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Chapter 1: Problem 388

Simplify. $$ \frac{-\frac{9}{16}}{\frac{33}{40}} $$

Short Answer

Expert verified
\( -\frac{15}{22} \)

Step by step solution

01

Invert the Denominator

To simplify the fraction \( \frac{-\frac{9}{16}}{\frac{33}{40}} \), start by taking the reciprocal (invert) of the denominator. This means you will flip \( \frac{33}{40} \) to become \( \frac{40}{33} \). Now your problem becomes \( -\frac{9}{16} \times \frac{40}{33} \).
02

Multiply the Numerators

Next, multiply the numerators of the fractions. So you'd calculate \( -9 \times 40 \), which equals \( -360 \).
03

Multiply the Denominators

Now, multiply the denominators of the fractions. So you'd calculate \( 16 \times 33 \), which equals \( 528 \).
04

Simplify the Resulting Fraction

You now have \( \frac{-360}{528} \). To simplify this fraction, find the greatest common divisor (GCD) of \( 360 \) and \( 528 \). The GCD is \( 24 \). Divide both the numerator and the denominator by \( 24 \): \( \frac{-360 \div 24}{528 \div 24} = \frac{-15}{22} \).

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Key Concepts

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

Reciprocal
The first important concept in our problem is the reciprocal. The reciprocal of a fraction is obtained by flipping its numerator and denominator. This concept is key when dealing with division of fractions.
For example, the reciprocal of \(\frac{33}{40}\) is \(\frac{40}{33}\).
By flipping, we switch the position of the numerator (top number) and the denominator (bottom number).
This allows us to turn the division problem into a multiplication problem.
Multiplying Fractions
Once we have the reciprocal, multiplying fractions becomes straightforward. We simply multiply the numerators together and the denominators together.
For example, in our problem, we have \(-\frac{9}{16} \times \frac{40}{33}\). Here鈥檚 how to do it:
  • Multiply the numerators: \(-9 \times 40 = -360\)
  • Multiply the denominators: \(\frac{16 \times 33 = 528}\)
This results in the new fraction: \(\frac{-360}{528}\). Now, we need to simplify our new fraction to get the final answer.
Greatest Common Divisor (GCD)
To simplify a fraction, we use the concept of the greatest common divisor (GCD). The GCD of two numbers is the largest positive integer that divides both numbers evenly.
In our example, we need to find the GCD of 360 and 528. The GCD here is 24.
So we divide both the numerator and the denominator by 24:
  • \(\frac{-360}{24} = -15\)
  • \(\frac{528}{24} = 22\)
Our simplified fraction is \(\frac{-15}{22}\).< This gives us the final simplified form of the expression.
Understanding how to find the GCD and use it for simplification is very important in fraction operations.

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Most popular questions from this chapter

Helen bought a 24-pound turkey for her family's Thanksgiving dinner and wants to know what time to put the turkey in to the oven. She wants to allow 20 minutes per pound cooking time. Calculate the length of time needed to roast the turkey: (a) First, by multiplying \(24 \cdot 20\) to find the total number of minutes and then multiplying the answer by \(\frac{1}{60}\) to convert minutes into hours. (b) Next, by multiplying \(24\left(20 \cdot \frac{1}{60}\right)\). (C) Which of the properties of real numbers says that your answers to parts (a), where you multiplied \((24 \cdot 20) \frac{1}{60}, \quad\) and \(\quad(b),\) where you multiplied \(24\left(20 \cdot \frac{1}{60}\right),\) should be equal?

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