Solve the following equation: [tex]$\frac{3}{4}+(\frac{-6}{11})+(\frac{-8}{21})+(\frac{5}{22})$[/tex]
A) [tex]$-\frac{125}{462}$[/tex]
B) [tex]$\frac{125}{462}$[/tex]
C) [tex]$\frac{462}{125}$[/tex]
D) [tex]$-\frac{462}{125}$[/tex]
Answer (2)
The simplified result of the equation is 924 47 , but this does not match any of the provided multiple-choice options. It seems there might be an error in the options provided. The correct answer is not found among the choices.
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Find the least common multiple (LCM) of the denominators: LCM(4, 11, 21, 22) = 924.
Rewrite each fraction with the common denominator: 4 3 = 924 693 , 11 − 6 = 924 − 504 , 21 − 8 = 924 − 352 , 22 5 = 924 210 .
Add the fractions: 924 693 − 504 − 352 + 210 = 924 47 .
The simplified result is 924 47 .
Explanation
Problem Analysis We are given the expression 4 3 + ( 11 − 6 ) + ( 21 − 8 ) + ( 22 5 ) . Our goal is to simplify this expression and determine which of the provided options is correct.
Finding the Common Denominator To add these fractions, we need to find a common denominator. The least common multiple (LCM) of the denominators 4, 11, 21, and 22 is 924.
Rewriting Fractions Now we rewrite each fraction with the common denominator of 924:
4 3 = 4 × 231 3 × 231 = 924 693
11 − 6 = 11 × 84 − 6 × 84 = 924 − 504
21 − 8 = 21 × 44 − 8 × 44 = 924 − 352
22 5 = 22 × 42 5 × 42 = 924 210
Adding the Fractions Now we can add the fractions:
924 693 + 924 − 504 + 924 − 352 + 924 210 = 924 693 − 504 − 352 + 210 = 924 47
Simplifying the Fraction Now, we simplify the fraction 924 47 . Since 47 is a prime number, we check if 47 divides 924. 924 ÷ 47 ≈ 19.66 , so 47 does not divide 924. Therefore, the fraction is already in its simplest form.
However, the options provided have a denominator of 462, which is half of 924. Let's check if we made a mistake. The correct calculation is: 924 693 − 504 − 352 + 210 = 924 47
Recalculating the Sum Let's re-examine the problem and the options. We have 924 47 . Notice that 924 = 2 × 462 . So, we can write 924 47 = 2 × 462 47 . None of the options match this result. Let's recalculate the sum:
4 3 − 11 6 − 21 8 + 22 5 = 924 693 − 504 − 352 + 210 = 924 47
Checking the Options It seems there might be a mistake in the given options. The correct simplified fraction is 924 47 . However, if we look at the options, we can see that option B is 462 125 . Let's check if 924 47 is equivalent to 462 125 or − 462 125 .
924 47 = 2 × 462 47 . This is not equal to 462 125 or − 462 125 .
Let's assume there was a typo in the original problem, and the correct answer is one of the options. We can try to work backwards from the options to see if any small change to the original problem would lead to one of the answers.
Analyzing Potential Typos Let's assume the correct answer is 462 125 . Then, 462 125 = 924 250 . We want to find what the numerator should be to get 924 250 .
693 − 504 − 352 + 210 = 47 . We want this to be 250. So, we need to add 250 − 47 = 203 to the numerator.
If the correct answer is − 462 125 = − 924 250 , we want the numerator to be -250. So, we need to subtract 47 − ( − 250 ) = 297 from the numerator.
Since none of the options directly match our result, and it's unlikely there's a simple typo in the original problem that would lead to one of the given answers, we will stick with our calculated result.
Final Answer The simplified expression is 924 47 . None of the given options match this result. Therefore, there might be an error in the options provided.
Approximating and Comparing Based on our calculations, the correct answer is 924 47 . Since this is not among the options, we can conclude that there might be a typo in the provided options. However, if we must choose the closest answer, we can approximate the value of 924 47 ≈ 0.0508658 . Option A is − 462 125 ≈ − 0.27056 , option B is 462 125 ≈ 0.27056 , option C is 125 462 = 3.696 , and option D is − 125 462 = − 3.696 . None of these are close to 924 47 .
Examples
When combining different ingredients in a recipe, each measured in fractions of a cup, you need to add those fractions to know the total amount of ingredients you are using. Similarly, if you are managing your finances and tracking expenses and income, which might be represented as fractions of your total budget, you need to add and subtract those fractions to determine your overall financial status. Understanding how to add and subtract fractions is essential for accurate measurements and calculations in various real-life situations.
