 20$$ years and Haroon's age $$= 28$$ years

Solution

Question 4

$$\displaystyle \frac {9}{17}$$

$$\displaystyle \frac {35}{9}$$

$$7$$

$$9$$

Solution

Question 5

$$x=4$$

$$x=8$$

$$x=5$$

$$x=12$$

Solution

Question 6

$$t = -2$$

$$t = 1$$

$$t = 6$$

$$t = -9$$

Solution

Question 7

$$14$$

$$16$$

$$19$$

$$12$$

Solution

Question 8

$$20\text{ km}$$

$$25\text{ km}$$

$$30\text{ km}$$

$$32\text{ km}$$

Solution

Question 9

The present age of granddaughter is $$5$$ years and that of grandfather is $$50$$ years.

The present age of granddaughter is $$4$$ years and that of grandfather is $$40$$ years.

The present age of granddaughter is $$7$$ years and that of grandfather is $$70$$ years.

The present age of granddaughter is $$6$$ years and that of grandfather is $$60$$ years.

Solution

Question 10

The age of girl is $$5$$ years and that of grandmother is $$60$$ years

The age of girl is $$6$$ years and that of grandmother is $$72$$ years

The age of girl is $$7$$ years and that of grandmother is $$84$$ years

The age of girl is $$8$$ years and that of grandmother is $$96$$ years

Solution

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