Olympiad Test: Linear Equations - Class 8 MCQ

Let the weight of the original loaf be x.

David cuts the loaf into two equal halves, so each half weighs x/2.

One half is cut into smaller pieces, and each of the small pieces weighs 20 grams. David ends up with 7 pieces of bread in total: 6 small pieces (from one half) and the remaining half as a whole piece.

The total weight of the 6 small pieces is:

6 × 20 = 120 grams
This 120 grams is half of the original loaf, so the other half also weighs 120 grams. Therefore, the total weight of the original loaf is:

120 + 120 = 240 grams
Thus, the weight of the original loaf is 240 grams.

Let's analyze the sequence and determine how many even numbers satisfy the conditions:

Sequence: 3, 8, 4, 1, 5, 7, 2, 8, 3, 4, 8, 9, 3, 9, 4, 2, 1, 5, 8, 2

Step-by-Step Analysis:

1. Number 8 (2nd position):

   • Preceding number: 3 → 8 is not divisible by 3.
   • Following number: 4 → Condition not met, ignore this 8.

2. Number 4 (3rd position):

   • Preceding number: 8 → 4 is not divisible by 8.
   • Following number: 1 → Condition not met, ignore this 4.

3. Number 2 (7th position):

   • Preceding number: 7 → 2 is not divisible by 7.
   • Following number: 8 → Condition not met, ignore this 2.

4. Number 8 (8th position):

   • Preceding number: 2 → 8 is divisible by 2.
   • Following number: 3 → 8 is not divisible by 3.
   • Condition met: Include this 8.

5. Number 4 (10th position):

   • Preceding number: 3 → 4 is not divisible by 3.
   • Following number: 8 → Condition not met, ignore this 4.

6. Number 8 (11th position):

   • Preceding number: 4 → 8 is divisible by 4.
   • Following number: 9 → 8 is not divisible by 9.
   • Condition met: Include this 8.

7. Number 4 (15th position):

   • Preceding number: 9 → 4 is not divisible by 9.
   • Following number: 2 → Condition not met, ignore this 4.

8. Number 2 (16th position):

   • Preceding number: 4 → 2 is not divisible by 4.
   • Following number: 1 → Condition not met, ignore this 2.

9. Number 8 (19th position):

   • Preceding number: 5 → 8 is not divisible by 5.
   • Following number: 2 → Condition not met, ignore this 8.

10. Number 2 (20th position):

    • This is the last number in the sequence, so no following number to check.

The numbers that satisfy the conditions are the 8 at positions 8 and 11.

Thus, the correct answer is (b) Two.

Mary counts down from 34 while Thomas counts up from 1, calling only the odd numbers. Both count at the same speed.

• Mary's sequence: 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22...
• Thomas's sequence: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25...

The common number they both call at the same time is 23.

A linear equation in one variable must involve only one variable.
Let's check each option:

Option A: 33z + 5 = 0 - This is a linear equation in one variable (z).

Option B: 33(x + y) = 0 - This is not a linear equation in one variable because it involves two variables, x and y.

Option C: 3x + 5 = 0 - This is a linear equation in one variable (x).

Option D: y + 4 = 0 - This is a linear equation in one variable (y).

Thus, the equation in Option B is not a linear equation in one variable.

Let the speed of the slower car be x km/h. Then, the speed of the faster car is x + 10 km/h (since one car is 10 km/h slower than the other).

In three hours, the slower car covers a distance of 3x km, and the faster car covers a distance of 3(x + 10) km.

The total distance between the two cars after 3 hours is 20 km, which is the sum of the distances covered by both cars minus the total distance between the milestones (230 km).
Therefore, we can write the equation:

3x + 3(x + 10) = 230 - 20
    
Simplifying the equation:

3x + 3x + 30 = 210
6x + 30 = 210
6x = 180
x = 30
Thus, the speed of the slower car is 30 km/h, and the speed of the faster car is:

30 + 10 = 40 km/h
So, the speeds of the two cars are 30 km/h and 40 km/h, respectively.

Let’s assume the original number of men is M.

The provisions last for 21 days for M men.

