/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 366 Multiply. $$ \left(-\frac{33... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 366

Multiply. $$ \left(-\frac{33}{60}\right)\left(-\frac{40}{88}\right) $$

Short Answer

Expert verified
\(\frac{1}{4}\)

Step by step solution

01

Simplify the fractions

Simplify each fraction before multiplying. For \(\frac{-33}{60}\), we find the greatest common divisor (GCD) of 33 and 60, which is 3. Thus, \(\frac{-33}{60} = \frac{-33 \div 3}{60 \div 3} = \frac{-11}{20}\). Similarly, for \(\frac{-40}{88}\), the GCD of 40 and 88 is 8. Thus, \(\frac{-40}{88} = \frac{-40 \div 8}{88 \div 8} = \frac{-5}{11}\).
02

Multiply the simplified fractions

Now we multiply the simplified fractions: \(\frac{-11}{20} \times \frac{-5}{11}\). Since both fractions are negative, the product will be positive. Multiply the numerators together and the denominators together: \(\frac{-11 \times -5}{20 \times 11} = \frac{55}{220}\).
03

Simplify the result

Simplify the resulting fraction \(\frac{55}{220}\). The GCD of 55 and 220 is 55. So, \(\frac{55}{220} = \frac{55 \div 55}{220 \div 55} = \frac{1}{4}\).

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Key Concepts

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

Simplifying Fractions
When working with fractions, it's important to simplify them. Simplifying a fraction makes it easier to understand and work with. You do this by dividing the numerator (top number) and the denominator (bottom number) by their Greatest Common Divisor (GCD).

For example, in the fraction \(\frac{-33}{60}\), the GCD of 33 and 60 is 3. You can simplify the fraction by dividing both the numerator and the denominator by 3: \(\frac{-33}{60} = \frac{-33 \div 3}{60 \div 3} = \frac{-11}{20}\).
Similarly, for the fraction \(\frac{-40}{88}\), the GCD is 8. So, \(\frac{-40}{88} = \frac{-40 \div 8}{88 \div 8} = \frac{-5}{11}\).

Now, the fractions are simpler and easier to work with.
Greatest Common Divisor (GCD)
Understanding the concept of the Greatest Common Divisor (GCD) is key when simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCD helps in reducing fractions to their simplest form.

For instance, to find the GCD of 33 and 60:

  • List the factors of each number: 33 (1, 3, 11, 33) and 60 (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60).
  • Identify the greatest number common to both lists: Here it's 3.

Similarly, for 40 and 88, the GCD is 8.

By using the GCD, you can simplify fractions effectively, as shown in the previous simplification steps.
Multiplying Fractions
Multiplying fractions might seem tricky, but it's straightforward once you remember a simple rule. To multiply fractions, you multiply the numerators together and the denominators together.

For example, when you multiply \(\frac{-11}{20}\) and \(\frac{-5}{11}\), you do the following:

  • Multiply the numerators: \(-11 \times -5 = 55\).
  • Multiply the denominators: \(\frac{20 \times 11}{220}\).

The result is \(\frac{55}{220}\).

However, you might need to simplify the result. For \(\frac{55}{220}\), the GCD of 55 and 220 is 55. So, you divide both the numerator and denominator by 55 to get \(\frac{1}{4}\).
The final simplified answer is \(\frac{1}{4}\).

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Most popular questions from this chapter

Simplify using the distributive property. $$ -9(9 a+4) $$

Some people think that \(65^{\circ}\) to \(75^{\circ}\) Fahrenheit is the ideal temperature range. (a) What is your ideal temperature range? Why do you think so? (b) Convert your ideal temperatures from Fahrenheit to Celsius.

Trader Joe's grocery stores sold a bottle of wine they called "Two Buck Chuck" for $$\$ 1.99 .\( They sold a case of 12 bottles for \)\$ 23.88\(. To find the cost of 12 bottles at \)\$ 1.99,\( notice that 1.99 is \)2-0.01\( (a) Multiply 12(1.99) by using the distributive property to multiply \)12(2-0.01)$$. (b) Was it a bargain to buy "Two Buck Chuck" by the case?

Simplify using the distributive property. $$ \frac{1}{5}(4 m+20) $$

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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