All questions of Numbers for Class 8 Exam

To solve this problem, let's assume the tens digit of the two-digit number as 'x' and the units digit as 'x+2'.

Let's break down the information given in the problem:

1. The units digit exceeds the tens digit by 2:
The units digit is 'x+2' and the tens digit is 'x'. Therefore, we can write the equation: (x+2) - x = 2. Simplifying, we get 2 = 2, which is true. This condition is satisfied.

2. The product of the number and the sum of its digits is equal to 144:
The given number is represented as 10x + (x+2), since it is a two-digit number. The sum of its digits is x + (x+2) = 2x + 2.

The product of the number (10x + (x+2)) and the sum of its digits (2x + 2) is equal to 144. So, we can write the equation: (10x + (x+2)) * (2x + 2) = 144.

Expanding the equation, we get: (11x + 2) * (2x + 2) = 144.
Simplifying further, we get: 22x^2 + 26x + 4 = 144.
Rearranging the equation, we have: 22x^2 + 26x - 140 = 0.

Now we need to factorize this quadratic equation. Dividing each term by 2, we get: 11x^2 + 13x - 70 = 0.

Factoring this equation, we find: (11x - 14)(x + 5) = 0.

Setting each factor to zero, we get two possible solutions:
11x - 14 = 0, which gives x = 14/11.
x + 5 = 0, which gives x = -5.

Since x represents the tens digit, it cannot be negative. Therefore, x = 14/11.

However, the tens digit must be a whole number, so we discard the solution x = 14/11.

Hence, the only possible solution is x = 2.

Therefore, the tens digit is 2 and the units digit is 4 (2+2).

So, the number is 24, which is option A.

Given: One-third of one-fourth of a number is 15.

Let's assume the number as 'x'.

One-third of one-fourth of 'x' can be represented as (1/3) * (1/4) * x.

According to the given information, (1/3) * (1/4) * x = 15.

Solving the equation, we get (1/12) * x = 15.

Now, we need to find three-tenths of 'x', which can be represented as (3/10) * x.

We can find the value of 'x' using the equation we obtained earlier.

(1/12) * x = 15

Multiplying both sides of the equation by 12 to isolate 'x', we get:

x = 15 * 12

x = 180

Now, substitute the value of 'x' in the expression (3/10) * x to find the answer:

(3/10) * 180 = 54

Therefore, three-tenths of the number is 54.

Hence, the correct answer is option 'D' - 54.

Let's assume the three numbers to be a, b, and c.

Given:
a^2 + b^2 + c^2 = 138 ...(1)
ab + bc + ac = 131 ...(2)

We need to find the sum of these three numbers, which is a + b + c.

To simplify the problem, let's first square equation (2):
(ab + bc + ac)^2 = 131^2
a^2b^2 + b^2c^2 + a^2c^2 + 2ab^2c + 2abc^2 + 2a^2bc = 17161

Now, let's subtract equation (1) from this:
(a^2b^2 + b^2c^2 + a^2c^2 + 2ab^2c + 2abc^2 + 2a^2bc) - (a^2 + b^2 + c^2) = 17161 - 138
a^2b^2 + b^2c^2 + a^2c^2 + 2ab^2c + 2abc^2 + 2a^2bc - a^2 - b^2 - c^2 = 17023

Rearranging the terms, we get:
a^2(b^2 - 1) + b^2(c^2 - 1) + c^2(a^2 - 1) + 2abc(ab + bc + ac - a - b - c) = 17023

Now, let's substitute equation (2) into this equation:
a^2(b^2 - 1) + b^2(c^2 - 1) + c^2(a^2 - 1) + 2abc(131 - a - b - c) = 17023

Simplifying further:
a^2b^2 - a^2 + b^2c^2 - b^2 + a^2c^2 - c^2 + 262abc - 2abc(a + b + c) = 17023

Rearranging and factoring:
(a^2b^2 + b^2c^2 + a^2c^2 - a^2 - b^2 - c^2) + abc(262 - 2a - 2b - 2c) = 17023

