Write as a single fraction.
[tex]1+\frac{2 c+3 d}{2 c}-\frac{4 c+5 d}{5 c}[/tex]
Simplify your answer as much as possible.
Answer (2)
Find the common denominator for all terms, which is 10 c .
Rewrite each term with the common denominator: 10 c 10 c + 10 c 5 ( 2 c + 3 d ) − 10 c 2 ( 4 c + 5 d ) .
Combine the numerators: 10 c 10 c + 5 ( 2 c + 3 d ) − 2 ( 4 c + 5 d ) .
Simplify the numerator by combining like terms: 10 c 12 c + 5 d .
The final expression is 10 c 12 c + 5 d .
Explanation
Understanding the Problem We are given the expression 1 + 2 c 2 c + 3 d − 5 c 4 c + 5 d . Our goal is to write this expression as a single, simplified fraction.
Finding the Common Denominator To combine these terms into a single fraction, we need to find a common denominator. The denominators are 1 , 2 c , and 5 c . The least common denominator (LCD) is 10 c .
Rewriting with Common Denominator Now, we rewrite each term with the common denominator 10 c :
1 = 10 c 10 c
2 c 2 c + 3 d = 5 ( 2 c ) 5 ( 2 c + 3 d ) = 10 c 10 c + 15 d
5 c 4 c + 5 d = 2 ( 5 c ) 2 ( 4 c + 5 d ) = 10 c 8 c + 10 d
Combining the Fractions Substitute these back into the original expression:
10 c 10 c + 10 c 10 c + 15 d − 10 c 8 c + 10 d
Combining Numerators Now, combine the numerators over the common denominator:
10 c 10 c + ( 10 c + 15 d ) − ( 8 c + 10 d )
Distributing the Negative Sign Distribute the negative sign in the numerator:
10 c 10 c + 10 c + 15 d − 8 c − 10 d
Simplifying the Numerator Combine like terms in the numerator:
10 c ( 10 c + 10 c − 8 c ) + ( 15 d − 10 d ) = 10 c 12 c + 5 d
Final Answer The simplified expression as a single fraction is 10 c 12 c + 5 d .
Examples
Fractions are used in everyday life, such as when cooking, measuring ingredients, or calculating proportions. For example, if you are baking a cake and need to increase the recipe by 50%, you would multiply each ingredient by 1 + 2 1 = 2 3 . Simplifying complex fractional expressions helps in accurately adjusting recipes or understanding financial ratios.
The expression 1 + 2 c 2 c + 3 d − 5 c 4 c + 5 d simplifies to 10 c 12 c + 5 d when combined into a single fraction. This was achieved by finding a common denominator and rewriting each term accordingly. After combining and simplifying, we arrive at the final answer.
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