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Magazine Chapter 7: Problem 12
Solve. If no solution exists, state this. $$\frac{5}{8}-\frac{3}{5}=\frac{x}{10}$$
Short Answer
Expert verified
The solution is \(x = \frac{1}{4}\).
Step by step solution
01
Identify the fractions
The given equation is \(\frac{5}{8} - \frac{3}{5} = \frac{x}{10}\). The fractions present are \(\frac{5}{8}\), \(\frac{3}{5}\), and \(\frac{x}{10}\).
02
Find a common denominator for the left side
To subtract \(\frac{5}{8}\) and \(\frac{3}{5}\), first find a common denominator. The least common multiple (LCM) of 8 and 5 is 40.
03
Convert fractions to have the common denominator
Convert \(\frac{5}{8}\) and \(\frac{3}{5}\) to have the denominator 40: \[ \frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \] \[ \frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40} \]
04
Subtract the fractions
Subtract \(\frac{25}{40}\) and \(\frac{24}{40}\): \[ \frac{25}{40} - \frac{24}{40} = \frac{1}{40} \]
05
Equation simplification
The equation simplifies to \(\frac{1}{40} = \frac{x}{10}\).
06
Cross-multiply to solve for x
Cross-multiply to solve for \(x\): \[ 1 \times 10 = 40 \times x \] \[ 10 = 40x \] Now, solve for \(x\) by dividing both sides by 40: \[ x = \frac{10}{40} = \frac{1}{4} \]
07
Verify the solution
Substitute \(x = \frac{1}{4}\) back into the original equation to verify: \[ \frac{5}{8} - \frac{3}{5} = \frac{1}{10} \] Both sides are equal, so the solution is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
common denominator
When working with fractions, finding a common denominator is essential for performing operations like addition or subtraction.
The common denominator is a shared multiple of all the denominators involved.
To identify it, look for the least common multiple (LCM) of the denominators.
For example, in our equation \(\frac{5}{8} - \frac{3}{5} = \frac{x}{10}\), the denominators are 8 and 5.
The LCM of 8 and 5 is 40.
This means that 40 is the smallest number that both 8 and 5 can divide into without leaving a remainder.
Changing both fractions to have this common denominator is crucial for accurate computation.
The common denominator is a shared multiple of all the denominators involved.
To identify it, look for the least common multiple (LCM) of the denominators.
For example, in our equation \(\frac{5}{8} - \frac{3}{5} = \frac{x}{10}\), the denominators are 8 and 5.
The LCM of 8 and 5 is 40.
This means that 40 is the smallest number that both 8 and 5 can divide into without leaving a remainder.
Changing both fractions to have this common denominator is crucial for accurate computation.
subtraction of fractions
Subtracting fractions becomes straightforward once you have a common denominator.
Convert each fraction to an equivalent fraction with this common denominator.
For instance, in our problem: \[ \frac{5}{8} = \frac{25}{40} \] \[ \frac{3}{5} = \frac{24}{40} \]
Once both fractions have the same denominator (40), subtraction is simple: \[ \frac{25}{40} - \frac{24}{40} = \frac{1}{40} \]
The denominators remain the same, and we perform the operation on the numerators.
Convert each fraction to an equivalent fraction with this common denominator.
For instance, in our problem: \[ \frac{5}{8} = \frac{25}{40} \] \[ \frac{3}{5} = \frac{24}{40} \]
Once both fractions have the same denominator (40), subtraction is simple: \[ \frac{25}{40} - \frac{24}{40} = \frac{1}{40} \]
The denominators remain the same, and we perform the operation on the numerators.
cross-multiplication
Cross-multiplication is a useful technique for solving equations involving fractions.
In the simplified form of our original equation, we have \ \frac{1}{40}=\frac{x}{10} \.
Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other and setting the products equal: \[ 1 \times 10 = 40 \times x \]
This simplifies to\ \[ 10 = 40x \]
and then you can solve for \(x\).
Finally, divide both sides by 40 to isolate \(x\) and get: \[ x = \frac{10}{40} = \frac{1}{4} \].
In the simplified form of our original equation, we have \ \frac{1}{40}=\frac{x}{10} \.
Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other and setting the products equal: \[ 1 \times 10 = 40 \times x \]
This simplifies to\ \[ 10 = 40x \]
and then you can solve for \(x\).
Finally, divide both sides by 40 to isolate \(x\) and get: \[ x = \frac{10}{40} = \frac{1}{4} \].
least common multiple
The least common multiple (LCM) is the smallest multiple that is exactly divisible by each of a set of numbers.
In fraction problems, the LCM serves as the common denominator.
For denominators 8 and 5, list the multiples of each:
The smallest multiple they share is 40, making it the LCM.
Using the LCM as a common denominator ensures that the fractions are comparable and can be easily subtracted or added.
In fraction problems, the LCM serves as the common denominator.
For denominators 8 and 5, list the multiples of each:
- Multiples of 8: 8, 16, 24, 32, 40, 48...
- Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40...
The smallest multiple they share is 40, making it the LCM.
Using the LCM as a common denominator ensures that the fractions are comparable and can be easily subtracted or added.
