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Chapter 7: Problem 61

In the following exercises, simplify. $$ -36 \cdot 11 \cdot \frac{4}{9} $$

Short Answer

Expert verified
-176

Step by step solution

01

Simplify the fraction

Simplify the fraction \( \frac{4}{9} \) first to check if there are any common factors with other numbers but \( \frac{4}{9} \) is already in its simplest form.
02

Multiply the numbers

Multiply \( -36 \) by \( 11 \) first: \[ -36 \times 11 = -396 \]
03

Multiply the result by the fraction

Now, multiply \( -396 \) by \( \frac{4}{9} \): \[ -396 \times \frac{4}{9} = \frac{-396 \times 4}{9} = \frac{-1584}{9} \]
04

Simplify the final fraction

Simplify the fraction \( \frac{-1584}{9} \) by performing the division: \[ \frac{-1584}{9} = -176 \]

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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 whenever possible. A fraction is simplified when the numerator and denominator have no common factors other than 1. For example, in the exercise, the fraction \( \frac{4}{9} \) is already in its simplest form. To check for simplification, always look for the greatest common divisor (GCD) of the numerator and the denominator.
If the GCD is 1, the fraction cannot be simplified any further. Here are steps to simplify fractions:
  • Find the GCD of the numerator and the denominator.
  • Divide both numerator and denominator by the GCD.
Keeping fractions in their simplest form makes calculations easier and the results more readable.
multiplying integers
Multiplying integers is a key foundational skill in prealgebra. When multiplying integers, keep in mind the rules regarding positive and negative signs:
  • Positive 脳 Positive = Positive
  • Negative 脳 Negative = Positive
  • Positive 脳 Negative = Negative
In the given exercise, you multiply \( -36 \) by \( 11 \):
\( -36 \times 11 = -396 \). Start with the multiplication of the absolute values: \( 36 \times 11 = 396 \).
Then, apply the rule that a positive number multiplied by a negative number results in a negative number. Hence, \( 36 \times -11 = -396 \). Consistency with these basic rules ensures accurate results.
multiplying fractions
Multiplying fractions involves a straightforward process. Multiply the numerators together and the denominators together. Here鈥檚 how to do it:
  • Multiply the numerators: \( a \times c \)
  • Multiply the denominators: \( b \times d \)
For the exercise, you have \( -396 \times \frac{4}{9} \). First, convert the integer to a fraction: \( \frac{-396}{1} \).
Then, multiply:
\[ \frac{-396}{1} \times \frac{4}{9} = \frac{-396 \times 4}{1 \times 9} = \frac{-1584}{9} \] Finally, simplify the resulting fraction.
In this case, dividing \( -1584 \) by \( 9 \) gives \( -176 \). So, \( -396 \times \frac{4}{9} = -176 \).

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