Linear Equations in One Variable Test

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Linear Equations in One Variable Test - 46

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Weekly Quiz Competition

Question 1
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The sum of $$2$$ consecutive numbers is $$13$$. Find the numbers.

A
$$6$$ and $$7$$.

B
$$5$$ and $$7$$.

C
$$6$$ and $$5$$.

D
$$7$$ and $$9$$.

Solution

Let the two consecutive numbers be $$x$$ and $$x+1$$
According to the question
$$x+x+1=13$$
$$2x+1=13$$
$$2x=13-1$$
$$2x=12$$
$$x=\dfrac{12}{2}$$
$$x=6$$
$$x+1=6+1=7$$
Therefore the two consecutive numbers are $$6$$ and $$7$$.
Question 2
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A man travelled two-fifth of his journey by train, one-third by bus, one-fourth by car and the remaining $$3\ km$$ on foot. What is the length of his total journey?

A
$$180$$

B
$$150$$

C
$$120$$

D
$$140$$

Solution

Let the length of his total journey be $$x$$
So,
$$\dfrac{2}{5}x+\dfrac{1}{3}x+\dfrac{1}{4}x+3=x$$
solving
$$x\left(\dfrac{2}{5}+\dfrac{1}{3}+\dfrac{1}{4}\right)-x=-3$$
$$x\left(\dfrac{2}{5}+\dfrac{1}{3}+\dfrac{1}{4}-1\right)=-3$$
$$x\left(\dfrac{2\times 12+20+1\times 15-60}{60}\right)=-3$$
$$x\left(\dfrac{24+20+15-60}{60}\right)=-3$$
$$x\left(\dfrac{-1}{60}\right)=-3$$
$$\dfrac{x}{60}=3$$
$$\therefore x=180\ km$$
Question 3
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The sum of three consecutive even numbers is $$276$$. Find the numbers.

A
$$96,92,94$$

B
$$91,92,94$$

C
$$90,94,98$$

D
$$90,92,94$$

Solution

Solution:- (D) $$90, 92, 94$$
Let the numbers be $$2a, 2a + 2, 2a + 4$$.
Now according to the question-
$$2a + \left( 2a + 2 \right) + \left(2a + 4 \right) = 276$$
$$2a + 2a + 2 + 2a + 4 = 276$$
$$6a + 6 = 276$$
$$a + 1 = \cfrac{276}{6}$$
$$a = 46 - 1 = 45$$
Therefore, the numbers are-
$$2a = 2 \times 45 = 90$$
$$2a + 2 = 2 \times 45 + 2 = 92$$
$$2a + 4 = 2 \times 45 + 4 = 94$$
Hence the numbers are $$90, 92$$ and $$94$$.
Question 4
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If twelve less than $$3$$ times a number is $$27$$, then the number is ___

A
$$29$$

B
$$39$$

C
$$65$$

D
$$13$$

Solution

Let $$x$$ be a number.

$$\Rightarrow\,3x-12=27$$

$$\Rightarrow\,3x=27+12$$

$$\Rightarrow\,3x=39$$

$$\Rightarrow\,x=\dfrac{39}{3}=13$$

$$\therefore\,x=13$$
Question 5
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If one fourth, half and one third of the number are added to the number itself, then result is equal to $$25$$. Find the number.

A
$$10$$

B
$$11$$

C
$$12$$

D
$$13$$

Solution
Question 6
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If $$\dfrac{1}{8}+\dfrac{1}{9}=\dfrac{x}{10}$$, then $$x$$ is equal to

A
$$100$$

B
$$1.36$$

C
$$170$$

D
$$2.36$$

Solution

$$\dfrac{1}{8} + \dfrac{1}{9} = \dfrac{x}{10}$$

$$\implies 10\left(\dfrac{1}{8} + \dfrac{1}{9} \right) = x$$

$$\implies 10 \left(\dfrac{9 + 8}{72} \right) = x$$

$$\implies \dfrac{170}{72} = x$$

$$\implies x = 2.36$$
Question 7
1 / -0

If the number exceeds $$\dfrac{2}{3}$$ of itself by $$7$$ ,then

A
$$21$$

B
$$44$$

C
$$2$$

D
$$7$$

Solution

Let the no. be $$x$$
$$2x/3=x-7\\\implies x=21$$
Question 8
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Find the solution to the equation $$\dfrac{4x}{5}-\dfrac{7}{4}=\dfrac{x}{5}+\dfrac{x}{4}$$.

