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Chapter 4: Problem 609

In the following exercises, perform the indicated operation. $$-5 \frac{7}{12} \cdot 4 \frac{4}{11}$$

Short Answer

Expert verified
\frac{-268}{11}

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For ewline $$-5 \frac{7}{12}:$$ ewline Multiply the whole number 5 by the denominator 12 and add the numerator 7: ewline \text{ } ewline \text{ } $$5 \times 12 + 7 = 60 + 7 = 67.ewline So, $$\text{-5} \frac{\text{7}}{\text{12}} = \frac{\text{-67}}{\text{12}}$$.
02

Convert the Second Mixed Number

Next, convert $$4 \frac{\text{4}}{\text{11}}$$ ewline Multiply the whole number 4 by the denominator 11 and add the numerator 4: ewline \text{ } ewline \text{ }$$4 \times 11 + 4 = 44 + 4 = 48.ewline So, $$4 \frac{\text{4}}{\text{11}} = \frac{\text{48}}{\text{11}}$$.
03

Multiply the Improper Fractions

Now multiply the improper fractions: ewline $$\frac{\text{-67}}{\text{12}} \times \frac{\text{48}}{\text{11}}$$ ewline Multiply the numerators: ewline \text{ } ewline \text{ }$$-67 \times 48 = -3216.ewline And multiply the denominators: ewline \text{ } ewline \text{ }$$12 \times 11 = 132.ewline So, the product is $$\frac{\text{-3216}}{\text{132}}$$.
04

Simplify the Fraction

Finally, simplify the fraction $$\frac{\text{-3216}}{\text{132}}$$ Divide both the numerator and the denominator by their greatest common divisor. Both -3216 and 132 are divisible by 12: $$\frac{\text{-3216} \text{/ 12}}{\text{132} \text{/ 12}} = \frac{\text{-268}}{\text{11}}.$$ So, the simplified answer is $$\frac{-268}{11}$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Converting Mixed Numbers
Mixed numbers are numbers that have both a whole number and a fraction. To multiply mixed numbers, we first convert them to improper fractions.
An improper fraction has a numerator larger than its denominator.

To convert a mixed number like \-5 \( \frac{7}{12} \) to an improper fraction:
  • Multiply the whole number (5) by the denominator (12)
  • Add the numerator (7) to this product
  • Keep the denominator the same
So, for 5 \( \times \ 12 + 7 = 60 + 7 = 67 \).

Thus, \-5 \( \frac{7}{12} = \frac{-67}{12}\).
Improper Fractions
Improper fractions are essential for multiplication because they help us work with simpler, more straightforward numbers.
An improper fraction has a larger numerator than the denominator.

For instance, we converted 4 \( \frac{4}{11} \) to \( \frac{48}{11}\):
  • Multiply the whole number (4) by the denominator (11)
  • Add the numerator (4) to the result
  • Keep the denominator the same
So, for 4 \( \times \ 11 + 4 = 44 + 4 = 48 \).

This makes 4 \( \frac{4}{11} = \frac{48}{11} \). Now, these improper fractions are ready for multiplication.
Multiplying Fractions
Multiplying fractions involves multiplying the numerators together and the denominators together.
We can follow these steps:
  • Multiply the numerators: \(-67 \times 48 = -3216\)
  • Multiply the denominators: \(12 \times 11 = 132\)

So, \( \frac{-67}{12} \times \frac{48}{11} = \frac{-3216}{132} \). It鈥檚 always a good idea to simplify the result when possible.
Simplifying Fractions
Simplifying a fraction means reducing it to its simplest form.
This involves finding the greatest common divisor (GCD) of the numerator and the denominator.
The GCD of -3216 and 132 is 12.
  • Divide the numerator by the GCD: \( -3216 \div 12 = -268 \)
  • Divide the denominator by the GCD: \( 132 \div 12 = 11 \)
So, \( \frac{-3216}{132} \) simplifies to \( \frac{-268}{11} \).

This final fraction is the simplified answer.

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Most popular questions from this chapter

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