Solving the Number Division Problem

The problem asks us to find a specific number based on information about its division by the sum of two other numbers. We are given the divisor (as a sum), the quotient (as two times the difference), and the remainder.

Let the unknown number be \(N\). According to the problem statement, when \(N\) is divided by the sum of 555 and 445, it gives a quotient and a remainder.

The fundamental relationship between the dividend, divisor, quotient, and remainder in a division operation is:

Dividend = Divisor × Quotient + Remainder

Let's break down the given information and calculate each component:

Step 1: Calculate the Divisor

The divisor is given as the sum of 555 and 445.

Divisor = Sum of 555 and 445

$$ \text{Divisor} = 555 + 445 $$

$$ \text{Divisor} = 1000 $$

So, the number is divided by 1000.

Step 2: Calculate the Difference

The quotient is based on the difference between 555 and 445.

Difference = Difference between 555 and 445

$$ \text{Difference} = 555 - 445 $$

$$ \text{Difference} = 110 $$

Step 3: Calculate the Quotient

The quotient is given as two times their difference (the difference calculated in Step 2).

Quotient = Two times the Difference

$$ \text{Quotient} = 2 \times \text{Difference} $$

$$ \text{Quotient} = 2 \times 110 $$

$$ \text{Quotient} = 220 $$

Step 4: Identify the Remainder

The remainder is directly given in the problem.

Remainder = 30

Step 5: Calculate the Number (Dividend)

Now we have all the parts needed to find the number \(N\). Using the division formula:

$$ \text{Number} = \text{Divisor} \times \text{Quotient} + \text{Remainder} $$

Substitute the values we found:

$$ N = 1000 \times 220 + 30 $$

First, perform the multiplication:

$$ 1000 \times 220 = 220000 $$

Now, add the remainder:

$$ N = 220000 + 30 $$

$$ N = 220030 $$

Thus, the number is 220030.

Let's verify this with the given options:

• 220030
• 22030
• 1220
• 1250

Our calculated number, 220030, matches the first option.

Here is a summary of the values:

The number that satisfies the given conditions is 220030.

Revision Table: Key Division Concepts

Understanding the relationship between dividend, divisor, quotient, and remainder is crucial for solving problems like this. Let's quickly review these terms:

• Dividend: The number being divided. In this problem, this is the unknown number \(N\).
• Divisor: The number by which the dividend is divided. Here, the divisor is the sum of 555 and 445, which is 1000.
• Quotient: The result of the division, showing how many times the divisor fits into the dividend. In this case, the quotient is two times the difference of 555 and 445, which is 220.
• Remainder: The amount left over after the division is complete. Here, the remainder is 30.

The formula connecting these is the foundation of division problems:

$$ \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} $$

Additional Information: Exploring Division with Remainders

Division with a remainder happens when one integer cannot be perfectly divided by another. The remainder is always less than the divisor. If the remainder is 0, it means the dividend is perfectly divisible by the divisor.

For example, if you divide 10 by 3:

• Dividend = 10
• Divisor = 3
• Quotient = 3 (because 3 goes into 10 three times)
• Remainder = 1 (because \(3 \times 3 = 9\), and \(10 - 9 = 1\))

Using the formula: \(10 = (3 \times 3) + 1\), which is true.

In our problem, the remainder of 30 is indeed less than the divisor 1000, which makes sense in the context of division.

This type of problem tests your ability to translate a word problem into a mathematical equation and apply the basic principles of arithmetic operations (addition, subtraction, multiplication, and division concepts).