Geeta Wants To Simplify The Following Problem To Get The Answers In Different Forms:${ 1110_2 + 10_5 + \frac{4 \times 10^3}{2 \times 10^2} }$(a) What Is The Difference Between 10 And { 10_2 $}$?(b) Write The Answer Of Geeta In

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Introduction

In mathematics, we often come across problems that involve mixed numbers, base conversion, and arithmetic operations. Geeta, a math enthusiast, wants to simplify the following problem to get the answers in different forms. In this article, we will explore the problem and provide step-by-step solutions to help Geeta achieve her goal.

Problem Statement

The problem statement is given as:

11102+105+4×1032×102{ 1110_2 + 10_5 + \frac{4 \times 10^3}{2 \times 10^2} }

This problem involves adding a binary number, a base-5 number, and a fraction. Geeta wants to simplify this expression and find the answers in different forms.

Part (a) - Difference between 10 and 10_2

To solve part (a), we need to find the difference between 10 and 10_2.

What is 10_2?

10_2 is a binary number, which means it is represented in base 2. To convert 10_2 to decimal, we need to understand the positional value of each digit.

In binary, the positional value of each digit is as follows:

  • 2^0 = 1
  • 2^1 = 2
  • 2^2 = 4
  • 2^3 = 8
  • ...

Using this positional value, we can convert 10_2 to decimal as follows:

10_2 = (1 × 2^1) + (0 × 2^0) = 2 + 0 = 2

What is the difference between 10 and 10_2?

Now that we know 10_2 is equal to 2, we can find the difference between 10 and 10_2.

Difference = 10 - 2 = 8

Therefore, the difference between 10 and 10_2 is 8.

Part (b) - Simplifying the Expression

To solve part (b), we need to simplify the expression:

11102+105+4×1032×102{ 1110_2 + 10_5 + \frac{4 \times 10^3}{2 \times 10^2} }

Simplifying 1110_2

First, let's convert 1110_2 to decimal.

1110_2 = (1 × 2^3) + (1 × 2^2) + (1 × 2^1) + (0 × 2^0) = 8 + 4 + 2 + 0 = 14

Simplifying 10_5

Next, let's convert 10_5 to decimal.

10_5 = (1 × 5^1) + (0 × 5^0) = 5 + 0 = 5

Simplifying the Fraction

Now, let's simplify the fraction:

4×1032×102{ \frac{4 \times 10^3}{2 \times 10^2} }

To simplify this fraction, we can divide the numerator and denominator by their greatest common divisor (GCD).

GCD(4 × 10^3, 2 × 10^2) = 2 × 10^2

Dividing the numerator and denominator by 2 × 10^2, we get:

4×1032×102=2×1011{ \frac{4 \times 10^3}{2 \times 10^2} = \frac{2 \times 10^1}{1} }

Simplifying further, we get:

2×1011=20{ \frac{2 \times 10^1}{1} = 20 }

Adding the Simplified Terms

Now that we have simplified each term, we can add them together:

14 + 5 + 20 = 39

Therefore, the simplified expression is 39.

Conclusion

Introduction

In our previous article, we explored the problem of simplifying a mixed number and base conversion problem. Geeta, a math enthusiast, wanted to simplify the expression:

11102+105+4×1032×102{ 1110_2 + 10_5 + \frac{4 \times 10^3}{2 \times 10^2} }

We broke down the problem into smaller parts, converted binary and base-5 numbers to decimal, simplified a fraction, and added the simplified terms together. In this Q&A article, we will answer some common questions related to the problem.

Q: What is the difference between 10 and 10_2?

A: The difference between 10 and 10_2 is 8. To find this difference, we converted 10_2 to decimal and subtracted it from 10.

Q: How do I convert a binary number to decimal?

A: To convert a binary number to decimal, you need to understand the positional value of each digit. In binary, the positional value of each digit is as follows:

  • 2^0 = 1
  • 2^1 = 2
  • 2^2 = 4
  • 2^3 = 8
  • ...

Using this positional value, you can convert a binary number to decimal by multiplying each digit by its positional value and adding the results.

Q: How do I convert a base-5 number to decimal?

A: To convert a base-5 number to decimal, you need to understand the positional value of each digit. In base-5, the positional value of each digit is as follows:

  • 5^0 = 1
  • 5^1 = 5
  • 5^2 = 25
  • 5^3 = 125
  • ...

Using this positional value, you can convert a base-5 number to decimal by multiplying each digit by its positional value and adding the results.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator. Once you have found the GCD, you can divide both the numerator and denominator by the GCD to simplify the fraction.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. To find the GCD of two numbers, you can use the Euclidean algorithm or list the factors of each number and find the greatest common factor.

Q: How do I add simplified terms together?

A: To add simplified terms together, you need to combine the like terms. Like terms are terms that have the same variable and exponent. Once you have combined the like terms, you can add the simplified terms together.

Conclusion

In this Q&A article, we answered some common questions related to the problem of simplifying a mixed number and base conversion problem. We covered topics such as converting binary and base-5 numbers to decimal, simplifying fractions, and adding simplified terms together. We hope that this article has been helpful in clarifying any questions you may have had about the problem.

Additional Resources

If you are interested in learning more about binary and base-5 numbers, fractions, and other math topics, we recommend checking out the following resources:

  • Khan Academy: Khan Academy offers a wide range of free online math courses and resources.
  • Mathway: Mathway is an online math problem solver that can help you with a wide range of math topics.
  • Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you with math, science, and other topics.

We hope that this article has been helpful in your math journey!