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Chapter 1: Problem 365

Multiply. $$ \left(-\frac{63}{84}\right)\left(-\frac{44}{90}\right) $$

Short Answer

Expert verified
The result is \(\frac{11}{30}\).

Step by step solution

01

Simplify the fractions

Simplify each fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.For \(-\frac{63}{84}\): The GCD of 63 and 84 is 21. Divide both numerator and denominator by 21:\[-\frac{63 \div 21}{84 \div 21} = -\frac{3}{4}\]For \(-\frac{44}{90}\): The GCD of 44 and 90 is 2. Divide both numerator and denominator by 2:\[-\frac{44 \div 2}{90 \div 2} = -\frac{22}{45}\]
02

Multiply the simplified fractions

To multiply fractions, multiply the numerators together and the denominators together:\[\left(-\frac{3}{4}\right)\left(-\frac{22}{45}\right) = \frac{-3 \cdot -22}{4 \cdot 45} = \frac{66}{180}\]
03

Simplify the resulting fraction

Simplify \(\frac{66}{180}\) by finding the GCD of 66 and 180, which is 6.Divide both numerator and denominator by 6:\[\frac{66 \div 6}{180 \div 6} = \frac{11}{30}\]

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Key Concepts

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

simplifying fractions
Simplifying fractions is an important skill in mathematics. When you simplify a fraction, you reduce it to its simplest form. The simplest form of a fraction means you cannot divide the numerator (the top number) and the denominator (the bottom number) by any number other than 1, and still get whole numbers.

To simplify a fraction, follow these steps:
  • Find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by this GCD.
Consider the fraction \(-\frac{63}{84}\). To simplify it:
  • The GCD of 63 and 84 is 21.
  • Divide the numerator and denominator by 21: \(-\frac{63}{84} \rightarrow -\frac{63 \div 21}{84 \div 21} = -\frac{3}{4}\).

This tells us that \(-\frac{63}{84} = -\frac{3}{4}\) in simplest form.
greatest common divisor
The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that divides two or more numbers without leaving a remainder.

When simplifying fractions, the GCD helps to reduce both the numerator and the denominator to their smallest possible values.
  • For example, in the fraction \(-\frac{44}{90}\), the GCD of 44 and 90 is 2.

Here's how you determine the GCD:
  • List the factors of both numbers.
  • Identify the largest common factor.
For 44 and 90:
  • Factors of 44: 1, 2, 4, 11, 22, 44
  • Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

The largest number common to both lists is 2, so the GCD is 2. Using this, you can simplify the fraction:
  • \(-\frac{44}{90} \rightarrow -\frac{44 \div 2}{90 \div 2} = -\frac{22}{45}\).

This makes calculations easier and clearer.
numerator and denominator multiplication
When multiplying fractions, you multiply the numerators together and the denominators together.

Here鈥檚 the process:
  • Write down both fractions to be multiplied.
  • Multiply the numerators (top numbers) together.
  • Multiply the denominators (bottom numbers) together.
  • Simplify if possible.
For example, to multiply \(-\frac{3}{4}\) and \(-\frac{22}{45}\):
  • Multiply the numerators: \(-3 \times -22 = 66\).
  • Multiply the denominators: \(\frac{4 \times 45 = 180}\).

The result is \(\frac{66}{180}\).

Next, simplify the fraction \(\frac{66}{180}\).
  • Find the GCD of 66 and 180, which is 6.
  • Divide the numerator and the denominator by 6: \(\frac{66 \div 6}{180 \div 6} = \frac{11}{30}\).

The final simplified fraction is \(\frac{11}{30}\).

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Most popular questions from this chapter

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