All questions of Linear Equations in One Variable for Class 10 Exam

Total Notes and Their Value
To solve the problem, we need to set up a system of equations based on the given information about the notes.
Given Data
- Total number of notes = 324
- Total amount = Rs. 12450
- Denominations = Rs. 20 and Rs. 50
Let’s Define Variables
- Let x = number of Rs. 20 notes
- Let y = number of Rs. 50 notes
Setting Up Equations
1. From the total number of notes:
- x + y = 324
2. From the total value of the notes:
- 20x + 50y = 12450
Simplifying the Equations
- From the first equation, we can express y in terms of x:
- y = 324 - x
- Now substitute y in the second equation:
- 20x + 50(324 - x) = 12450
Expanding and Simplifying
- Distributing the 50:
- 20x + 16200 - 50x = 12450
- Combining like terms:
- -30x + 16200 = 12450
- Rearranging gives:
- -30x = 12450 - 16200
- -30x = -3750
- Dividing both sides by -30 results in:
- x = 125
Conclusion
The number of Rs. 20 notes is 125, which corresponds to option 'C'.
Verification
- If x = 125, then:
- y = 324 - 125 = 199
- Checking the total value:
- Total value = 20(125) + 50(199) = 2500 + 9950 = 12450, which is correct.
Thus, the answer is confirmed!

Let's assume the present age as 'x'.

Given that 80% of the age 6 years ago is the same as 60% of the age after 10 years, we can write the equation as:

0.8(x - 6) = 0.6(x + 10)

Simplifying this equation:

0.8x - 4.8 = 0.6x + 6

Subtracting 0.6x from both sides:

0.2x - 4.8 = 6

Adding 4.8 to both sides:

0.2x = 10.8

Dividing both sides by 0.2:

x = 54

Therefore, the present age is 54.

To find the product of the digits of the present age, we can multiply the digits together:

Product = 5 * 4 = 20

Hence, the product of the digits of the present age is 20, which corresponds to option B.

Let's assume the two numbers as x and y.
We are given that the sum of the two numbers is 34, so we can write the equation as:
x + y = 34 ...(1)

We are also given that the product of the two numbers is 288, so we can write the equation as:
xy = 288 ...(2)

To find the sum of their cubes, we need to find the values of x and y first.

Solving the equations:
From equation (1), we can express y in terms of x:
y = 34 - x

Substituting this value of y in equation (2), we get:
x(34 - x) = 288

Expanding the equation:
34x - x^2 = 288

Rearranging the equation:
x^2 - 34x + 288 = 0

Now we can solve this quadratic equation to find the values of x.

Factoring the equation:
(x - 16)(x - 18) = 0

Setting each factor equal to zero:
x - 16 = 0 or x - 18 = 0

Solving for x, we have two possible solutions:
x = 16 or x = 18

Now we can find the corresponding values of y:
For x = 16, y = 34 - 16 = 18
For x = 18, y = 34 - 18 = 16

So the two numbers are 16 and 18.

Finding the sum of their cubes:
16^3 + 18^3 = 4096 + 5832 = 9928

Hence, the sum of their cubes is 9928, which corresponds to option (b).

Given: y^2 = y - 7

To find: The value of y^3

Solution:
To find the value of y^3, we need to multiply y^2 by y.

y^2 * y = (y - 7) * y

Expanding the equation:

y^3 = y * y - 7 * y

Simplifying further:

y^3 = y^2 - 7y

But we already know that y^2 = y - 7.

Substituting this value into the equation:

y^3 = (y - 7) - 7y

Simplifying again:

y^3 = y - 7 - 7y

Combining like terms:

y^3 = -6y - 7

Therefore, the value of y^3 is -6y - 7.

Answer:
The correct option is (A) 8y - 7.

To find the product of two consecutive numbers, we need to let the first number be represented by x, and the second number be represented by x+1.

Given that four times the first number is 10 more than thrice the second number, we can form the equation:

4x = 3(x+1) + 10

Now, let's solve the equation step by step:

4x = 3x + 3 + 10 (Distribute 3 to x+1)
4x - 3x = 13 (Combine like terms)
x = 13 (Divide both sides by x)

So, the first number is 13 and the second number is 13+1 = 14.

To find the product of these two consecutive numbers:

Product = 13 * 14 = 182

Therefore, the correct answer is option B) 182.

Understanding the Problem
To solve the problem, we need to set up equations based on the information given about the ages of the father and son.
Step 1: Define Variables
- Let F = Father's current age
- Let S = Son's current age
Step 2: Set Up Equations
1. In 10 years:
- F + 10 = 2(S + 10)
- This implies that in 10 years, the father will be twice as old as his son.
2. Five years ago:
- F - 5 = 3(S - 5)
- This indicates that five years ago, the father was three times as old as his son.
Step 3: Simplify the Equations
From the first equation:
- F + 10 = 2S + 20
- Rearranging gives: F = 2S + 10 - 10
- So, F = 2S + 10
From the second equation:
- F - 5 = 3S - 15
- Rearranging gives: F = 3S - 15 + 5
- So, F = 3S - 10
Step 4: Solve the System of Equations
Now we have two expressions for F:
1. F = 2S + 10
2. F = 3S - 10
Set them equal to each other:
- 2S + 10 = 3S - 10
Rearranging gives:
- 10 + 10 = 3S - 2S
- 20 = S
Now substitute S back to find F:
- F = 2(20) + 10 = 40 + 10 = 50
Conclusion
Thus, the current ages are:
- Father's age = 50 years
- Son's age = 20 years
The correct answer is option 'A': 50, 20.

To solve this problem, let's assume the first odd number as x.

First Odd Number: x
Second Odd Number: x + 2 (since the numbers are consecutive odd numbers)
Third Odd Number: x + 4 (since the numbers are consecutive odd numbers)

According to the given information:
1/3rd of the first number is 2 more than 1/5th of the third number.

Mathematically, this can be represented as:

1/3 * x = (1/5 * (x + 4)) + 2

Now, let's solve this equation step by step.

Step 1: Simplify the equation

1/3 * x = 1/5 * (x + 4) + 2

Step 2: Multiply both sides of the equation by the least common multiple (LCM) of 3 and 5, which is 15, to eliminate the fractions.

15 * (1/3 * x) = 15 * (1/5 * (x + 4) + 2)

5x = 3(x + 4) + 30

Step 3: Distribute on the right side of the equation

5x = 3x + 12 + 30

Step 4: Combine like terms

5x = 3x + 42

Step 5: Subtract 3x from both sides of the equation

5x - 3x = 42

2x = 42

Step 6: Divide both sides of the equation by 2

x = 42/2

x = 21

Now that we have the value of x, we can find the second number by adding 2 to the first number.

Second Odd Number = x + 2 = 21 + 2 = 23

Therefore, the second number is 23, which matches with option B.