Calculate Total Provisions:

• Total provisions can be represented as:
  • Total Provisions = Number of Men × Number of Days
  • So, Total Provisions = M × 21

Determine the New Scenario:

• Now, only one-seventh of the original number of men will be consuming the provisions.
  • New Number of Men = M/7

Set Up the Equation for the New Scenario:

• Let’s denote the number of days the provisions will last in the new scenario as D.
• Using the proportionality:
  • New Number of Men × New Number of Days = Total Provisions
  • So, (M/7) × D = M × 21

Simplify the Equation:

• Divide both sides of the equation by M to eliminate M: D/7 = 21

Solve for D: Multiply both sides by 7:

• D = 21 × 7
• D = 147

Thus, the provisions will last for 147 days when only one-seventh of the original number of men is using them.

Let 1 person do 1 unit of work in 1 day
When 25 people are employed, amount of work in 1 day = 25 units
Amount of work in 16 days = 25 x 16 = 400
After 16 days, 5 more people are employed and work is completed in 15 days
So work done in next 15 days = 30 x 15 = 450
If those 5 people are not employed , No. of days needed to complete 450 units of work = 450/25 = 18
But they were supposed to complete the work by 16 days
Hence 2 extra days are needed by 25 people.

We have,
 

⇒ 

553x − 41 = 38x + 34
⇒ 15x = 75

 ⇒ x = 75/15 = 5

Let the number be x. Then, One-third of x = x/3, Quarter of x = x/4
One-twelfth of x = x/12
Average of third, quarter and one-twelfth of 

According to question, we have 


⇒ 
⇒ 
⇒ 36x − 4x − 3x − x = 36 × 56 ⇒ 28x = 36 × 56
⇒ 

Hence, the number is 72. 

Let the number be x.
According to question, we have
 

 

Hence, the number is 15.

We are given the equation:
6(3x + 2) - 5(6x - 1) = 6(x - 3) - 5(7x - 6) + 12x
Step 1: Expand both sides of the equation
On the left-hand side:
6(3x + 2) = 18x + 12
-5(6x - 1) = -30x + 5
So, the left-hand side becomes:
18x + 12 - 30x + 5 = -12x + 17
On the right-hand side:
6(x - 3) = 6x - 18
-5(7x - 6) = -35x + 30
So, the right-hand side becomes:
6x - 18 - 35x + 30 + 12x = -17x + 12
Step 2: Set up the simplified equation
-12x + 17 = -17x + 12
Step 3: Solve for x
Move all terms involving x to one side and constants to the other side:
-12x + 17x = 12 - 17
5x = -5
Now, solve for x:
x = -5 / 5
x = -1
Thus, the value of x is -1.

Given the ratio between the two numbers is 5 : 8.

Let the numbers be 5x and 8x respectively.

Given the sum of two numbers is 299.

So, 5x + 8x = 299

⇒ 13x = 299

⇒ x = 299/13

⇒ x = 23

∴ One number = 5x = 5 × 23 = 115 and Second Number = 8x = 8 × 23 = 184

Thus, the product of two numbers = 184 × 115 = 21160

Let the number be x.
We are given that two-thirds of the number is 20 less than the original number. This can be written as:
(2/3)x = x - 20
Step 1: Multiply both sides of the equation by 3 to eliminate the fraction:
2x = 3(x - 20)
Step 2: Expand the equation:
2x = 3x - 60
Step 3: Move all terms involving x to one side:
2x - 3x = -60
-x = -60
Step 4: Solve for x:
x = 60
Thus, the number is 60.

Let the number be x.
We are given that the seventh part of the number exceeds its eighth part by 1. Mathematically, this can be written as:

Step 1: Find a common denominator for the fractions.
The least common denominator of 7 and 8 is 56. So, rewrite the fractions with denominator 56:

Now substitute these into the equation:

Step 2: Simplify the equation.

Step 3: Solve for x.
Multiply both sides by 56:
x = 56
Thus, the required number is 56.

Let the original number be 10a + b, where a is the tens digit and b is the ones digit.
Step 1: Given conditions
We are told that the sum of the digits is 9:
a + b = 9  (Equation 1)
We are also told that if 27 is subtracted from the original number, the digits are interchanged:
10a + b - 27 = 10b + a  (Equation 2)
Step 2: Solve the system of equations
Simplifying Equation 2:
10a + b - 27 = 10b + a
10a + b - a - 10b = 27
9a - 9b = 27
a - b = 3  (Equation 3)
Step 3: Solve for a and b
Now, we have the system of equations:
a + b = 9
a - b = 3
Adding these two equations:
(a + b) + (a - b) = 9 + 3
2a = 12
a = 6
Now, substitute a = 6 into Equation 1:
6 + b = 9
b = 3
Step 4: Find the original number
The original number is:
10a + b = 10 × 6 + 3 = 63
Thus, the original number is 63.