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Chapter 2: Problem 62

Review operations with real numbers. Simplify. $$ \frac{1}{2} \div\left(-\frac{3}{4}\right) $$

Short Answer

Expert verified
\( \frac{-2}{3} \)

Step by step solution

01

Rewrite the Division as Multiplication

To simplify the expression \(\frac{1}{2} \div \left(-\frac{3}{4}\right)\), rewrite the division as multiplication by the reciprocal. This gives \( \frac{1}{2} \times \left(-\frac{4}{3}\right) \).
02

Perform the Multiplication

Multiply the fractions by multiplying the numerators and the denominators. \( \frac{1}{2} \times \left(-\frac{4}{3}\right) = \frac{1 \times -4}{2 \times 3} = \frac{-4}{6}\).
03

Simplify the Result

Simplify the fraction \( \frac{-4}{6} \). The greatest common divisor of 4 and 6 is 2, so divide both the numerator and the denominator by 2: \( \frac{-4}{6} = \frac{-4 \div 2}{6 \div 2} = \frac{-2}{3}\).

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Key Concepts

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

Division of Fractions
When you divide fractions, you are essentially multiplying by the reciprocal of the second fraction.
For example, in our exercise, we start with \(\frac{1}{2} \div \left(-\frac{3}{4}\right)\).
To solve this, we convert the division into a multiplication by the reciprocal.
The reciprocal of a fraction is found by swapping its numerator and denominator.
So, \(-\frac{3}{4}\) becomes \(-\frac{4}{3}\).
Now, it's simply: \(\frac{1}{2} \times \left(-\frac{4}{3}\right)\).
This process makes working with fractions much easier and more intuitive.
Multiplication by Reciprocal
Multiplying by the reciprocal is a key step in dividing fractions.
The reciprocal of a fraction is a new fraction where the numerator and denominator are switched.
For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
If the fraction is negative, like \(-\frac{3}{4}\), its reciprocal remains negative, so it becomes \(-\frac{4}{3}\).
By multiplying \(\frac{1}{2}\) by \(-\frac{4}{3}\), we are still following the rules of fraction multiplication:
  • Multiply the numerators together: \{1 \times -4} = -4\
  • Multiply the denominators together: \{2 \times 3} = 6\
This results in \(\frac{-4}{6}\). Such steps transform division problems into simpler multiplication problems.
Fraction Simplification
Simplifying fractions is about making them as simple as possible.
This means finding the greatest common divisor (GCD) of the numerator and the denominator, then dividing both by that number.
Let's simplify \(\frac{-4}{6}\):
1. Find the GCD of 4 and 6, which is 2.
2. Divide both the numerator and the denominator by 2:
\(\frac{-4}{6} = \frac{-4 \div 2}{6 \div 2} = \frac{-2}{3}\)
Simplification makes the fraction easier to read and understand.
Remember to always check if your fraction can be reduced further.

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Most popular questions from this chapter

Use an inequality and the five-step process to solve each problem. As a rule of thumb, debt payments (other than mortgages) should be less than \(8 \%\) of a consumer's monthly gross income. Oliver makes \(\$ 54,000\) per year and has a \(\$ 100\) student-loan payment every month. What size car payment can he afford?

Use an inequality and the five-step process to solve each problem. The average price of a movie ticket can be estimated by the equation \(P=0.169 Y-333.04,\) where \(Y\) is the year and \(P\) is the average price, in dollars. For what years will the average price of a movie ticket be at least \(\$ 7 ?\) (Include the year in which the \(\$ 7\) ticket first occurs.

Use an inequality and the five-step process to solve each problem. Following the guidelines of the Food and Drug Administration, Dale tries to eat at least 5 servings of fruits or vegetables each day. For the first \(\operatorname{six}\) days of one week, she had \(4,6,7,4,6,\) and 4 servings. How many servings of fruits or vegetables should Dale eat on Saturday, in order to average at least 5 servings per day for the week?

Use an inequality and the five-step process to solve each problem. Rhetoric Advertising is a directmail company. It determines that for a particular campaign, it can use any envelope with a fixed width of \(3 \frac{1}{2}\) in. and an area of at least \(17 \frac{1}{2}\) in \(^{2} .\) Determine (in terms of an inequality) those lengths that will satisfy the company constraints.

Use an inequality and the five-step process to solve each problem. Toni can be paid in one of two ways: Plan \(A:\) A salary of \(\$ 400\) per month, plus a commission of \(8 \%\) of gross sales; Plan \(B:\) A salary of \(\$ 610\) per month, plus a commission of \(5 \%\) of gross sales. For what amount of gross sales should Toni select plan A?
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