How Many Times Does 15 Go Into 135

Determining how many times 15 goes into 135 is a fundamental arithmetic problem that can be approached through various methods, including division, repeated subtraction, and multiplication. Understanding this process is essential for grasping more complex mathematical concepts. This article will explore these methods in detail, provide examples, and offer insights into the underlying principles, ensuring a comprehensive understanding of this basic yet crucial mathematical operation.

Introduction

The question "How many times does 15 go into 135?" is essentially asking us to divide 135 by 15. Division is one of the four basic arithmetic operations, the others being addition, subtraction, and multiplication. It is the inverse operation of multiplication. In simple terms, division helps us determine how many times one number (the divisor) is contained within another number (the dividend). In this case, 135 is the dividend, and 15 is the divisor. The result we obtain is called the quotient.

Understanding how to solve this problem is crucial for several reasons:

Basic Arithmetic Proficiency: It reinforces understanding of division, multiplication, and related concepts.
Problem-Solving Skills: It enhances problem-solving abilities applicable in various real-life scenarios.
Foundation for Advanced Math: It lays the groundwork for more complex mathematical topics such as algebra, calculus, and statistics.

In the following sections, we will explore different methods to solve this problem, providing a step-by-step explanation for each.

Method 1: Long Division

Long division is a standard algorithm used to divide larger numbers, especially when mental calculation is challenging. It provides a systematic way to break down the division process into manageable steps. Here’s how to apply long division to find how many times 15 goes into 135:

Steps for Long Division

Set up the Division Problem: Write the division problem in the long division format. The dividend (135) goes inside the division symbol, and the divisor (15) goes outside.
```
   _______
15 | 135
```
Divide the First Digit(s): Look at the first digit of the dividend (1). Can 15 go into 1? No, it cannot because 1 is less than 15.

Next, consider the first two digits of the dividend (13). Can 15 go into 13? Again, no, because 13 is less than 15.

Now, consider the first three digits of the dividend (135). Can 15 go into 135? Yes, it can.
Estimate the Quotient: Estimate how many times 15 goes into 135. A good starting point is to round 15 to 20 and think: How many times does 20 go into 135? Since 20 x 6 = 120 and 20 x 7 = 140, we can estimate that 15 goes into 135 around 9 times.
Multiply and Subtract:
- Write 9 above the 5 in 135, indicating that 9 is our estimated quotient.
```
   9
15 | 135
```
- Multiply the divisor (15) by the estimated quotient (9): 15 x 9 = 135.
- Write the result (135) under the dividend (135) and subtract.
```
   9
15 | 135
   135
   ---
```

Check the Remainder: Subtract 135 from 135.

       9
    15 | 135
       135
       ---
        0
    ```

The remainder is 0, which means 15 goes into 135 exactly 9 times.

Write the Final Answer: The quotient is 9, so 15 goes into 135 exactly 9 times.

Example of Long Division

Let’s go through the steps again to reinforce understanding:

Set up:
```
   _______
15 | 135
```
Estimate: We estimate that 15 goes into 135 about 9 times.

Multiply and Subtract:

Remainder: The remainder is 0.
Answer: The quotient is 9.

Therefore, 15 goes into 135 exactly 9 times.

Method 2: Repeated Subtraction

Repeated subtraction is a more intuitive method for understanding division, especially for those who are new to the concept. It involves repeatedly subtracting the divisor from the dividend until you reach zero or a number less than the divisor. The number of times you subtract is the quotient.

Steps for Repeated Subtraction

Start with the Dividend: Begin with the number 135.
Subtract the Divisor: Subtract 15 from 135.
- 135 - 15 = 120 (1 subtraction)
Continue Subtracting: Keep subtracting 15 from the result until you reach zero or a number less than 15.
- 120 - 15 = 105 (2 subtractions)
- 105 - 15 = 90 (3 subtractions)
- 90 - 15 = 75 (4 subtractions)
- 75 - 15 = 60 (5 subtractions)
- 60 - 15 = 45 (6 subtractions)
- 45 - 15 = 30 (7 subtractions)
- 30 - 15 = 15 (8 subtractions)
- 15 - 15 = 0 (9 subtractions)
Count the Subtractions: Count how many times you subtracted 15 from 135 to reach zero. In this case, you subtracted 15 a total of 9 times.
Final Answer: The number of times you subtracted (9) is the answer. So, 15 goes into 135 exactly 9 times.

Example of Repeated Subtraction

Here’s a summary of the repeated subtraction process:

135 - 15 = 120
120 - 15 = 105
105 - 15 = 90
90 - 15 = 75
75 - 15 = 60
60 - 15 = 45
45 - 15 = 30
30 - 15 = 15
15 - 15 = 0

We subtracted 15 nine times from 135 to reach 0. Therefore, 15 goes into 135 exactly 9 times.

