Problem 332
Question
In the following exercises, simplify. $$ \frac{2}{5}+\frac{5}{12}+\left(-\frac{2}{5}\right) $$
Step-by-Step Solution
Verified Answer
\( \frac{5}{12} \)
1Step 1: Identify Like Terms
Group the fractions with the same denominators together. Here, \(\frac{2}{5}\) and \(-\frac{2}{5}\) can be grouped.
2Step 2: Combine Like Terms
Add \(\frac{2}{5}\) and \(-\frac{2}{5}\): \(\frac{2}{5} + (-\frac{2}{5}) = 0\).
3Step 3: Simplify the Expression
Now, add the result from the previous step to the remaining fraction: \(\frac{0}{1} + \frac{5}{12} = \frac{5}{12}\).
Key Concepts
Combining Like TermsCommon DenominatorsFraction Addition
Combining Like Terms
When simplifying fractions, it's essential to first identify and combine like terms.
Like terms in fractions have the same denominators.
In simple terms, these are fractions that are parts of the same whole.
For example, in the given exercise, \(\frac{2}{5}\) and \(-\frac{2}{5}\) are like terms because they share the same denominator (5).
When combining like terms, you simply add or subtract the numerators, keeping the denominator the same.
This process makes the expression easier to manage.
Like terms in fractions have the same denominators.
In simple terms, these are fractions that are parts of the same whole.
For example, in the given exercise, \(\frac{2}{5}\) and \(-\frac{2}{5}\) are like terms because they share the same denominator (5).
When combining like terms, you simply add or subtract the numerators, keeping the denominator the same.
This process makes the expression easier to manage.
Common Denominators
A common denominator is a shared multiple of the denominators of two or more fractions.
When adding or subtracting fractions, finding a common denominator is an essential step.
In the exercise, \(\frac{2}{5}\) and \(-\frac{2}{5}\) already have common denominators, making them easy to combine.
But \(\frac{5}{12}\) has a different denominator.
Once the fractions with the same denominators are combined, attention shifts to any remaining fractions.
In this case, since combining \(\frac{2}{5}\) and \(-\frac{2}{5}\) results in 0, there's only \(\frac{5}{12}\) left.
Here, no additional adjustment for common denominators is needed.
When adding or subtracting fractions, finding a common denominator is an essential step.
In the exercise, \(\frac{2}{5}\) and \(-\frac{2}{5}\) already have common denominators, making them easy to combine.
But \(\frac{5}{12}\) has a different denominator.
Once the fractions with the same denominators are combined, attention shifts to any remaining fractions.
In this case, since combining \(\frac{2}{5}\) and \(-\frac{2}{5}\) results in 0, there's only \(\frac{5}{12}\) left.
Here, no additional adjustment for common denominators is needed.
Fraction Addition
Adding fractions involves adjusting them to the same denominator before combining the numerators.
You can think of it as bringing different pieces of pie to the same size before adding them together.
In the provided solution, since \(\frac{2}{5}\) and \(-\frac{2}{5}\) already cancel each other out, the next step only involves simplifying the remaining fraction.
\(\frac{0}{1} + \frac{5}{12} = \frac{5}{12}\).
You can see how the problem becomes straightforward after combining like terms.
Remembering to find common denominators and simplify fractions can make these tasks easier and quicker.
You can think of it as bringing different pieces of pie to the same size before adding them together.
In the provided solution, since \(\frac{2}{5}\) and \(-\frac{2}{5}\) already cancel each other out, the next step only involves simplifying the remaining fraction.
\(\frac{0}{1} + \frac{5}{12} = \frac{5}{12}\).
You can see how the problem becomes straightforward after combining like terms.
Remembering to find common denominators and simplify fractions can make these tasks easier and quicker.
Other exercises in this chapter
Problem 330
In the following exercises, simplify. $$ 27 c+16-27 c $$
View solution Problem 331
In the following exercises, simplify. $$ \frac{1}{2}+\frac{7}{8}+\left(-\frac{1}{2}\right) $$
View solution Problem 333
In the following exercises, simplify. $$ 10(0.1 d) $$
View solution Problem 334
In the following exercises, simplify. $$ 100(0.01 p) $$
View solution