A
$$5$$

B
$$6$$

C
$$7$$

D
$$0$$

Solution

Given, $$\dfrac{4x}{5}-\dfrac{7}{4}=\dfrac{x}{5}+\dfrac{x}{4}$$.

$$\Rightarrow \left(\dfrac{4x}{5}-\dfrac{7}{4}\right)= \left(\dfrac{x}{5}+\dfrac{x}{4}\right)$$

$$\Rightarrow\left(\dfrac{(4\times4x)-(7\times5)}{20}\right)=\left(\dfrac{5x+4x}{20}\right)$$ ... [Cross-multiplying the denominators]

$$\Rightarrow\left(\dfrac{16x-35}{20}\right)=\left(\dfrac{9x}{20}\right)$$

$$\Rightarrow \,16x-35=9x$$.

Transposing $$x$$ terms to one side, we get,
$$\Rightarrow \,16x-9x=35$$

$$\Rightarrow \,7x=35$$

$$\Rightarrow\,x=\dfrac{35}{7}=5$$

$$\therefore\,x=5$$.

Hence, option $$A$$ is correct.
Question 9
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Which of the following is a linear equation in one variable?

A
$$2x+1=y-3$$

B
$$2t-1=3t+5$$

C
$$2x-1=x^{2}$$

D
$$x^{2}-x+1=0$$

Solution

$$\textbf{Step - 1: Checking all options}$$
$$A) 2x+1=y-3$$
$$\text{We can see that the given equation has two variables x and y in and }$$
$$\text{hence is not a linear equation in}$$ $$\text{one variable}$$

$$B) 2t-1=3t+5$$
$$\text{We can clearly see that the given equation has only one variable t and}$$
$$\text{hence is a linear equation in one }$$$$\text{variable.}$$

$$C) 2x-1=x^2$$
$$\text{We can see that the given equation is not linear as its variable has}$$
$$\text{a power greater than one}$$

$$D) x^2-x+1=0$$
$$\text{We can see that the given equation is not linear as its variable has}$$
$$\text{a power greater than one}$$
$$\textbf{Thus, the equation }\mathbf{ 2t-1=3t+5}\textbf{ is a linear equation in one variable.}$$
Question 10
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Rashmi has a sum of $$Rs. x$$. She spend $$Rs. 800$$ on grocery, $$Rs. 600$$ on cloths and $$Rs. 500$$ on education and received as $$Rs. 200$$ as a gift. How much money (in $$Rs$$) is left with her?

A
$$x - 1700$$

B
$$x - 1900$$

C
$$x + 200$$

D
$$x - 2100$$

Solution

Total money $$=Rs. x$$
Money spent $$=Rs. 800$$ on grocery
Money spent $$=Rs. 600$$ on cloths
Money spent $$=Rs. 500$$ on education
Money left with Rashmi
$$=x-(Rs. 800+ Rs. 600+ Rs. 500)$$
$$=x-Rs. 1900$$
She received a gift of $$=Rs. 200$$
$$\therefore $$ Money left $$=x-1900+200$$
$$=x-1700$$
Hence correct option is $$A$$.

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Linear Equations in One Variable Test - 46

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Linear Equations in One Variable Test - 46

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SHARING IS CARING

Question 1

$$6$$ and $$7$$.

$$5$$ and $$7$$.

$$6$$ and $$5$$.

$$7$$ and $$9$$.

Solution

Question 2

$$180$$

$$150$$

$$120$$

$$140$$

Solution

Question 3

$$96,92,94$$

$$91,92,94$$

$$90,94,98$$

$$90,92,94$$

Solution

Question 4

$$29$$

$$39$$

$$65$$

$$13$$

Solution

Question 5

$$10$$

$$11$$

$$12$$

$$13$$

Solution

Question 6

$$100$$

$$1.36$$

$$170$$

$$2.36$$

Solution

Question 7

$$21$$

$$44$$

$$2$$

$$7$$

Solution

Question 8

$$5$$

$$6$$

$$7$$

$$0$$

Solution

Question 9

$$2x+1=y-3$$

$$2t-1=3t+5$$

$$2x-1=x^{2}$$

$$x^{2}-x+1=0$$

Solution

Question 10

$$x - 1700$$

$$x - 1900$$

$$x + 200$$

$$x - 2100$$

Solution

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