Question

Write down the reciprocal of (a) 18 (b) \(\frac{1}{11}\) (c) \(\frac{3}{8}\) (d) \(\frac{2 x}{3 y}\)

Short answer

Question: Find the reciprocals of the following numbers and expressions: a) 18 b) \( \frac{1}{11} \) c) \( \frac{3}{8} \) d) \( \frac{2x}{3y} \) Answer: a) \( \frac{1}{18} \) b) 11 c) \( \frac{8}{3} \) d) \( \frac{3y}{2x} \)

Step-by-step solution

Finding the reciprocal of 18

To find the reciprocal of 18, we need to take the fraction \(\frac{1}{18}\). This is because 18 multiplied by \(\frac{1}{18}\) would result in 1.

Finding the reciprocal of \(\frac{1}{11}\)

To find the reciprocal of \(\frac{1}{11}\), we need to take the fraction \(\frac{11}{1}\). The reciprocal of a fraction \(\frac{a}{b}\) is given by \(\frac{b}{a}\). In this case, a = 1 and b = 11, so the reciprocal is \(\frac{11}{1}\), which is equal to 11.

Finding the reciprocal of \(\frac{3}{8}\)

To find the reciprocal of \(\frac{3}{8}\), we need to take the fraction \(\frac{8}{3}\). The reciprocal of a fraction \(\frac{a}{b}\) is given by \(\frac{b}{a}\). In this case, a = 3 and b = 8, so the reciprocal is \(\frac{8}{3}\).

Finding the reciprocal of \(\frac{2x}{3y}\)

To find the reciprocal of \(\frac{2x}{3y}\), we need to take the fraction \(\frac{3y}{2x}\). The reciprocal of a fraction \(\frac{a}{b}\) is given by \(\frac{b}{a}\). In this case, a = 2x and b = 3y, so the reciprocal is \(\frac{3y}{2x}\).

Key concepts

Reciprocal of a Number

Understanding the concept of the reciprocal of a number is essential in arithmetic and algebra. The reciprocal—also known as the multiplicative inverse—of any number is the value that, when multiplied with the original number, will give a product of 1. For instance, to find the reciprocal of a whole number, such as 18, you would write it as a fraction with the number 18 as the denominator and 1 as the numerator, resulting in \( \frac{1}{18} \).

This concept can be easily remembered with a simple rule: to obtain the reciprocal of a number 'n', express it as \( \frac{1}{n} \). Thus, the reciprocal of 18 is \( \frac{1}{18} \) because \( 18 \times \frac{1}{18} = 1 \). It's a straightforward yet powerful concept with broad applications, particularly when dealing with division and fractions in various mathematical problems.

Reciprocal of a Fraction

When it comes to fractions, the reciprocal or multiplicative inverse still follows the basic principle of producing a product of 1 when multiplied by the original fraction. The process, however, involves flipping the fraction. Essentially, if you have a fraction \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \). For example, the reciprocal of \( \frac{1}{11} \) is \( \frac{11}{1} \) or simply 11, since \( \frac{1}{11} \times \frac{11}{1} = 1 \).

Similarly, for the fraction \( \frac{3}{8} \), its reciprocal is \( \frac{8}{3} \) because flipping the numerator and denominator gives us the fraction that, when multiplied with the original, results in 1. This concept is immensely useful for divining by fractions since division can be easily converted into multiplication by the reciprocal of a divisor.

Algebraic Fractions Reciprocals

The principle of finding reciprocals extends to algebraic fractions as well, which are fractions that contain variables. Algebraic fractions follow the same rule of swapping the numerator and the denominator to find the reciprocal. If an algebraic fraction is given by \( \frac{2x}{3y} \), its reciprocal is \( \frac{3y}{2x} \). This means that \( \frac{2x}{3y} \times \frac{3y}{2x} = 1 \), proving they are multiplicative inverses.

Understanding reciprocals of algebraic fractions is crucial for solving equations and simplifying expressions, especially when working with complex rational expressions. Remember, the variables behave just as numbers do; thus, their positions are switched in the same order to calculate the reciprocal. Grasping this concept empowers students to tackle algebra with confidence.