Problem 22 - Algebra 2

Question

REVIEW Anna took brownies to a club meeting. She gave half of her brownies to Sarah. Sarah gave a third of her brownies to Rob. Rob gave a fourth of his brownies to Trina. If Trina has 3 brownies, how many brownies did Anna have in the beginning? $$ \begin{array}{l}{\mathrm{F} 12} \\ {\mathrm{G} 36} \\ {\mathrm{H} 72} \\\ {\mathrm{J} 144}\end{array} $$

Step-by-Step Solution

Answer

Anna originally had 72 brownies.

Step 1: Understand Trina's Share

Trina has 3 brownies, which represent a fourth of the brownies Rob had.

Step 2: Calculate Rob's Share

If Trina's 3 brownies are a fourth of Rob's total brownies, then Rob had 4 times 3 brownies. Hence, Rob had \(3 \times 4 = 12\) brownies.

Step 3: Understand Sarah's Share

Rob received a third of Sarah's brownies. Thus, Sarah had 3 times the number of brownies Rob had.

Step 4: Calculate Sarah's Share

Sarah shared \(12\) brownies with Rob (which is a third of her total). Therefore, Sarah had \(12 \times 3 = 36\) brownies.

Step 5: Understand Anna's Share

Sarah received half of Anna's brownies. Therefore, Anna originally had 2 times the number of brownies Sarah had.

Step 6: Calculate Anna's Original Share

Since Sarah ended up with \(36\) brownies from Anna, Anna originally had \(36 \times 2 = 72\) brownies.

Key Concepts

Fractions in Everyday Scenarios

Fractions are an essential part of mathematics, representing parts of a whole. They are very useful when dividing things into equal parts. For instance, when Anna divided her brownies, she used fractions to share them equally among her friends. Fractions make it easier to understand how much each person gets if something is distributed evenly.
In the brownie problem, we see fractions come into play several times. Anna gave away half of her brownies to Sarah, so Sarah received \(\frac{1}{2}\) of Anna's total. Similarly, Sarah passed on \(\frac{1}{3}\) of her share to Rob, and Rob gave Trina \(\frac{1}{4}\) of what he had. Understanding these divisions is crucial to solve how many brownies Anna initially had.
Using fractions involves looking at them as parts-per-container or total. When solving these problems, make sure to multiply the given quantity by the reciprocal of the fraction it represents to find the original total.

Basics of Algebra in Problem Solving

Algebra is a branch of mathematics that helps us solve problems using symbols and letters to represent numbers. It provides a way to work with unknowns in a systematic manner. When solving problems like the brownie distribution, algebra helps in finding unknowns by setting up equations and expressions.
In our case, we analyze each person's share in terms of algebraic expressions. By knowing Trina's share and working backward, we set up an equation to determine the total number of brownies Anna had. Each step involved calculating an unknown using known quantities and fractional relationships.

• For Trina, she had 3 brownies, which were \(\frac{1}{4}\) of Rob's. To find out Rob's total, we multiply: \(3 \times 4 = 12\).
• Similarly, Rob had \(\frac{1}{3}\) of Sarah's share, making Sarah's total \(12 \times 3 = 36\).
• Finally, Sarah had \(\frac{1}{2}\) of Anna's brownies, so Anna initially had \(36 \times 2 = 72\).

Understanding how algebra is used in sequencing these steps allows us to track back to the starting point methodically.

Understanding Arithmetic Sequences

An arithmetic sequence involves a series of numbers with a constant difference between consecutive terms, much like a progression. Although the problem we're solving doesn't rely directly on arithmetic sequences, the reasoning process is similar to following a sequence of logical steps.
The brownie problem can be likened to a reverse arithmetic sequence, where instead of adding or subtracting a constant, we multiply by the reciprocal of a given fraction because each person in the sequence received a defined portion. By reversing the process of fractional division, we piece together each previous amount leading back to Anna.
Think of it like peeling back layers of a sequence, where each layer (person) represents a fraction of the previous one. This approach not only helps in the brownie problem but strengthens your ability to follow and expand sequences logically. Though not an arithmetic sequence per se, the logical layering mirrors this powerful mathematical concept.