/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Subtract and simplify. $$ \f... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 30

Subtract and simplify. $$ \frac{7}{12}-\frac{2}{9} $$

Short Answer

Expert verified
\(\frac{13}{36}\)

Step by step solution

01

Find a common denominator

To subtract fractions, we need a common denominator. The denominators here are 12 and 9. The least common multiple (LCM) of 12 and 9 is 36.
02

Convert each fraction

Convert each fraction to have the common denominator of 36:For \(\frac{7}{12}\):\(\frac{7}{12}\) becomes \(\frac{7 \times 3}{12 \times 3} = \frac{21}{36}\)For \(\frac{2}{9} \):\(\frac{2}{9}\) becomes \(\frac{2 \times 4}{9 \times 4} = \frac{8}{36}\)
03

Subtract the numerators

Now subtract the two fractions:\(\frac{21}{36} - \frac{8}{36} = \frac{21 - 8}{36} = \frac{13}{36}\)
04

Simplify the fraction (if necessary)

Check if the resulting fraction can be simplified. \(\frac{13}{36}\) is already in its simplest form because 13 and 36 have no common factors (other than 1).

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Key Concepts

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

Common Denominator
When subtracting fractions, the first crucial step is finding a common denominator. The common denominator is essential because it allows the fractions to be easily compared and subtracted. Without a common denominator, it would be like trying to subtract apples from oranges.
To find a common denominator, you need the Least Common Multiple (LCM) of the denominators. In our exercise, the denominators are 12 and 9. The LCM of 12 and 9 is 36. Why 36? Because 36 is the smallest number that both 12 and 9 can divide into without leaving a remainder. This way, both fractions can be expressed in terms of 36, making it possible to do the subtraction.
Least Common Multiple
The Least Common Multiple, or LCM, is a key concept when dealing with fractions, especially in operations like addition and subtraction. The LCM of two numbers is the smallest multiple that is evenly divisible by both numbers.
In our example, the LCM of the denominators 12 and 9 needs to be found. Prime factorization can help:
  • 12 = 2虏 * 3
  • 9 = 3虏
For the LCM, take the highest power of each prime number: we get 2虏 and 3虏. Thus, LCM = 4 * 9 = 36.
Using the LCM allows us to change both fractions so they have the same denominator, making further operations straightforward.
Fraction Simplification
Once you have performed the subtraction, the last step is to simplify the fraction if possible. Simplification means reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1.
For the fraction \(\frac{13}{36}\): Check the greatest common divisor (GCD) of 13 and 36. Since 13 is a prime number and does not divide 36, the fraction is already in its simplest form. Hence, simplifying is not necessary here, but it鈥檚 always good practice to check!- Simplified fractions are easier to read and understand, providing the most concise answer.

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