Problem 46 - Basic College Mathematics

Question

How many \(\frac{2}{3}\) -cup cereal bowls can be filled from 10 cups of cornflakes?

Step-by-Step Solution

Answer

15 servings

Step 1: Understand the problem

The task is to determine how many \(\frac{2}{3}\) -cup servings of cereal can be made from 10 cups of cornflakes.

Step 2: Set up the division

We need to divide the total amount of cornflakes (10 cups) by the serving size (\(\frac{2}{3}\) cups). The mathematical expression for this is: \[ \frac{10}{\frac{2}{3}} \]

Step 3: Simplify the division

Dividing by a fraction is the same as multiplying by its reciprocal. So, we convert the division to multiplication: \[ 10 \times \frac{3}{2} \]

Step 4: Perform the multiplication

Multiply the numbers: \[ 10 \times 1.5 = 15 \]

Step 5: Interpret the result

The result indicates that there are 15 servings of \(\frac{2}{3}\) cups each in 10 cups of cornflakes.

Key Concepts

Division of Fractions

Division of fractions might seem confusing at first, but it's a straightforward process once you get the hang of it. The task is to determine how many times one fraction fits into another number or fraction. In this problem, we are trying to find out how many \(\frac{2}{3}\) cups of cereal can be filled from 10 cups of cornflakes.
We start by setting up our division problem as shown in the solution: \[ \frac{10}{\frac{2}{3}} \] This expression means we are dividing 10 by \(\frac{2}{3}\). To proceed, we need to understand the role of reciprocals in division.

Reciprocals in Fractions

When we divide by a fraction, we use its reciprocal. The reciprocal of a fraction is found by flipping its numerator and denominator. So, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
Now, instead of dividing by \(\frac{2}{3}\), we multiply by its reciprocal. This changes our problem from:

• \[ \frac{10}{\frac{2}{3}} \]
• to
• \[ 10 \times \frac{3}{2} \]

By converting the division into multiplication using the reciprocal, it simplifies our calculation process. This is a key step in working with fraction division, as it makes the arithmetic simpler and more intuitive.

Multiplication of Fractions

Once we have our new expression: \[ 10 \times \frac{3}{2} \], we move to the multiplication step. Multiplying whole numbers by fractions follows a simple method:

• Multiply the whole number by the numerator of the fraction.
• Then divide by the denominator of the fraction.

In our example, 10 is multiplied by 3 (the numerator):

\[ 10 \times 3 = 30 \]
Then, we divide the result by 2 (the denominator):

\[ 30 \div 2 = 15 \]
This process tells us that 10 cups of cornflakes can be divided into 15 servings of \(\frac{2}{3}\) cups each. Remember, multiplcation is essentially repeated addition, so fraction multiplication logically extends this idea by incorporating the rules specific to fractions.