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Magazine Chapter 4: Problem 69
Divide the mixed fractions and express your answer as a mixed fraction. $$\left(-7 \frac{1}{2}\right) \div\left(-2 \frac{2}{5}\right)$$
Short Answer
Expert verified
The result is \(3 \frac{1}{8}\).
Step by step solution
01
Convert Mixed Fractions to Improper Fractions
First, we need to convert each mixed fraction into an improper fraction. A mixed fraction like \(-7 \frac{1}{2}\) can be converted by multiplying the whole part by the denominator and adding the numerator: \(-7 \cdot 2 + 1 = -14 + 1 = -15\). So, \(-7 \frac{1}{2} = -\frac{15}{2}\). Similarly, convert \(-2 \frac{2}{5}\): \(-2 \cdot 5 + 2 = -10 + 2 = -8\). \(-2 \frac{2}{5} = -\frac{12}{5}\).
02
Divide the Improper Fractions
Dividing fractions involves multiplying by the reciprocal of the second fraction. So, we compute:\[-\frac{15}{2} \div -\frac{12}{5} = -\frac{15}{2} \times -\frac{5}{12}\]Both fractions have the same negative sign, so multiplying two negatives result in a positive number. Thus, the division reduces to:\[\frac{15}{2} \times \frac{5}{12}\]
03
Simplify the Product
Next, simplify the resulting expression by canceling out common factors:\[\frac{15 \times 5}{2 \times 12} = \frac{75}{24}\]By dividing both the numerator and the denominator by their greatest common divisor (3), we simplify to:\[\frac{75 \div 3}{24 \div 3} = \frac{25}{8}\]
04
Convert to Mixed Fraction
Now, convert the improper fraction \(\frac{25}{8}\) back to a mixed fraction. Divide 25 by 8:- The quotient is 3, so the whole number part is 3.- The remainder is 1, so the fraction part is \(\frac{1}{8}\).\(\frac{25}{8} = 3 \frac{1}{8}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mixed Fractions
A mixed fraction combines a whole number with a proper fraction. For example, in the exercise given, \(-7 \frac{1}{2}\) and \(-2 \frac{2}{5}\) are mixed fractions. The whole number part and fractional part make mixed fractions very useful when representing quantities greater than one.
To understand mixed fractions better, let's break it down:
To understand mixed fractions better, let's break it down:
- The whole number represents complete units.
- The fractional part represents the remaining part of a unit, which is less than one.
Improper Fractions
Improper fractions have numerators larger than or equal to their denominators. In the exercise, we convert mixed fractions like \(-7 \frac{1}{2}\) into improper fractions like \(-\frac{15}{2}\) to simplify division.
The steps to convert a mixed fraction to an improper fraction are:
The steps to convert a mixed fraction to an improper fraction are:
- Multiply the whole number by the denominator of the fractional part.
- Add this product to the numerator of the fractional part.
- Write the result over the original denominator.
- \(-7 \times 2 = -14\)
- Add \(-14 + 1 = -15\).
- So, \(-7 \frac{1}{2}\) becomes \(-\frac{15}{2}\).
Division of Fractions
Dividing fractions might seem complex, but it's made simpler by multiplying by the reciprocal.
Here's how division works in our example:
Here's how division works in our example:
- Identify the fractions: \(-\frac{15}{2}\) and \(-\frac{12}{5}\).
- Turn the division into multiplication by flipping the second fraction: find the reciprocal of \(-\frac{12}{5}\), which is \(-\frac{5}{12}\).
- Multiply the fractions: \(-\frac{15}{2} \times -\frac{5}{12}\).
Conversion of Fractions
Conversion between improper and mixed fractions is a vital skill. Once you finish calculations, it helps to represent the answer in an intuitive form, like a mixed fraction.
In the given exercise, we concluded with the improper fraction \(\frac{25}{8}\). To convert this back to a mixed fraction:
In the given exercise, we concluded with the improper fraction \(\frac{25}{8}\). To convert this back to a mixed fraction:
- Divide the numerator by the denominator: 25 divided by 8 gives 3 as the quotient.
- The remainder, 1, forms the fractional part: \(\frac{1}{8}\). This is the leftover part once the whole units are taken out.
- Therefore, \(\frac{25}{8} = 3 \frac{1}{8}\).
