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Chapter 6: Problem 41

Find the value of \(\frac{3}{5} \cdot \frac{15}{18} \div \frac{5}{9}\).

Short Answer

Expert verified
The value of the expression is \(\frac{9}{10}\).

Step by step solution

01

Address the Multiplication

The exercise involves first multiplying two fractions: \(\frac{3}{5}\) and \(\frac{15}{18}\). To do this, multiply the numerators together and the denominators together. This gives: \[\frac{3 \cdot 15}{5 \cdot 18} = \frac{45}{90}\].
02

Simplify the Fraction

Next, simplify the resulting fraction \(\frac{45}{90}\). Find the greatest common divisor (GCD) of 45 and 90, which is 45, and divide both the numerator and the denominator by the GCD: \[\frac{45 \div 45}{90 \div 45} = \frac{1}{2}\].
03

Address the Division

Now, divide the simplified product \(\frac{1}{2}\) by the fraction \(\frac{5}{9}\). Dividing by a fraction is equivalent to multiplying by its reciprocal. So, multiply \(\frac{1}{2}\) by \(\frac{9}{5}\): \[\frac{1}{2} \cdot \frac{9}{5} = \frac{1 \cdot 9}{2 \cdot 5} = \frac{9}{10}\].
04

Final Simplification

Simplify \(\frac{9}{10}\). Since the GCD of 9 and 10 is 1, the fraction is already in its simplest form.

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Key Concepts

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

Multiplying Fractions
When multiplying fractions, it's important to remember that you're simply working with two sets of numerators and denominators. This process is all about straightforward multiplication: you multiply the numerators together, and you do the same with the denominators.

For example, if you have
  • \(\frac{3}{5}\)
  • \(\frac{15}{18}\)
you would multiply the top numbers (3 and 15) and the bottom numbers (5 and 18):
  • \(3 \times 15 = 45\)
  • \(5 \times 18 = 90\)
Thus, \[\frac{3}{5} \cdot \frac{15}{18} = \frac{45}{90}.\]
Keep in mind that larger products don't necessarily result in complicated fraction values鈥攖hey just require simplification after this step.
Simplifying Fractions
Simplifying fractions means reducing them to their lowest terms so that they are easier to work with or interpret. The key is to find the greatest common divisor (GCD) of the numerator and the denominator.

Say you started with \(\frac{45}{90}\).
  • The GCD of 45 and 90 is 45.
  • Divide both the top and bottom of the fraction by 45: \(\frac{45 \div 45}{90 \div 45} = \frac{1}{2}.\)
The fraction \(\frac{1}{2}\) is in its simplest form.
Every multiple of 1 between two numbers can act as base values too鈥攁nd the simplicity is key not just in making math cleaner but often clearer, especially with more complex problems.
Dividing Fractions
Dividing fractions might sound tricky, but it鈥檚 actually a breeze with the right method: instead of dividing, you multiply by the reciprocal of the second fraction. But what is a reciprocal? It鈥檚 simply flipping the numerator and the denominator.

For instance, for \(\frac{1}{2} \div \frac{5}{9}\):
  • The reciprocal of \(\frac{5}{9}\) is \(\frac{9}{5}\).
  • Multiply the fraction by its reciprocal: \(\frac{1}{2} \cdot \frac{9}{5} = \frac{9}{10}.\)
The result \(\frac{9}{10}\) is already simplified since 9 and 10 have no common factors other than 1.
This method of turning division into multiplication through reciprocals truly simplifies the approach, allowing you to quickly solve problems without confusion.

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