Problem 79 - Prealgebra

Question

Some of the application problems below involve multiplication or division, while others involve addition or subtraction. Capacity One carton of milk contains \(\frac{1}{2}\) pint while another contains 4 pints. How much milk is contained in both cartons?

Step-by-Step Solution

Answer

The total milk in both cartons is 4 and a half pints.

Step 1: Identify the Problem Type

The problem asks for the total amount of milk contained in the two cartons. Since we are looking to combine the quantities from the two cartons, this problem involves addition.

Step 2: Convert Mixed Measurements into Common Units

Both amounts are given in pints: one carton contains \(\frac{1}{2}\) pint and the other contains 4 pints. As they are already in the same unit, we can proceed with addition directly.

Step 3: Add the Quantities

Add the quantities together to find the total amount of milk: \[ \frac{1}{2} + 4 = \frac{1}{2} + \frac{8}{2} = \frac{1 + 8}{2} = \frac{9}{2} \]

Step 4: Simplify the Result

Convert \(\frac{9}{2}\) into a mixed number: \(\frac{9}{2} = 4 \frac{1}{2}\). This shows that the total amount of milk is 4 and a half pints.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that include both a whole number and a fraction. They are useful for expressing quantities that are greater than a whole but not quite whole numbers in themselves. For instance, in our exercise, when we find the total milk to be \(\frac{9}{2}\), we convert this into a mixed number. This is because \(\frac{9}{2} = 4\frac{1}{2}\), which tells us there are four whole pints and a half pint remaining. To convert an improper fraction like \(\frac{9}{2}\) into a mixed number, follow these steps:

• Divide the numerator (9 in this case) by the denominator (2).
• The whole number result is the number of times the denominator goes into the numerator completely. For \(9 \div 2\), this is 4.
• The remainder is what is left and represents the fraction portion, so \(9 - (2 \times 4) = 1\), which means the leftover is \(\frac{1}{2}\).

Turning fractions into mixed numbers can often help with understanding and comparing sizes of quantities.

Unit Conversion in Measurements

Conversion between units is a fundamental concept in mathematics, particularly when working with measurements. In this exercise, we're dealing with pints, a unit of capacity. Ensuring that all the quantities are in the same unit is crucial to perform any arithmetic operation accurately. Sometimes, you might encounter measurements in different units that need to be converted to perform addition or subtraction. Here's how to handle such conversions:

• Identify the units involved and what you need to convert to. For example, if one carton was specified in gallons instead of pints, you would need to convert gallons to pints because 1 gallon equals 8 pints.
• Use multiplication or division to convert the units. For example, if you had one gallon, you'd multiply 1 by 8 to get 8 pints.

In our exercise, both cartons were already in pints, allowing us to skip this step and proceed directly with addition.

Fraction Simplification

Simplifying fractions ensures they are in their most easily understandable form. This process often involves reducing fractions to their smallest terms, or in our exercise, finding a way to express them as mixed numbers for easier readability.In simplifying a fraction like \(\frac{9}{2}\), as seen in this exercise, we didn't need to reduce it because it was more relevant to convert it into a mixed number, \(4\frac{1}{2}\). This demonstrates both simplification and clarity.General tips for fraction simplification include:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and denominator by the GCD to reduce the fraction.
• If the numerator is larger than the denominator, consider rewriting it as a mixed number.

Simplifying fractions helps in maintaining consistency and clarity in math problems.