Problem 45 - Basic College Mathematics

Question

How many \(\frac{2}{3}\) -cup sugar bowls can be filled from 16 cups of sugar?

Step-by-Step Solution

Answer

24

Step 1: Understand the Problem

Determine how many \( \frac{2}{3} \) -cup sugar bowls can be filled with 16 cups of sugar. This is a division problem where you divide the total amount of sugar by the amount each bowl can hold.

Step 2: Set Up the Equation

Write the equation as: \( 16 \div \frac{2}{3} \).

Step 3: Invert the Fraction

To divide by a fraction, multiply by its reciprocal. The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). So, the equation becomes \( 16 \times \frac{3}{2} \).

Step 4: Perform the Multiplication

Multiply 16 by \( \frac{3}{2} \): \( 16 \times \frac{3}{2} = \frac{48}{2} = 24 \).

Step 5: Conclusion

You can fill 24 \( \frac{2}{3} \) -cup sugar bowls with 16 cups of sugar.

Key Concepts

reciprocal

To understand division of fractions, you first need to know what a reciprocal is. The reciprocal of a number is essentially its 'flipped' version. For example, the reciprocal of a fraction like \(\frac{2}{3}\) is \(\frac{3}{2}\).
You get the reciprocal by swapping the numerator (top part) and the denominator (bottom part) of the fraction.

• Reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
• Reciprocal of \(\frac{5}{4}\) is \(\frac{4}{5}\).

Understanding reciprocals is crucial because they transform division problems into more manageable multiplication problems.
This concept will help simplify the process when dealing with fractions in division scenarios.

multiplication of fractions

Once you've found the reciprocal of a fraction, you can turn a division problem into a multiplication problem. Multiplying fractions involves multiplying the numerators together and then the denominators together.
For example, if you need to multiply \(\frac{3}{2}\) by 16, you first change 16 into fraction form, which is \(\frac{16}{1}\).
Then, you perform the multiplication:
\ ( \frac{16}{1} \times \frac{3}{2} = \frac{16\cdot3}{1\cdot2} = \frac{48}{2} = 24 \ )

• Multiply the top numbers (numerators) together.
• Multiply the bottom numbers (denominators) together.

After simplifying the fraction \( \frac{48}{2} \), you get 24.
Remember, multiplying fractions is straightforward compared to division, which is why converting division actions into multiplications using reciprocals is helpful.

fraction division

Fraction division might initially seem tricky, but it follows clear steps. Start by converting the division into a multiplication problem using the reciprocal. For instance, if you need to solve \( \frac{16}{1} \div \frac{2}{3} \), follow these steps:

• Find the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).
• Change the division to multiplication: \( 16 \times \frac{3}{2} \).
• Perform the multiplication: \( \frac{48}{2} = 24 \).

By following these steps, you convert a potentially challenging division problem into a simpler multiplication problem.
This makes it easier to handle and ensures accuracy in your solutions.
The key is understanding reciprocals and knowing how to shift from division to multiplication.