Problem 24 - Basic College Mathematics

Question

Divide and simplify. \(40 \div \frac{2}{3}\)

Step-by-Step Solution

Answer

The result is 60.

Step 1: Understand Division of Fractions

When dividing by a fraction, it's helpful to remember that dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Find the Reciprocal

The reciprocal of a fraction is created by swapping the numerator and the denominator. For the fraction \(\frac{2}{3}\), the reciprocal is \(\frac{3}{2}\).

Step 3: Change Division into Multiplication

Rewrite the division problem \(40 \div \frac{2}{3}\) as a multiplication problem with the reciprocal. This changes the expression to \(40 \times \frac{3}{2}\).

Step 4: Multiply

Multiply the whole number by the fraction: \(40 \times \frac{3}{2}\). This can be written as \(\frac{40 \times 3}{2}\).

Step 5: Simplify

Simplify the expression: \(40 \times \frac{3}{2} = \frac{120}{2} = 60\).

Key Concepts

reciprocal

A reciprocal is a key concept in the division of fractions. The reciprocal of a fraction is what you get when you switch its numerator and denominator. For instance, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).

Why does this matter? Because division by a fraction is the same as multiplying by its reciprocal. This means \(40 \div \frac{2}{3}\) becomes \(40 \times \frac{3}{2}\).

So, next time you see a fraction you need to divide by, just flip it to find its reciprocal and multiply instead. It’s much easier to handle!

multiplication of fractions

Multiplying fractions involves a few simple steps. When you multiply a whole number by a fraction, you treat the whole number as if it has a denominator of 1.

For example, in the problem, \(40 \div \frac{2}{3}\) turns into \(40 \times \frac{3}{2}\). Here is the process:

• Treat 40 as \( \frac{40}{1} \)
• Multiply the numerators: 40 \times 3 = 120
• Multiply the denominators: 1 \times 2 = 2

This gives you \(\frac{120}{2}\). Remember, multiplying fractions is straightforward once you know how to handle both the numerators and denominators.

simplification

Simplification is the final step in solving math problems involving fractions.

After multiplying, you often need to reduce the fraction to its simplest form. Simplify \(\frac{120}{2}\) by performing the division.

Divide the numerator by the denominator: \(\frac{120}{2} = 60\)

This is your final answer.

• Always check if a fraction can be simplified
• If either the numerator or the denominator has common factors, divide both by the greatest common factor

Simplification makes your answer cleaner and easier to understand.