Problem 88 - Prealgebra

Question

Divide. $$15 \div 3$$

Step-by-Step Solution

Answer

The result of 15 divided by 3 is 5.

Step 1: Understand Division

Division is the operation of dividing or sharing a number into equal parts. The problem given is trying to find out how many times the divisor (3) can fit into the dividend (15).

Step 2: Perform the Division

Divide 15 by 3. Count how many times the number 3 can be subtracted from 15 until what remains is less than 3. You can also think of it as counting up in multiples of 3 until you reach 15: 3, 6, 9, 12, 15. There are five 3s in 15.

Step 3: Determine the Quotient

The number of times 3 can be subtracted from 15, reaching zero, is 5. This result is called the quotient.

Step 4: Check the Result

Multiply the quotient (5) by the divisor (3) to check the result: 5 × 3 = 15. This confirms the division is done correctly as the product equals the original number we divided (15).

Key Concepts

Understanding the Quotient

In division, the quotient is the result you get after dividing one number by another. It tells you how many times the divisor can fit into the dividend without exceeding it. Consider the division problem where 15 is divided by 3. Here, the quotient is 5 because 3 fits into 15 exactly five times.

To find the quotient, you can either subtract the divisor repeatedly from the dividend or use multiplication. If you're visually or hands-on inclined, you might picture levels or groups forming until a remainder less than the divisor is left.

• Example: Keep subtracting 3 from 15:
  • 15 - 3 = 12
  • 12 - 3 = 9
  • 9 - 3 = 6
  • 6 - 3 = 3
  • 3 - 3 = 0
• This sequence shows that 3 has been subtracted five times.

This calculation confirms that the quotient is 5. Simply put, in the context of sharing or breaking down items, the quotient gives the answer to "how many are in each group?" when the items are distributed evenly.

Role of the Divisor

The divisor is the number by which we divide the dividend. It acts as a frame of reference that tells us how big each group will be when we divide. In our example, where 15 is divided by 3, the divisor is 3.

The divisor must never be zero because dividing by zero is undefined in mathematics. It's like asking how many ways we can divide something when there's no number to divide by. Meanwhile, the divisor helps guide the division process:

• In "15 divided by 3," 3 is the divisor, indicating that each group will have exactly 3 units.
• Understanding the divisor’s size compared to the dividend tells us if division will be simple or if there's a need for deeper calculations.

Always remember the role of the divisor: it defines each part or share in the total. This understanding makes dividing more intuitive especially when large numbers are involved.

Defining the Dividend

The dividend is the total you're working to divide. It's the starting quantity you want to break into smaller equal parts, as indicated by the divisor. In our division of 15 by 3, the number 15 is designated as the dividend.

The dividend can be thought of as the "whole" in the division problem. Its size compared to the divisor determines how many complete and equal parts you can make (which is the quotient).

• The process involves distributing this whole (dividend) equally using the divisor as a guide.
• The dividend is the number from which the subtraction begins when understanding division as repeated subtraction.

Breaking down the dividend using the divisor helps reach the quotient effectively. In simple terms, the dividend is the "what you have in total" before it gets distributed into parts defined by the divisor.