Method 3: Multiplication

Multiplication, being the inverse of division, can also be used to solve this problem. Instead of dividing 135 by 15, we can ask ourselves: "What number multiplied by 15 equals 135?"

Steps for Multiplication

Start with an Estimate: Begin by estimating what number multiplied by 15 would be close to 135.
Trial and Error: Use trial and error to find the correct number.
- Try 5: 15 x 5 = 75 (Too low)
- Try 10: 15 x 10 = 150 (Too high)
- Try 8: 15 x 8 = 120 (Close, but still too low)
- Try 9: 15 x 9 = 135 (Correct!)
Check Your Answer: Once you find a number that, when multiplied by 15, equals 135, you have your answer.
Final Answer: The number you found (9) is the answer. So, 15 goes into 135 exactly 9 times.

Example of Multiplication

Let's illustrate with the correct multiplication:

15 x 9 = 135

Therefore, 15 goes into 135 exactly 9 times.

Scientific Explanation

The mathematical operation we’re performing is based on the principles of arithmetic and number theory. Understanding the underlying scientific concepts can provide a deeper appreciation for the simplicity and elegance of mathematics.

Division as the Inverse of Multiplication

Division is fundamentally the inverse operation of multiplication. This means that if a x b = c, then c / b = a. In our case, 15 x 9 = 135, so 135 / 15 = 9. This relationship is crucial for understanding why multiplication can be used to solve division problems.

Euclidean Division

The problem we are solving is an example of Euclidean division, which states that for any two integers a and b (with b ≠ 0), there exist unique integers q and r such that:

a = b x q + r

where 0 ≤ r < |b|.

In our case:

a = 135 (dividend)
b = 15 (divisor)
q = 9 (quotient)
r = 0 (remainder)

So, 135 = 15 x 9 + 0.

Importance of Remainders

In some division problems, the divisor does not divide the dividend evenly, resulting in a remainder. The remainder represents the amount "left over" after dividing as much as possible. In our specific problem, the remainder is 0, indicating that 15 divides 135 perfectly.

Practical Applications

Understanding division and its relationship with multiplication is essential in various fields, including:

Finance: Calculating interest rates, splitting costs, and determining profit margins.
Engineering: Designing structures, calculating material requirements, and analyzing data.
Computer Science: Developing algorithms, optimizing code, and managing data.
Everyday Life: Cooking, shopping, planning events, and managing time.

FAQ

Q: What does it mean when we say "how many times does one number go into another?"

A: It means we are trying to find out how many units of one number (the divisor) are contained within another number (the dividend). Mathematically, this is equivalent to performing a division operation.

Q: Can repeated subtraction always be used to solve division problems?

A: Yes, repeated subtraction can always be used, but it may not be the most efficient method, especially for larger numbers. For smaller numbers, it provides an intuitive way to understand the concept of division.

Q: Why is it important to understand the relationship between multiplication and division?

A: Understanding that multiplication and division are inverse operations helps in verifying answers and provides a deeper understanding of arithmetic principles. It also allows for flexibility in problem-solving strategies.

Q: What if there is a remainder? How does that change the answer?

A: If there is a remainder, it means the divisor does not divide the dividend evenly. The remainder is the amount left over after dividing as much as possible. In the context of the question "how many times does one number go into another," the quotient represents the whole number of times the divisor fits into the dividend, and the remainder is what is left over.

Q: Are there any real-world applications of solving division problems like this?

A: Yes, there are numerous real-world applications, including:

Dividing Resources: Distributing a certain number of items among a group of people.
Calculating Averages: Finding the average of a set of numbers.
Measurement Conversion: Converting between different units of measurement.
Financial Planning: Budgeting and calculating expenses.

Q: How can I improve my division skills?

A: Practice is key. Start with simple division problems and gradually increase the complexity. Use different methods, such as long division and repeated subtraction, to reinforce your understanding. Also, try to relate division problems to real-life scenarios to make the learning process more engaging.

Conclusion

Determining how many times 15 goes into 135 is a fundamental arithmetic problem that can be solved using various methods, including long division, repeated subtraction, and multiplication. Each method offers a unique perspective on the division process, reinforcing the understanding of basic arithmetic principles.

Long division provides a systematic approach for solving division problems, especially with larger numbers.
Repeated subtraction offers an intuitive understanding of division by repeatedly subtracting the divisor from the dividend.
Multiplication, being the inverse of division, can be used to find the quotient by determining what number multiplied by the divisor equals the dividend.

Understanding these methods and the underlying mathematical concepts is crucial for building a strong foundation in mathematics and enhancing problem-solving skills applicable in various real-life scenarios. By mastering these techniques, one can approach division problems with confidence and accuracy.