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Chapter 7: Problem 90

To prepare for Section 7.2, review multiplication and division using fraction notation. Simplify. $$ \frac{7}{10} \div\left(-\frac{8}{15}\right)[1.7] $$

Short Answer

Expert verified
-\(\frac{21}{16}\)

Step by step solution

01

Understand the Sign Change

When dividing by a negative fraction, the result will be negative. Keep in mind that \(\frac{a}{b} \div -\frac{c}{d} = - \frac{a}{b} \div \frac{c}{d}\).
02

Reciprocal of the Divisor

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(-\frac{8}{15}\) is \(-\frac{15}{8}\). Thus the expression becomes: \[ \frac{7}{10} \times -\frac{15}{8} \]
03

Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \[ \frac{7 \times -15}{10 \times 8} = \frac{-105}{80} \]
04

Simplify the Fraction

Simplify the fraction \(\frac{-105}{80}\) by finding the greatest common divisor (GCD) of 105 and 80. The GCD is 5. Thus, \[ \frac{-105 \div 5}{80 \div 5} = \frac{-21}{16} \]

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Key Concepts

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

fraction notation
In mathematics, fractions are a way to represent parts of a whole. They consist of two numbers separated by a slash. The top number is called the numerator, and it represents how many parts we have. The bottom number is the denominator, and it shows how many parts make up a whole.
For example, in the fraction \(\frac{7}{10}\), 7 is the numerator and 10 is the denominator. This means we have 7 parts out of 10.
Fraction notation is essential in understanding mathematical operations like multiplication and division. Let's emphasize a few points:
  • The fraction line (or slash) means division. So, \(\frac{a}{b}\) literally means 'a divided by b'.
  • Fractions can also be proper (numerator smaller than denominator), improper (numerator larger than denominator), or mixed numbers (a whole number and a fraction).
  • In the given exercise, we have to handle both proper fractions (e.g., \(\frac{7}{10}\)) and negative fractions.
reciprocal of fractions
The reciprocal of a fraction is simply flipping the numerator and the denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). Reciprocals are very useful when dividing fractions.
To divide fractions, you multiply by the reciprocal of the divisor. Let's break it down using our exercise:
  • We started with the problem \(\frac{7}{10} \div -\frac{8}{15}\).
  • The reciprocal of \-\(\frac{8}{15}\) is \-\(\frac{15}{8}\).
So, instead of division, our problem becomes: \(\frac{7}{10} \times -\frac{15}{8}\). This method simplifies the operation and makes it consistent with multiplication rules.
Reciprocals are especially important for simplifying mathematical expressions, as shown in our example.
simplifying fractions
Simplifying fractions means making them as simple as possible. This usually involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number.
In our exercise, we ended up with \(\frac{-105}{80}\). To simplify this fraction:
  • First, we find the GCD of 105 and 80. The GCD is 5.
  • Next, divide both the numerator and the denominator by 5:
So, \(\frac{-105 \div 5}{80 \div 5} = \frac{-21}{16}\). Simplifying helps in understanding and working with fractions easily. It's like reducing the fraction to its simplest form.
Remember to check for any remaining common factors to ensure the fraction is fully simplified. The simpler the fraction, the easier it is to interpret and use in further calculations.

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Most popular questions from this chapter

Paloma drove \(200 \mathrm{km} .\) For the first \(100 \mathrm{km}\) of the trip, she drove at a speed of \(40 \mathrm{km} / \mathrm{h}\). For the second half of the trip, she traveled at a speed of \(60 \mathrm{km} / \mathrm{h}\). What was the average speed of the entire trip? (It was \(n o t\) \(50 \mathrm{km} / \mathrm{h} .)\)

Find the LCM. $$10 t^{3}, 5 t^{4}$$

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Escalators. Together, a \(100-\mathrm{cm}\) wide escalator and a \(60-\mathrm{cm}\) wide escalator can empty a 1575 -person auditorium in 14 min. The wider escalator moves twice as many people as the narrower one. How many people per hour does the \(60-\mathrm{cm}\) wide escalator move?

ListAndSell is an auction dropoff store that accepts items for sale on Internet auctions. For a fee, the items are listed on the auction and shipped when sold. The table below lists the amount that a seller receives from the sale of various items. $$\begin{array}{l|c}{\text { Item Selling Price }} & {\text { Amount Seller Receives }} \\ {\$ 75.00} & {\$ 41.42} \\ {100.00} & {55.50} \\ {200.00} & {111.85} \\ {400.00} & {240.55}\end{array}$$ a) Determine whether the amount that the seller receives varies directly or inversely as the selling price. b) Find an equation of variation that approximates the data. Use the data point \((200,111.85)\) c) Use the equation to predict the amount the seller receives if an item sells for \(\$ 150.00 .\)
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