Problem 80 - Prealgebra

Question

Some of the application problems below involve multiplication or division, while others involve addition or subtraction. Baking A recipe calls for \(\frac{2}{3}\) cup of flour and \(\frac{3}{4}\) cup of sugar. What is the total amount of flour and sugar called for in the recipe?

Step-by-Step Solution

Answer

The total amount of flour and sugar is \(1 \frac{5}{12}\) cups.

Step 1: Identify the Operation

This problem involves finding the total amount of ingredients, so we need to perform addition.

Step 2: Common Denominator for Fraction Addition

To add two fractions, they must have a common denominator. The denominators are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12.

Step 3: Convert Fractions

Convert each fraction to have a denominator of 12. \(\frac{2}{3} = \frac{8}{12}\) because \(2 \times 4 = 8\) and \(3 \times 4 = 12\). Similarly, \(\frac{3}{4} = \frac{9}{12}\) because \(3 \times 3 = 9\) and \(4 \times 3 = 12\).

Step 4: Add the Fractions

With a common denominator, add the numerators: \(\frac{8}{12} + \frac{9}{12} = \frac{17}{12}\).

Step 5: Simplify the Fraction

The fraction \(\frac{17}{12}\) is an improper fraction and can be converted into a mixed number. \(\frac{17}{12} = 1 \frac{5}{12}\) since 17 divided by 12 equals 1 with a remainder of 5.

Key Concepts

Least Common Multiple

When adding fractions, it is essential to have a common denominator, which ensures that the fractions are compatible for addition. The least common multiple (LCM) is the smallest number that each of the denominators can divide evenly into. For example, if you have the denominators 3 and 4, you find the multiples of each:

• Multiples of 3: 3, 6, 9, 12, 15, ...
• Multiples of 4: 4, 8, 12, 16, ...

The smallest multiple they share is 12, which becomes the least common denominator. This step is crucial because it allows you to rewrite each fraction with the same denominator, making the addition straightforward.

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). After adding

The whole number is the result of the division.

The numerator of the fraction is the remainder.

The denominator remains the same.

Using mixed numbers makes it simpler to visualize the quantity, especially in everyday contexts, such as cooking or measuring materials.

Fraction Simplification

Simplifying a fraction means reducing it to its lowest terms, making it easier to understand or use. A fraction is simplified when the numerator and the denominator have no common factors other than 1. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
For instance, with the fraction

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Other exercises in this chapter

Problem 79

Write each fraction as an equivalent fraction with denominator 36. $$\frac{5}{12}$$

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Problem 80

Combine. \(\frac{1}{2}+\frac{2}{3}\)

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Problem 80

Multiply. $$6 \cdot 6 \cdot 2$$

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Problem 80

Which of the following fractions is closest to the number 1? a. \(\frac{1}{2}\) b. \(\frac{1}{3}\) c. \(\frac{1}{4}\) d. \(\frac{1}{5}\)

View solution