Problem 17 - Algebra 1

Question

State the inverse operation. Multiply by \(\frac{2}{3}\).

Step-by-Step Solution

Answer

The inverse operation of 'multiplying by \(\frac{2}{3}\)' is 'multiplying by \(\frac{3}{2}\).'

Step 1: Identify the Operation

The given operation is Multiplication by \(\frac{2}{3}\).

Step 2: Identify the Inverse Operation

The inverse operation for multiplication is division. To find the inverse operation of multiplying by \(\frac{2}{3}\), we need to swap the operation from multiplication to division, which results in 'dividing by \(\frac{2}{3}\).'

Step 3: Apply the Fundamental Law

In division of fractions law, dividing by a fraction is the same as multiplying by the reciprocal of that fraction. The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\), so 'dividing by \(\frac{2}{3}\)' is the same as 'multiplying by \(\frac{3}{2}\).' The inverse operation of 'multiplying by \(\frac{2}{3}\)' is 'multiplying by \(\frac{3}{2}\).'

Key Concepts

Multiplication and Division

Multiplication and division are fundamental arithmetic operations used to calculate changes in quantities. They are inverse operations, which means they undo each other. If you multiply a number by a certain value, you can return to the original number by dividing by that same value. This relationship is useful in various mathematical contexts and helps us solve equations effectively.

When multiplying, you combine groups of numbers. For example, multiplying 4 by 3 results in 12, because you add the number 4 three times.

• 4 \(\times\) 3 = 12

Division, on the other hand, is about splitting a number into equal parts. For example, dividing 12 by 3 would result in 4, as 12 can be evenly split into three groups of 4.

• 12 \(\div\) 3 = 4

Recognizing that multiplication and division are inverses helps calculate unknowns by rearranging equations, making it crucial for solving complex problems.

Fractions

Fractions represent parts of a whole and are written with two numbers separated by a slash. The top number, called the numerator, indicates how many parts you have, while the bottom number, called the denominator, signifies how many equal parts the whole is split into. For example, in \(\frac{2}{3}\), 2 is the numerator and 3 is the denominator.

Understanding fractions is essential, particularly when performing arithmetic operations involving them, like multiplication and division. In multiplication with fractions, you simply multiply the numerators together and the denominators together. For instance, when you multiply \(\frac{2}{3}\) by \(\frac{3}{2}\), it looks like this:

• \(\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1\)

For division, you'll use the reciprocal of the divisor, as it is equivalent to multiplying by its reciprocal. A clear understanding of fractions allows you to handle complex operations and solve equations involving fractions.

Reciprocal

Reciprocal is a crucial concept in mathematics, especially when dealing with fractions and operations like division. The reciprocal of a fraction is obtained by flipping its numerator and denominator. This means for a fraction \(\frac{a}{b}\), its reciprocal will be \(\frac{b}{a}\). For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).

The reciprocal plays a key role when dividing by fractions. Dividing by a fraction is equivalent to multiplying by its reciprocal, simplifying the operation to multiplication. This technique is widely used to solve equations where division by a fraction is required.

• Division by \(\frac{a}{b}\) = Multiplication by \(\frac{b}{a}\).

Using reciprocals gives you a consistent way to handle division problems involving fractions, making calculations more straightforward and less error-prone. Understanding this concept makes arithmetic with fractions much more manageable.