Understanding the Division Problem

This problem asks us to find the dividend in a division equation based on given relationships between the divisor, quotient, and remainder, and the value of the remainder itself.

The core formula for division is:

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

We are given the following information:

• The remainder is 40.
• The divisor is 15 times the quotient.
• The divisor is 3 times the remainder.

Our goal is to use this information to find the values of the divisor and the quotient, and then use the division formula to find the dividend.

Step 1: Find the Divisor

We are given that the remainder is 40 and the divisor is 3 times the remainder. We can use this directly to calculate the divisor.

Using the relationship: \(\text{Divisor} = 3 \times \text{Remainder}\)

Substitute the given remainder value:

\(\text{Divisor} = 3 \times 40\)

\(\text{Divisor} = 120\)

So, the divisor is 120.

Step 2: Find the Quotient

We know the divisor is 120. We are also given that the divisor is 15 times the quotient. We can use this relationship to find the quotient.

Using the relationship: \(\text{Divisor} = 15 \times \text{Quotient}\)

Substitute the calculated divisor value:

\(120 = 15 \times \text{Quotient}\)

To find the quotient, divide both sides of the equation by 15:

\(\text{Quotient} = \frac{120}{15}\)

\(\text{Quotient} = 8\)

So, the quotient is 8.

Step 3: Find the Dividend

Now we have all the components needed to find the dividend using the main division formula:

• Divisor = 120
• Quotient = 8
• Remainder = 40 (given)

Using the formula: \(\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}\)

Substitute the values we found:

\(\text{Dividend} = 120 \times 8 + 40\)

First, calculate the product of the divisor and the quotient:

\(120 \times 8 = 960\)

Now, add the remainder to this product:

\(\text{Dividend} = 960 + 40\)

\(\text{Dividend} = 1000\)

Therefore, the dividend is 1000.

Summary of Calculation Steps

Here is a summary of the steps taken to find the dividend:

1. Start with the given remainder: Remainder = 40.
2. Use the relationship "divisor is 3 times the remainder" to find the divisor: Divisor = 3 * 40 = 120.
3. Use the relationship "divisor is 15 times the quotient" and the calculated divisor to find the quotient: 120 = 15 * Quotient \( \implies \) Quotient = 120 / 15 = 8.
4. Use the division formula Dividend = Divisor * Quotient + Remainder with the values found: Dividend = 120 * 8 + 40 = 960 + 40 = 1000.

Final Answer Determination

Based on our calculations, the dividend is 1000.

Revision Table: Key Concepts for Division Problems

Additional Information: Understanding Division Components

Division is one of the four basic arithmetic operations. It involves splitting a number (the dividend) into equal parts, determined by another number (the divisor). The result tells us how many times the divisor fits into the dividend (the quotient) and how much is left over (the remainder).

The relationship \(\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}\) is fundamental to understanding division. It shows that the original number (dividend) can be perfectly reconstructed by multiplying the number of times the divisor went into it (quotient) and adding the leftover part (remainder).

In problems like this, where relationships between the components are given, we use these relationships to find the missing values step-by-step until we have enough information to calculate the final required value, which was the dividend in this case.