Linear Equations in One Variable Test

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Linear Equations in One Variable Test - 47

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Question 1
1 / -0

The sum of two numbers is 72. if one number is double the other find the numbers.

A
$$24,48$$

B
$$25,50$$

C
$$23,46$$

D
None f these

Solution

Given that,
The sum of two numbers is $$72$$.
One of the numbers is double that of the other.
To find out,
The numbers.

Let one of the numbers be $$x$$.
Hence, the other number will be $$2x$$.
The sum of the two numbers is $$72$$.
Hence, $$x+2x=72$$
$$3x=72\\$$
$$x=\dfrac{72}{3}$$
$$=24$$
Hence, $$2x=2\times 24$$
$$=48$$
Hence, the numbers are $$24$$ and $$48$$.
Question 2
1 / -0

The sum of three consecutive integers is 72. What is the sum of the squares of the first number and the last number?

A
$$1201$$

B
$$576$$

C
$$1236$$

D
$$1154$$

Solution

Given that, the sum of three consecutive integers is $$72$$
To find out: The sum of squares of the first and last number.

Let the three consecutive numbers be $$x, \ x+1$$, and $$x+2$$
$$\therefore \ x +\left (x+1\right) + \left(x+2\right) = 72$$
$$\Rightarrow 3x + 3 = 72$$
$$\Rightarrow 3x = 69$$
$$\Rightarrow x =\dfrac{69}{3}= 23$$
Thus the numbers are $$23, 24, 25$$.

$$\therefore \ $$ The sum of squares of first and last numbers $$= 23^2 + 25^2= 529 +625 = 1154$$.

Hence, the sum of squares of the first and last number is $$1154$$.
Question 3
1 / -0

On increasing the salary of a man by $$25\%$$, it becomes $$Rs. 20000$$. What was his original salary?

A
$$Rs. 15000$$

B
$$Rs. 16000$$

C
$$Rs. 18000$$

D
$$Rs. 25000$$

Solution

Let original salary of him $$=x$$

The value increased $$=25\%$$ of $$Rs. x$$

$$=\left(\dfrac {25}{100} \right )\times x$$

$$=\left (\dfrac {25x}{100} \right)$$

$$=\left(\dfrac x4\right )$$

Salary after increment $$=(x+\left (\dfrac x4\right))$$

$$=\dfrac {(4x+x)}4$$

$$=\left(\dfrac {5x}4\right )$$

His increased salary $$=Rs. 20000$$
$$\left (\dfrac {5x}4 \right)=20000$$

$$x=\dfrac {(20000\times 4)}5$$

$$x=4000\times 4$$

$$x=Rs. 16000$$
Question 4
1 / -0

The sum of three consecutive multiples of $$7$$ is $$357$$. Find the smallest multiple.

A
$$112$$

B
$$126$$

C
$$119$$

D
$$116$$

Solution

Let us take three consecutive multiples of $$7$$ as,
$$7x,(7x+7),(7x+14)$$ [$$x$$ is a natural number]
$$7x+(7x+7)+(7x+14)=357$$
$$\Rightarrow 21x+21=357$$
$$\Rightarrow 21(x+1)=357$$
$$\Rightarrow \frac{21(x+1)}{21}=\frac{357}{21}$$ [Diving both sides by $$21$$]
$$\Rightarrow x+1=17$$
$$x=17-1$$
$$x=16$$
Hence, the smallest multiple of $$7$$ is $$7*16=112$$
Question 5
1 / -0

The digit in the ten's place of a two-digit number is $$3$$ more than the digit in the units place. Let the digit at the unit's place be $$b$$. Then the number is

A
$$11b+30$$

B
$$10b+30$$

C
$$11b+3$$

D
$$10b+3$$

Solution

We take unit place to be $$b$$,
So, the digit at ten's place = $$(3+b)$$,
Hence, the number = $$10(3+b)+b = 30+10b+b = 11b+30$$.
Question 6
1 / -0

Linear equation in one variable has

A
Only one variable with any power.

B
Only one term with a variable.

C
Only one variable with power $$1$$.

D
Only constant term.

