Problem 60 - Elementary Algebra

Question

The ingredients of a recipe includes \(\frac{3}{8}\) cup grated cheese. If an additional \(\frac{1}{4}\) cup of cheese is used, find the total amount of cheese.

Step-by-Step Solution

Answer

Total amount of cheese is \(\frac{5}{8}\) cups.

Step 1: Identify the given fractions

The fractions of cheese given are \(\frac{3}{8}\) cups and \(\frac{1}{4}\) cups.

Step 2: Find a common denominator

The denominators are 8 and 4. The least common multiple is 8. Convert \(\frac{1}{4}\) to have a denominator of 8.

Step 3: Convert \(\frac{1}{4}\)

To convert \(\frac{1}{4}\) to have a denominator of 8, multiply both the numerator and the denominator by 2: \[\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}\]

Step 4: Add the fractions

Now that both fractions have the same denominator, add the numerators: \[\frac{3}{8} + \frac{2}{8} = \frac{3 + 2}{8} = \frac{5}{8}\]

Step 5: Simplify (if necessary)

The fraction \(\frac{5}{8}\) is already in simplest form.

Key Concepts

Fractions

A fraction represents a part of a whole. It consists of a numerator (the top part) and a denominator (the bottom part). The fraction \(\frac{3}{8}\) means 3 parts out of 8 equal parts. Fractions can be added, subtracted, multiplied, or divided, but must often be adjusted so the denominators match before performing these operations.

Least Common Multiple

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more denominators. Finding the LCM is crucial when adding fractions because it provides a common denominator. For the fractions \(\frac{3}{8}\) and \(\frac{1}{4}\), the denominators are 8 and 4. The LCM of 4 and 8 is 8 because 8 is the smallest number that both 4 and 8 can divide into without a remainder.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, after adding \(\frac{3}{8}\) and \(\frac{2}{8}\) to obtain \(\frac{5}{8}\), you check if 5 and 8 have any common divisors besides 1. Since they do not, \(\frac{5}{8}\) is already in its simplest form.

Denominators

The denominator is the lower part of a fraction and it tells how many equal parts the whole is divided into. In the fraction \(\frac{3}{8}\), the denominator is 8, indicating that the whole is divided into 8 equal parts. When adding fractions, it's essential to have the same denominator to combine them easily. If the denominators are different, find the Least Common Multiple (LCM) to create a common base for the addition.