Operations and Applications of Rational Numbers

Questions and Answers

Explain the step-by-step process of dividing rational numbers (2/3) by (1/5).

Multiply by the reciprocal: (2/3) * (5/1) = 10/3

Provide the result of adding (2/3) and (3/4) as a simplified rational number.

2/3

What is the product of (2/3) and (4/5) expressed as a simplified rational number?

8/15

How is the speed of an object calculated using rational numbers if a car travels 120 miles in 4 hours?

<p>30 miles per hour</p> Signup and view all the answers

Explain how rational numbers are used in expressing the debt-to-equity ratio of a company.

<p>Debt-to-equity ratio of a company is expressed as a ratio of two rational numbers (e.g., 15/4).</p> Signup and view all the answers

Study Notes

Operations with Rational Numbers

Rational numbers are a subset of real numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. The set of rational numbers is represented by the symbol ℚ. Operations on rational numbers involve addition, subtraction, multiplication, and division.

Addition of Rational Numbers

When adding rational numbers, there are two cases to consider:

Same denominators: In this case, simply add the numerators and keep the same denominator. For example, (2/8) + (3/8) = (2 + 3)/8 = 5/8.
Different denominators: To add rational numbers with different denominators, you need to find a common denominator. For example, when adding (2/5) and (3/4), the common denominator is (2 * 4) = 8. So, (2/5) + (3/4) = (2 * 3)/(5 * 4) = 6/20.

Subtraction of Rational Numbers

Subtraction of rational numbers follows the same principles as addition. To subtract (a/b) - (c/d), find the common denominator and subtract the numerators while retaining the common denominator. For example, (2/8) - (3/8) = (2 - 3)/8 = -1/8.

Multiplication of Rational Numbers

Multiplication of rational numbers is straightforward: multiply the numerators and the denominators separately. For example, (2/3) * (4/5) = (2 * 4)/(3 * 5) = 8/15.

Division of Rational Numbers

To divide rational numbers, reverse the division sign and multiply the dividend by the reciprocal of the divisor. For example, (2/3) / (1/5) = (2/3) * (5/1) = (2 * 5)/(3 * 1) = 10/3.

Examples of Operations on Rational Numbers

Here are some examples of operations on rational numbers:

(2/3) + (3/4) = (2 * 4)/(3 * 4) = 8/12 = 2/3
(2/3) - (1/2) = (2 * 2)/(3 * 2) = 4/6 = 2/3
(2/3) * (4/5) = (2 * 4)/(3 * 5) = 8/15
(2/3) / (1/2) = (2/3) * (2/1) = (2 * 2)/(3 * 1) = 4/3

Applications of Rational Numbers

Rational numbers have various applications in different fields:

Fractional Measurements

Rational numbers are used to express fractional measurements, such as the length of a piece of wood, the duration of a song, or the amount of water in a container. For example, 3/4 of a pie can be divided into three equal parts.

Financial Calculations

In finance, rational numbers are used to represent financial proportions, such as the ratio of assets to liabilities or the percentage of an investment that is returned. For example, a company's debt-to-equity ratio can be expressed as (6000000000)/(4000000000) = 15/4.

Mathematical Problems

Rational numbers are used to solve mathematical problems, such as finding the slope of a line, calculating the area of a triangle, or determining the speed of an object. For example, if a car travels 120 miles in 4 hours, the average speed is (120)/(4) = 30 miles per hour.

In conclusion, rational numbers play a significant role in mathematics and various practical applications. Understanding the operations and properties of rational numbers is essential for solving problems in different fields.