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Chapter 5: Problem 21

Write down the reciprocal of the following: (a) \(\frac{1}{3}+\frac{1}{2}\) (b) \(\frac{x+y}{13}\) (c) \(\frac{2 R+1}{R-1}\) (d) \(4 !\)

Short Answer

Expert verified
(a) \( \frac{6}{5} \), (b) \( \frac{13}{x+y} \), (c) \( \frac{R-1}{2R+1} \), (d) \( \frac{1}{24} \)

Step by step solution

01

Calculate the sum of fractions

For part (a), add the fractions \( \frac{1}{3} \) and \( \frac{1}{2} \) by finding a common denominator: \( \frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6} \). The sum is \( \frac{5}{6} \).
02

Find the reciprocal of the sum (a)

The reciprocal of a fraction is achieved by flipping its numerator and denominator. Thus, the reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5} \).
03

Determine the reciprocal of expression (b)

The expression \( \frac{x+y}{13} \) has a reciprocal found by flipping the fraction: \( \frac{13}{x+y} \).
04

Calculate the reciprocal of expression (c)

The reciprocal of \( \frac{2R+1}{R-1} \) is found similarly by flipping the fraction: \( \frac{R-1}{2R+1} \).
05

Evaluate factorial expression for (d)

Calculate \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
06

Find factorial reciprocal (d)

The reciprocal of \( 4! \) is \( \frac{1}{24} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Addition
Adding fractions is a foundational skill in mathematics. To combine fractions, you need to have a common denominator. This denominator is a shared multiple of the individual denominators. For instance, to add \(\frac{1}{3}\) and \(\frac{1}{2}\), the common denominator is 6. Think of it as a small party where each fraction must bring the same treat. Here, \(\frac{1}{3}\) becomes \(\frac{2}{6}\) and \(\frac{1}{2}\) converts to \(\frac{3}{6}\). Now they are speaking the same language! So they add up to \(\frac{5}{6}\).
  • Ensure that denominators are the same before adding fractions.
  • Convert each fraction to equivalent fractions with a common denominator.
  • Combine the numerators once the denominators are equal.
Understanding this process makes fraction addition straightforward and easy to stick to your math memory bank.
Reciprocal Calculation
The reciprocal of a fraction is quite simply its flipped version, where the numerator becomes the denominator and vice versa.
For any fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). It's like a fun switcheroo! Here鈥檚 how you can find the reciprocal:
  • Identify the numerator and denominator.
  • Flip them to form the reciprocal.
  • For example, the reciprocal of \(\frac{5}{6}\) is \(\frac{6}{5}\).
Reciprocals are essential in various arithmetic operations, such as division of fractions. Learning this concept will help you navigate through math equations more smoothly.
Factorial Reciprocal
Factorials are a way to describe a number multiplied by each whole number less than itself, often represented by an exclamation mark. It's a major player in permutations and combinations. For example, \(4!\) means \(4\times3\times2\times1 = 24\). Once you have the number from a factorial, its reciprocal is just \(\frac{1}{\text{factorial}}\).
For instance, the reciprocal of \(4!\) is \(\frac{1}{24}\). Here鈥檚 how it works:
  • Calculate the factorial value.
  • Simplify \(\frac{1}{\text{factorial value}}\).
Understanding the simplicity of finding the reciprocal of a factorial helps in various mathematical contexts, especially where inverses are involved.
Common Denominator
Finding a common denominator is crucial when adding or comparing fractions. It ensures that all fractions are on equal footing, much like making sure everyone speaks the same language at a meeting.

The lowest common denominator (LCD) is the smallest number that all denominators can divide into cleanly.
Here is how to do it:
  • List some multiples of each denominator you鈥檙e working with.
  • Identify the smallest multiple that appears in each list- that鈥檚 your common denominator!
  • Convert each fraction to an equivalent fraction with the LCD as the new denominator.
Understanding how to find and use a common denominator not only makes fraction addition and subtraction easier, but also improves your skills in higher mathematical operations.

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Most popular questions from this chapter

Transpose each of the following formulae to make the given variable the subject: (a) \(x=\frac{c}{y}\), for \(y\) (b) \(x=\frac{c}{y}\), for \(c\) (c) \(k=\frac{2 n+5}{n+3}\), for \(n\) (d) \(T=2 \pi \sqrt{\frac{R-L}{g}}\), for \(R\)

Evaluate without using a calculator: (a) \(\left(\frac{2}{3}\right)^{2}\) (b) \(\left(\frac{2}{5}\right)^{3}\) (c) \(\left(\frac{1}{2}\right)^{2}\) (d) \(\left(\frac{1}{2}\right)^{3}\) (e) \(0.1^{3}\)

Evaluate using a calculator: (a) \(7^{3}\) (b) \((14)^{3.2}\)

Simplify \(\frac{2 x-5}{10}-\frac{3 x-2}{15}\)

Explain why \(x^{2}\) is a factor of \(4 x^{2}+3 y x^{3}+5 y x^{4}\) but \(y\) is not. Factorise \(4 x^{2}+3 y x^{3}+5 y x^{4}\)
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