Chapter 10: Problem 40
Subtract \(\frac{3 x}{10 y}-\frac{4 y}{15 x}\). Write your answer in lowest terms. A. \(\frac{9 x-8 y}{15 x y}\) B. \(\frac{9 x^2-8 y^2}{15 x y}\) C. \(\frac{9 x-8 y}{30 x y}\) D. \(\frac{9 x^2-8 y^2}{30 x y}\)
Short Answer
Expert verified
Option C: \(\frac{9x - 8y}{30xy}\)
Step by step solution
01
- Find a Common Denominator
Identify the denominators in the fractions. The denominators are \(10y\) and \(15x\). The least common multiple (LCM) of these denominators is \(30xy\).
02
- Rewrite Fractions with Common Denominator
Rewrite both fractions with the common denominator \(30xy\). Multiply the numerator and denominator of the first fraction \(\frac{3x}{10y}\) by 3 to get \(\frac{9x}{30xy}\). Multiply the numerator and denominator of the second fraction \(\frac{4y}{15x}\) by 2 to get \(\frac{8y}{30xy}\).
03
- Subtract the Fractions
Subtract the fractions: \(\frac{9x}{30xy} - \frac{8y}{30xy}\). Since they have the same denominator, subtract the numerators: \(\frac{9x - 8y}{30xy}\).
04
- Simplify the Expression
The fraction \(\frac{9x - 8y}{30xy}\) is already in its lowest terms as there are no common factors in the numerator and the denominator.
05
- Verify the Answer
Compare the simplified expression \(\frac{9x - 8y}{30xy}\) with the given options. The correct option is C: \(\frac{9x - 8y}{30xy}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
common denominator
A `common denominator` in fractions means the same bottom number in both fractions. This allows you to easily add or subtract them. In the given exercise, the fractions \( \frac{3x}{10y} \) and \( \frac{4y}{15x} \) have different denominators: 10y and 15x.
To find a common denominator, we need to determine the `least common multiple (LCM)` of these denominators. The LCM is the smallest number that both denominators can divide into without leaving a remainder.
In this case, the denominators are 10y and 15x. We need to combine both denominators' factors. Hence, the LCM of 10y and 15x is 30xy. This will be our new common denominator, which we will use to rewrite both fractions.
To find a common denominator, we need to determine the `least common multiple (LCM)` of these denominators. The LCM is the smallest number that both denominators can divide into without leaving a remainder.
In this case, the denominators are 10y and 15x. We need to combine both denominators' factors. Hence, the LCM of 10y and 15x is 30xy. This will be our new common denominator, which we will use to rewrite both fractions.
subtracting fractions
Once we have a `common denominator`, we can easily subtract the fractions by aligning them to the same denominator. The fractions \( \frac{3x}{10y} \) and \( \frac{4y}{15x} \) need to be rewritten with the common denominator 30xy.
To do this, we multiply the numerator and denominator of the first fraction \( \frac{3x}{10y} \) by 3, to get \( \frac{9x}{30xy} \). Similarly, we multiply the numerator and denominator of the second fraction \( \frac{4y}{15x} \) by 2, to get \( \frac{8y}{30xy} \).
Now we have \( \frac{9x}{30xy} - \frac{8y}{30xy} \), which can be easily subtracted as they share the same denominator. Simply subtract the numerators and keep the denominator the same: \( \frac{9x - 8y}{30xy} \).
To do this, we multiply the numerator and denominator of the first fraction \( \frac{3x}{10y} \) by 3, to get \( \frac{9x}{30xy} \). Similarly, we multiply the numerator and denominator of the second fraction \( \frac{4y}{15x} \) by 2, to get \( \frac{8y}{30xy} \).
Now we have \( \frac{9x}{30xy} - \frac{8y}{30xy} \), which can be easily subtracted as they share the same denominator. Simply subtract the numerators and keep the denominator the same: \( \frac{9x - 8y}{30xy} \).
least common multiple
The `least common multiple (LCM)` is pivotal for operations involving fractions. To understand it better, let's break down how we find the LCM of 10y and 15x.
Step 1: Factorize both denominators.
- 10y = 2 \( \cdot \) 5 \( \cdot \) y
- 15x = 3 \( \cdot \) 5 \( \cdot \) x
Step 2: Identify the highest power of each prime factor.
- For 2, the highest power is 2 from 10y.
- For 3, the highest power is 3 from 15x.
- For 5, it appears in both denominators, so take 5.
- For x and y, take x and y independently.
Therefore, multiply these together to find the LCM: 2 \( \cdot \) 3 \( \cdot \) 5 \( \cdot \) x \( \cdot \) y = 30xy.
The LCM of 10y and 15x is 30xy, which serves as our common denominator.
Step 1: Factorize both denominators.
- 10y = 2 \( \cdot \) 5 \( \cdot \) y
- 15x = 3 \( \cdot \) 5 \( \cdot \) x
Step 2: Identify the highest power of each prime factor.
- For 2, the highest power is 2 from 10y.
- For 3, the highest power is 3 from 15x.
- For 5, it appears in both denominators, so take 5.
- For x and y, take x and y independently.
Therefore, multiply these together to find the LCM: 2 \( \cdot \) 3 \( \cdot \) 5 \( \cdot \) x \( \cdot \) y = 30xy.
The LCM of 10y and 15x is 30xy, which serves as our common denominator.
fraction simplification
After subtracting the numerators, we need to check if the resulting fraction can be simplified further. Fraction simplification involves reducing the fraction to its lowest terms.
In our exercise, we ended up with \( \frac{9x - 8y}{30xy} \). To simplify it, we need to check if the numerator and the denominator have any common factors.
In this case, the numerator 9x - 8y and the denominator 30xy do not share any common factors beyond what we already have. Thus, the fraction is already in its simplest form.
The simplified result \( \frac{9x - 8y}{30xy} \) matches with one of the options provided in the exercise, confirming our result is correct.
In our exercise, we ended up with \( \frac{9x - 8y}{30xy} \). To simplify it, we need to check if the numerator and the denominator have any common factors.
In this case, the numerator 9x - 8y and the denominator 30xy do not share any common factors beyond what we already have. Thus, the fraction is already in its simplest form.
The simplified result \( \frac{9x - 8y}{30xy} \) matches with one of the options provided in the exercise, confirming our result is correct.