Problem 27 - Algebra 1

Question

Find the product. $$(-4)(-7)\left(\frac{3}{7}\right)$$

Step-by-Step Solution

Answer

The product of (-4)(-7)(3/7) is 12.

Step 1 - Multiplication of negative numbers

Look at the two negative numbers: -4 and -7. The multiplication of -4 and -7 is 28 because the product of two negative numbers is a positive number.

Step 2 - Multiplication of result with a fraction

Next, multiply the result from Step 1, 28, with the fraction \(\frac{3}{7}\). This equals to 12 because when multiplying by a fraction, multiply as if it is a normal number. So, \(\frac{3}{7}\) times 28 simplifies to 12.

Key Concepts

Understanding Negative Numbers in Multiplication

Negative numbers can be a bit tricky, but once you understand the basic rules, they become much easier to handle. In mathematics, negative numbers are values less than zero. They often represent a lack, decrease, or opposite direction. When it comes to multiplication, there are some simple rules to remember:

• The product of two negative numbers is a positive number. This rule is important because it shows that multiplying negatives effectively "cancels out" the negative sign, resulting in a positive.
• If one number is negative and the other is positive, the product is negative. This rule shows that the presence of an odd number of negative factors results in a negative product.

The example given in the exercise (-4) multiplied by (-7) shows the first rule in action. Even though both numbers are negative, their multiplication yields a positive 28. Understanding this logic will aid significantly in becoming comfortable with negative number calculations.

Multiplying Fractions

Fractions represent parts of a whole and can sometimes appear daunting, yet they follow quite straightforward multiplication rules. Here is how to approach them:

• First, multiply the numerators (the top parts of the fractions).
• Then, multiply the denominators (the bottom parts).
• If applicable, simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor.

However, when multiplying a whole number by a fraction, you can think of the whole number as being over 1 for simplicity. In the step mentioned in the solution, multiplying 28 by \(\frac{3}{7}\) involves:

• Multiplying 28 (or \(\frac{28}{1}\) if you prefer a fractional representation) by 3, giving 84.
• Then, because the divisor is 7, divide 84 by 7 to simplify to 12.

This process demonstrates that multiplying by a fraction is much like scaling a number by that fraction's value.

Grasping Integer Multiplication

Integer multiplication is a fundamental operation that opens the door to more complex mathematical concepts. Integers include whole numbers, zero, and their negative counterparts. Here are some key points:

• The product of two integers is always an integer. This makes predictions of result types straightforward.
• Understanding multiplication as repeated addition can simplify concepts for beginners—such as seeing 3 times 4 being the same as adding 3, four times.

The exercise combines integer multiplication principles with fractions and negatives, illustrating that these core concepts relate and interact regularly in problem-solving. Remembering the properties of integer operations helps in strategic planning for calculations, paving the way for success in both basic and advanced mathematical tasks.