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Chapter 9: Problem 26

Convert each measurement as indicated. $$ 3 \frac{1}{4} \text { quarts to cups } $$

Short Answer

Expert verified
3 1/4 quarts is equivalent to 13 cups.

Step by step solution

01

Understand Units of Measurement

1 quart is equivalent to 4 cups. Therefore, to convert quarts to cups, we need to multiply the number of quarts by 4.
02

Convert Mixed Number to Improper Fraction

The given value is a mixed number: \(3 \frac{1}{4}\). To convert this to an improper fraction, multiply the whole number 3 by 4 and then add the numerator of the fraction: \(3 \times 4 + 1 = 12 + 1 = 13\). So, \(3 \frac{1}{4} = \frac{13}{4}\).
03

Convert Quarts to Cups

Since each quart is 4 cups, multiply \(\frac{13}{4}\) quarts by 4: \(\frac{13}{4} \times 4\).
04

Simplify the Multiplication

Simplify the expression: \(\frac{13}{4} \times 4 = 13\), because \(4\) in the numerator and denominator cancel out.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mixed Numbers
A mixed number is a combination of a whole number and a proper fraction. In our problem, we start with the mixed number \(3 \frac{1}{4}\). This means we have 3 whole quarts and an additional quarter of a quart. Mixed numbers are useful for expressing quantities that are more than a single whole but not quite the next whole number.
  • Whole Number: It represents the complete units, in this case, 3 quarts.
  • Proper Fraction: The fraction part \(\frac{1}{4}\) tells us about the remaining incomplete part of the unit.

This concept is especially important in measurements when you need to convey both complete and partial units.
Improper Fractions
When dealing with calculations, improper fractions are often preferred, as they allow easier manipulation. An improper fraction is where the numerator is greater than or equal to the denominator. For example, converting \(3 \frac{1}{4}\) into an improper fraction means finding how many total parts we have based on the fraction's denominator.

To convert a mixed number to an improper fraction:
  • Multiply the whole number by the fraction's denominator.
  • Add the result to the numerator of the fraction part.
  • Write this sum as the new numerator, keeping the original denominator.

For \(3 \frac{1}{4}\), multiply 3 by 4 to get 12, then add 1 to get 13. Thus, \(3 \frac{1}{4}\) is converted to \(\frac{13}{4}\). This allows us to execute consistent arithmetic operations easily.
Simplification of Fractions
Simplifying fractions is essential for reducing fractions to their simplest form, which helps with clearer understanding and easier calculations. In multiplication, simplification often involves canceling out factors in the numerator and denominator. After converting quarts to cups, we have the operation \(\frac{13}{4} \times 4\).

Here are steps for simplification:
  • Identify any common factors between numerators and denominators across the fraction.
  • Cancel the common factor. In our example, the 4 in the numerator and 4 in the denominator cancels each other out.
  • The simplified result is \(13\), representing the direct conversion of quarts to cups.

Simplification is a crucial step to ensure the final result is easy to interpret and correctly scaled to the problem's requirements.

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