Substituting equation (1) into this equation:
138 + abc(262 - 2a - 2b - 2c) = 17023

Simplifying:
abc(262 - 2a - 2b - 2c) = 16885

Dividing both sides by 2:
abc(131 - a - b - c) = 8442

Now, let's substitute equation (2) into this equation:
abc(131 - a - b - c) = (ab + bc + ac)(131 - a - b - c)
abc(131 - a - b - c) = 131(ab + bc + ac) - (a^2 + b^2 + c^2)

Substituting the given values:
abc(131 - a - b - c) = 131(131) - 138
abc(131 - a - b

To solve this problem, let's consider the two-digit number as "10x + y," where x represents the tens digit and y represents the units digit.

1. Product of the digits is 8:
We know that the product of the digits is 8, so we can write the equation as:
x * y = 8

2. When 18 is added to the number, the digits are reversed:
When we add 18 to the number, the digits are reversed. So, the new number can be written as "10y + x." We can express this as an equation:
10x + y + 18 = 10y + x

Now, let's solve these equations step by step:

From equation 1, we have:
x * y = 8

From equation 2, we can simplify it by combining like terms:
10x - x + y - 10y = -18
9x - 9y = -18

Dividing both sides of the equation by 9, we get:
x - y = -2

Now, we have a system of two equations:
x * y = 8
x - y = -2

We can solve this system by substitution or elimination method:

Substitution Method:
From the second equation, we can express x as:
x = y - 2

Substituting this value of x into the first equation, we get:
(y - 2) * y = 8
y^2 - 2y - 8 = 0

Factoring the quadratic equation, we get:
(y - 4)(y + 2) = 0

So, y can be either 4 or -2. But since we are dealing with a two-digit number, y cannot be -2.

Therefore, y = 4. Substituting this value back into the equation x = y - 2, we get:
x = 4 - 2 = 2

Hence, the two-digit number is 24, which is option B.

Let's assume the two-digit number as XY, where X represents the tens digit and Y represents the units digit.

Given conditions:
1) The sum of the digits is 15: X + Y = 15
2) The difference between the digits is 3: X - Y = 3

To find the value of X and Y, we can solve these two equations simultaneously.

Adding the two equations, we get:
(X + Y) + (X - Y) = 15 + 3
2X = 18
X = 9

Substituting the value of X in the first equation, we get:
9 + Y = 15
Y = 15 - 9
Y = 6

Therefore, the two-digit number is 96.

Now, let's check the options given:
a) 69: The sum of the digits is 6 + 9 = 15, but the difference between the digits is 9 - 6 = 3, which does not satisfy the given conditions.
b) 78: The sum of the digits is 7 + 8 = 15, but the difference between the digits is 8 - 7 = 1, which does not satisfy the given conditions.
c) 96: The sum of the digits is 9 + 6 = 15, and the difference between the digits is 9 - 6 = 3. This option satisfies both the given conditions.
d) Cannot be determined: This option is incorrect as we have determined that the two-digit number is 96.

Therefore, the correct answer is option c) 96.

Interchanging the digits of a two-digit number and adding the new number to the original number:

Let's consider a two-digit number, AB, where A represents the tens digit and B represents the units digit. The value of this number can be represented as 10A + B.

If we interchange the digits, the new number will be BA, which can be represented as 10B + A.

When we add the original number (10A + B) to the new number (10B + A), the resulting number can be calculated as:

(10A + B) + (10B + A) = 11A + 11B = 11(A + B)

Divisibility by 11:

To determine if the resulting number is divisible by 11, we need to check if (A + B) is divisible by 11.

If (A + B) is divisible by 11, then 11(A + B) will also be divisible by 11.

Example:

Let's consider the number 34.

If we interchange the digits, we get the number 43.

When we add 34 and 43, we get 77, which is divisible by 11 because 7 + 7 = 14, and 14 is divisible by 11.

Conclusion:

In general, whenever we interchange the digits of a two-digit number and add the new number to the original number, the resulting number will always be divisible by 11.

Therefore, the correct answer is option D) 11.