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Magazine Chapter 2: Problem 5
Simplify. $$\left(\frac{5}{6}\right)^{2}-\frac{5}{9}$$
Short Answer
Expert verified
\( \frac{5}{36} \)
Step by step solution
01
Calculate the square of the fraction
According to the order of operations, you should firstly handle operations inside parentheses. You have a square operation to perform on a fraction \( \frac{5}{6} \). The square of a fraction can be found by squaring the numerator and the denominator separately. So, \( \left(\frac{5}{6}\right)^{2} = \frac{5^2}{6^2} =\frac{25}{36} \).
02
Subtract the fraction
After obtaining the square of the fraction, you will then subtract the fraction \( \frac{5}{9} \) from \( \frac{25}{36} \). But before subtracting, make sure that the fractions have the same denominator. The fraction \( \frac{25}{36} \) can be subtracted from \( \frac{5}{9} \) only when they have the same denominator.
03
Find a common denominator
To find a common denominator, you can usually just multiply the two denominators together. In this case, however, 36 is a multiple of 9. So, you can convert \( \frac{5}{9} \) into a fraction with 36 as denominator. To do this, multiply both the numerator and the denominator by 4. Therefore, \( \frac{5}{9} = \frac{5*4}{9*4} = \frac{20}{36}\). Now, both fractions have the same denominator.
04
Subtract the fractions
Now that both fractions have the same denominator, you can subtract the fractions: \( \frac{25}{36} - \frac{20}{36} \). This operation is performed only on the numerators while the denominator remains the same. The result is \( \frac{5}{36} \). This fraction is already in its simplest form and cannot be simplified further.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominator
When dealing with fractions, finding a common denominator is crucial to perform addition or subtraction. A common denominator is a shared multiple of the denominators of the fractions involved. This allows the fractions to be compared or combined together in a straightforward way.
For example, consider the fractions \( \frac{25}{36} \) and \( \frac{5}{9} \). To subtract these, they need to have the same denominator. Since 36 is already a multiple of 9, we don't need to find a completely new denominator; instead, we convert \( \frac{5}{9} \) so that it has a denominator of 36:
For example, consider the fractions \( \frac{25}{36} \) and \( \frac{5}{9} \). To subtract these, they need to have the same denominator. Since 36 is already a multiple of 9, we don't need to find a completely new denominator; instead, we convert \( \frac{5}{9} \) so that it has a denominator of 36:
- Multiply the numerator and denominator of \( \frac{5}{9} \) by 4.
- This conversion gives us \( \frac{20}{36} \).
Order of Operations
The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed. This ensures consistency and accuracy in solving mathematical expressions.
- Remember the acronym PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
- Parentheses: Simplify expressions inside parentheses first.
- Exponents: Before any addition or subtraction, handle the square \((\frac{5}{6})^2\), which equals \( \frac{25}{36} \).
Subtraction of Fractions
Subtracting fractions involves a straightforward approach once you have a common denominator.
- With a common denominator established, subtract only the numerators.
- Subtract the numerators: 25 - 20 = 5.
- Keep the denominator the same: 36.