Solution

$$\textbf{Step - 1: Defining}$$
$$\text{A linear equation is a relation between variables of degree 1. }$$
$$\text{A linear equation in one variable is a relation of a variable of power 1}$$

$$\textbf{Hence a linear equation in one variable has only one variable with power 1}$$
Question 7
1 / -0

A linear equation in one variable has

A
Only one solution

B
Two solution

C
More than two solutions

D
No solution

Solution

$$\textbf{Step - 1: Defining}$$
$$\text{A linear equation is a relation between variables of degree 1. }$$
$$\text{A linear equation in one variable is a relation of a variable of power 1}$$
$$\implies \text{It has only one solution}$$
$$\textbf{Hence option A is correct}$$
Question 8
1 / -0

If the mean of $$5, 7, 9, x $$ is $$9$$ then the value of $$ x $$ is

A
$$11$$

B
$$15$$

C
$$18$$

D
$$16$$

Solution

Mean is defined as summation of all entities divided by number of entities.
According to the given condition,
$$9=\dfrac{5+7+9+x}{4}$$
$$21+x=36$$
$$x=36-21\\\ \ \ =15$$
Question 9
1 / -0

A container is $$\dfrac{1}{8}$$ full of water. After $$10$$ cups of water are added, the container is $$\dfrac{3}{4}$$ full. What is the volume of the container, in cups?

A
$$13\dfrac{1}{3}$$

B
$$13\dfrac{1}{2}$$

C
$$15$$

D
$$16$$

E
$$40$$

Solution

Let the container full of water be $$\dfrac {1}{8}x$$.
After $$10$$ cups added, the container is $$\dfrac {3}{4}$$ full.
$$\dfrac {x}{8}+10=\dfrac {3x}{4}$$
$$\Rightarrow \dfrac {6x}{8}-\dfrac {x}{8}=10$$
$$\Rightarrow 5x=80$$
$$\Rightarrow x=16$$
Therefore, volume of container in cups is $$16$$.
Question 10
1 / -0

Lakshmi is a cashier in a bank. She has notes of denomination of Rs. 20, Rs. 10 and Rs. 5. The ratio of number of these notes is $$1 : 2 : 3$$. The total cash with Lakshmi is Rs. $$1100$$. How many notes of Rs. $$20$$ denomination does she have?

A
$$10$$

B
$$20$$

C
$$30$$

D
None of the above.

Solution

Let the common multiple be $$x$$.
Therefore, the number of notes of each denomination are:
Rs. $$20 = 1(x) = x$$ notes
Rs. $$10 = 2(x) = 2x$$ notes
Rs. $$5 = 3(x) = 3x$$ notes

According to the given condition, total amount with Lakshmi is Rs. $$1100$$.
$$20\left( x \right) + 10\left( 2x \right) +5 \left( 3x \right) =1100$$
$$\Longrightarrow 20x + 20x + 15x = 1100$$
$$\Longrightarrow\displaystyle 55x=1100$$
$$\Longrightarrow\displaystyle x=20$$
Therefore, Lakshmi has $$20$$ notes of Rs. $$20$$ denomination.

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Linear Equations in One Variable Test - 47

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Linear Equations in One Variable Test - 47

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Question 1

$$24,48$$

$$25,50$$

$$23,46$$

None f these

Solution

Question 2

$$1201$$

$$576$$

$$1236$$

$$1154$$

Solution

Question 3

$$Rs. 15000$$

$$Rs. 16000$$

$$Rs. 18000$$

$$Rs. 25000$$

Solution

Question 4

$$112$$

$$126$$

$$119$$

$$116$$

Solution

Question 5

$$11b+30$$

$$10b+30$$

$$11b+3$$

$$10b+3$$

Solution

Question 6

Only one variable with any power.

Only one term with a variable.

Only one variable with power $$1$$.

Only constant term.

Solution

Question 7

Only one solution

Two solution

More than two solutions

No solution

Solution

Question 8

$$11$$

$$15$$

$$18$$

$$16$$

Solution

Question 9

$$13\dfrac{1}{3}$$

$$13\dfrac{1}{2}$$

$$15$$

$$16$$

$$40$$

Solution

Question 10

$$10$$

$$20$$

$$30$$

None of the above.

Solution

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