Number Systems

To solve the problem, we need to find a two-digit number based on the following conditions: the sum of its digits is 10, and after subtracting 18, the resulting number has equal digits.

Let the two-digit number be represented as , where is the tens digit and is the units digit. The conditions can be expressed as:

1.

2. results in a number with equal digits, say . Therefore, .

By rearranging the second equation, we have:

.

Since the result must be a two-digit number with equal digits, can only be 1 to 9. Let's start by solving for .

Substituting the sum condition into the equation obtained, we organize as:

Substitute into :

Testing valid values for , we try to find integer solutions for and :

When , , thus:

Then

Thus the calculated number is: .

Hence, the correct number is 73.

To find the number of ducks in the zoo, we use the given information: There are 160 heads and 450 legs. Let the number of ducks be and the number of rabbits be . Since each duck and rabbit has one head, we have the equation:

Each duck has 2 legs and each rabbit has 4 legs. Thus, the second equation is:

We can simplify the second equation by dividing it by 2:

Now, we have the system of equations:

Subtract equation 1 from equation 2:

Substitute back into equation 1:

Therefore, the number of ducks in the zoo is 95.

A 6-digit number formed by consecutive natural numbers can be represented as: . The sum of these digits will be:

This expression is always divisible by 3 because:

- Any integer when multiplied by 6 results in a number divisible by 3, as 6 itself is divisible by 3.

- The constant 15 is divisible by 3.

Therefore, is divisible by 3, making the original number always divisible by 3.

To find the value of , we must first solve for and .

Step 1: Find

The mean of the numbers 3, 4, 9, 2k, 10, 8, 6, and is 8. The formula for the mean is:

For these numbers:

Simplify and solve for :

Step 2: Find

The mode of the numbers 2, 2, 3, , , 4, 4, 5, and 6 is given as 4. Mode is the number that appears most frequently.

For 4 to be the mode, it must appear more frequently than any other number. Consider permutations of :

Since 4 is the mode, means .

Step 3: Calculate

With and , calculate:

Thus, the value of is 2.

To determine the difference between the greatest and smallest of the given fractions, we need to compare each of them. The fractions are , , , , and . To make comparison easy, convert each fraction to a common denominator or compare them in decimal form.

Converting each fraction to decimal:

By comparing these decimal values, we can identify the smallest and greatest fractions:

• Smallest:
• Greatest:

Next, calculate the difference between the greatest and smallest fractions:

Difference =

To subtract, use a common denominator (8):

So,

Therefore, the difference between the greatest and smallest fractions is .

To solve the problem, we need to find a two-digit number where the product of its digits is 27 and adding 9 to it reverses the digits.

1. Understanding the Concepts:

- Two-digit number: Can be expressed as , where is the tens digit and is the units digit.
- Product of digits:
- Digit reversal condition: Adding 9 to the number reverses the digits:

2. Set Up the Equations:

From the condition:
Rewriting: Also given:

3. Substituting and Solving:

Substitute into the product equation: Solving the quadratic: Since this doesn't give an integer, we check integer factor pairs of 27:

• 1 × 27
• 3 × 9
• 9 × 3
• 27 × 1

Try → , and check reversal: Number = 10×3 + 9 = 39
39 + 9 = 48 → digits reversed → Valid

Final Answer:

The number is 39.