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Chapter 1: Problem 732

Simplify. $$ -36 \cdot 11 \cdot \frac{4}{9} $$

Short Answer

Expert verified
-176

Step by step solution

01

Simplify the Fraction

The first step is to simplify the fraction \(\frac{4}{9}\). This fraction is already in its simplest form, so no further simplification is needed.
02

Multiply the Numbers

Next, multiply the numbers \-36\, \-11\, and \frac{4}{9}\ together. Start by multiplying the integers: \(-36 \times 11 = -396\).
03

Multiply by the Fraction

Now, multiply the result by the fraction \(\frac{4}{9}\): \-396 \times \frac{4}{9}\. This can be expressed as: \-396 \times \frac{4}{9} = \frac{-396 \times 4}{9} = \frac{-1584}{9}\.
04

Simplify the Final Fraction

Finally, simplify the fraction \(\frac{-1584}{9}\) if possible. Dividing \-1584\ by \9\ gives: \-176\.

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Key Concepts

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

fraction simplification
Fraction simplification is the process of reducing a fraction to its simplest form. This means expressing the fraction so that the numerator and denominator share no common factors other than 1. For example, \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\) because 4 and 8 can both be divided by their greatest common factor, 4.

In the exercise: \(\frac{4}{9}\) is already in simplest form, so no further action is required. When simplifying, always check if the numerator and denominator have a common factor.
multiplying integers
Multiplying integers involves straightforward arithmetic but always pay attention to the signs (positive or negative) of the numbers. The product rule for multiplying integers states:
  • A positive multiplied by a positive = positive
  • A negative multiplied by a negative = positive
  • A positive multiplied by a negative or a negative by a positive = negative
For instance, \(-36 \times 11 = -396\). Here, a negative times a positive results in a negative product.
fractions in multiplication
Multiplying fractions involves multiplying the numerators together and the denominators together. The general formula is:
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
In the exercise, we have an integer \(-396\) multiplied by a fraction \(\frac{4}{9}\). Think of \(-396\) as \(\frac{-396}{1}\). Then, the multiplication looks like this:
\(\frac{-396}{1} \times \frac{4}{9} = \frac{-396 \times 4}{1 \times 9}\) which simplifies to \(\frac{-1584}{9}\).
reducing fractions
Reducing fractions is the step of simplifying a fraction to its lowest terms. To reduce the fraction \(\frac{-1584}{9}\), divide both the numerator and the denominator by their greatest common divisor (GCD). Here, \-1584 divided by 9 = -176. Hence, the fraction reduces to:
\(\frac{-1584}{9} = -176\).
Always check for common factors to ensure the fraction cannot be simplified further.

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