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Chapter 5: Problem 16

Multiply. Write the product in lowest terms. $$\frac{15}{33} \cdot \frac{22}{27}$$

Short Answer

Expert verified
Question: Find the product of the two fractions 15/33 and 22/27 and write the result in the lowest terms. Answer: The product of the two fractions 15/33 and 22/27 in the lowest terms is 10/27.

Step by step solution

01

Multiply the fractions

To multiply the fractions, multiply the numerators together and the denominators together: $$\frac{15}{33} \cdot \frac{22}{27} = \frac{15 \cdot 22}{33 \cdot 27}.$$
02

Calculate the numerator and denominator

Multiply the numerators (15 and 22) and denominators (33 and 27) and write the resulting fraction: $$\frac{15 \cdot 22}{33 \cdot 27} = \frac{330}{891}.$$
03

Simplify the fraction

Now we attempt to simplify the fraction by finding the GCD of the numerator (330) and the denominator (891). The GCD of 330 and 891 is 33. Divide both the numerator and the denominator by the GCD to find the simplified fraction: $$\frac{330}{891} = \frac{330 \div 33}{891 \div 33} = \frac{10}{27}$$ Thus, the product of the two fractions in the lowest terms is: $$\frac{15}{33} \cdot \frac{22}{27} = \frac{10}{27}.$$

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Key Concepts

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

Simplifying Fractions
When you work with fractions, especially when multiplying them, simplifying to the lowest terms is essential. Simplifying means reducing the fraction so that the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1. This makes the fraction easier to understand and compare.
To simplify a fraction, look for the greatest common divisor (GCD) of both the numerator and the denominator. Then, divide both by the GCD.
  • For instance, take the fraction \(\frac{330}{891}\). First, determine the GCD of 330 and 891, which is 33.
  • Divide both the numerator and the denominator by 33: \(\frac{330 \div 33}{891 \div 33} = \frac{10}{27}\).
By understanding how to simplify fractions, you'll master turning complex-looking problems into something much more manageable.
Greatest Common Divisor
The greatest common divisor (GCD) is a key concept when simplifying fractions. It鈥檚 the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Using the GCD helps shrink the fraction down to its simplest form.
To find the GCD of two numbers, follow these steps:
  • List all the factors of each number.
  • Identify the greatest factor that appears in both lists. This is your GCD.
For \(330\) and \(891\), the GCD is 33. Dividing each by this GCD simplifies the fraction efficiently.
Understanding the GCD helps in reducing fractions and in recognizing equivalent fractions, making multiplication results more straightforward.
Numerical Multiplication
Multiplying fractions involves a simple process where you multiply straight across the numerators and denominators. This means taking the two top numbers from each fraction and multiplying them together, just as you do with the bottom numbers.
For example, multiplying \(\frac{15}{33}\) by \(\frac{22}{27}\) involves:
  • Multiplying the numerators: \(15 \times 22 = 330\).
  • Multiplying the denominators: \(33 \times 27 = 891\).
This results in the fraction \(\frac{330}{891}\).
The true trick lies in multiplying accurately and then simplifying the resulting fraction properly. Accurate numerical multiplication sets the stage for accurate simplification and eventually finding the product of fractions in its lowest form.

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Most popular questions from this chapter

Estimate and find the actual sum expressed as a mixed mumber in simplest form. $$6 \frac{2}{5}-2 \frac{3}{4}$$

Add or subtract. $$-5 \frac{1}{2}+2 \frac{1}{8}$$

Simplify. $$\left(\frac{-2 x y^{4}}{z^{3}}\right)^{5}$$

Multiply. $$\left(\frac{-7}{15} x^{4}\right)\left(\frac{5}{-21} x^{2}\right) \quad$$

Solve. If Fernando has a test average (mean) of 90 or higher, he will be exempt from the final exam. He has taken four out of the five tests for the course. His scores on those four tests are \(85,88,94,\) and \(86 .\) What is the lowest he can score on the fifth test to be exempt from the final exam?
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