Solving a Division Problem: Finding the Dividend

In this problem, we are given information about the relationships between the divisor, quotient, and remainder in a division sum. We need to find the value of the dividend.

We are provided with the following details:

• The divisor is 13 times the quotient.
• The divisor is 6 times the remainder.
• The remainder is 39.

The standard formula for a division sum is:

Dividend = Divisor $\times$ Quotient + Remainder

Step-by-Step Calculation to Find the Dividend

Let's use the given information and the division formula to find the dividend.

Step 1: Find the Divisor

We know that the divisor is 6 times the remainder and the remainder is 39.

Divisor = 6 $\times$ Remainder

Divisor = 6 $\times$ 39

Let's calculate 6 $\times$ 39:

\(6 \times 39 = 6 \times (40 - 1) = 6 \times 40 - 6 \times 1 = 240 - 6 = 234\)

So, the divisor is 234.

Step 2: Find the Quotient

We know that the divisor is 13 times the quotient. We have just found the divisor is 234.

Divisor = 13 $\times$ Quotient

234 = 13 $\times$ Quotient

To find the quotient, we divide the divisor by 13:

Quotient = $\frac{234}{13}$

Let's perform the division:

\(234 \div 13\)

13 goes into 23 once with a remainder of 10. Bring down the 4 to make 104.

13 goes into 104 eight times (\(13 \times 8 = 104\)).

So, $\frac{234}{13} = 18$.

Thus, the quotient is 18.

Step 3: Find the Dividend

Now we have the values for the divisor, quotient, and remainder:

• Divisor = 234
• Quotient = 18
• Remainder = 39

Using the division formula: Dividend = Divisor $\times$ Quotient + Remainder

Dividend = 234 $\times$ 18 + 39

First, calculate 234 $\times$ 18:

\(234 \times 18 = 234 \times (20 - 2) = 234 \times 20 - 234 \times 2\)

\(234 \times 20 = 4680\)

\(234 \times 2 = 468\)

\(4680 - 468 = 4212\)

So, 234 $\times$ 18 = 4212.

Now, add the remainder:

Dividend = 4212 + 39

\(4212 + 39 = 4251\)

Therefore, the dividend is 4251.

Summary of the Division Calculation

Let's verify: \(4251 \div 234\). \(234 \times 18 = 4212\). \(4251 - 4212 = 39\). The remainder is 39, which matches the given information.

The calculated divisor (234) is 6 times the remainder (39), \(234 = 6 \times 39\). This is correct.

The calculated divisor (234) is 13 times the quotient (18), \(234 = 13 \times 18\). This is also correct.

The dividend is 4251.

Revision Table: Key Concepts in Division

Additional Information: Understanding Division Relationships

Division is one of the four basic arithmetic operations, along with addition, subtraction, and multiplication. It is essentially the process of splitting a number into equal parts or groups.

The relationship Dividend = Divisor $\times$ Quotient + Remainder is fundamental to understanding division. This formula allows us to check the result of a division or to find one of the components if the others are known, as we did in this problem.

For example, if you divide 10 by 3:

• Dividend = 10
• Divisor = 3
• Quotient = 3 (since \(3 \times 3 = 9\))
• Remainder = 1 (since \(10 - 9 = 1\))

Using the formula: \(10 = 3 \times 3 + 1\), which is \(10 = 9 + 1\), a true statement.

Problems like the one discussed help reinforce the understanding of these relationships and how to use them to solve for unknown values in a division context.