Problem 67 - Elementary Algebra

Question

A recipe for frosted cookies includes \(\frac{1}{2}\) pound of butter for the cookies and \(\frac{1}{8}\) pound of butter for the frosting. Find the total amount of butter in this recipe.

Step-by-Step Solution

Answer

The total amount of butter is \( \frac{5}{8} \) pound.

Step 1: Identify the amounts of butter needed

Recognize the individual amounts of butter required for both the cookies and the frosting. For the cookies, it requires \( \frac{1}{2} \) pound of butter, and for the frosting, it requires \( \frac{1}{8} \) pound of butter.

Step 2: Find a common denominator

To add fractions, first, find a common denominator. The denominators here are 2 and 8. The least common multiple of 2 and 8 is 8.

Step 3: Convert the fractions

Convert \( \frac{1}{2} \) to a fraction with a denominator of 8. This would be \( \frac{4}{8} \), since \( \frac{1}{2} = \frac{4}{8} \).

Step 4: Add the fractions

Add \( \frac{4}{8} \) and \( \frac{1}{8} \) to find the total amount of butter. This is done by adding the numerators together: \( \frac{4}{8} + \frac{1}{8} = \frac{5}{8} \).

Step 5: Simplify the result if necessary

Check to see if the fraction can be simplified. In this case, \( \frac{5}{8} \) is already in its simplest form.

Key Concepts

Common Denominator

When adding fractions, like in our recipe example with \( \frac{1}{2} \) pound of butter for cookies and \( \frac{1}{8} \) pound for frosting, you need a common denominator. This means finding a number that both denominators can divide into evenly.

In this case, the numbers are 2 and 8. The common denominator will help make the fractions comparable. You can think of it like finding a common language for two people to speak to each other.

By converting both fractions to have the same denominator, you can add them together easily.

Least Common Multiple

To find a common denominator, you first need to determine the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into without leaving a remainder.

For our denominators 2 and 8, the LCM is 8. This is because 8 is the smallest number that both 2 and 8 will divide evenly into.

Here’s how you can find the LCM step-by-step:

• List the multiples of each denominator. For 2, you have 2, 4, 6, 8, etc. For 8, you have 8, 16, 24, etc.


• The smallest common multiple in these lists is 8.


• Thus, the LCM of 2 and 8 is 8.

This number helps us transform our fractions to have the same denominator, simplifying the addition process.

Fraction Conversion

Once we have a common denominator (the LCM), we can convert our fractions to have this denominator. In our example, we need to convert \( \frac{1}{2} \) to an equivalent fraction with a denominator of 8.

Here's how to do it:

• First, determine what number you need to multiply the denominator 2 by to get 8. In this case, it’s 4, since \( 2 \times 4 = 8 \).


• Next, multiply both the numerator and the denominator of \( \frac{1}{2} \) by 4, resulting in \( \frac{4}{8} \).

Now you can easily add this converted fraction (\( \frac{4}{8} \)) to \( \frac{1}{8} \). Both fractions now speak the same 'language' by having the same denominator.

Simplifying Fractions

After adding the fractions, the last step is to check if the result can be simplified. Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1.

In the case of our recipe, after converting and adding the fractions, we got \( \frac{4}{8} + \frac{1}{8} = \frac{5}{8} \). To simplify \( \frac{5}{8} \), you look for the greatest common divisor (GCD) of 5 and 8. The GCD here is 1 because 5 and 8 share no other common factors.

This means \( \frac{5}{8} \) is already in its simplest form, and there’s no further reduction needed.

To sum up:

• Always check the final fraction to see if the numerator and denominator can be divided by any common number.


• If they can, divide both by this number to simplify.


• If not, the fraction is already simplified.