Problem 46 - Algebra 2

Question

ACT/SAT In a jar of red and white gumballs, the ratio of white gumballs to red gumballs is \(5 : 4\) . If the jar contains a total of 180 gumballs, how many of them are red? A 45 B 64 C 80 D 100

Step-by-Step Solution

Answer

There are 80 red gumballs in the jar.

Step 1: Understand the Ratio

The ratio of white gumballs to red gumballs is given as 5:4. This means for every 5 white gumballs, there are 4 red ones. Let the number of white gumballs be 5x and the number of red gumballs be 4x, where x is a positive integer.

Step 2: Set Up the Equation

The total number of gumballs is 180, which is the sum of white and red gumballs. Therefore, we can set up the equation: \[ 5x + 4x = 180 \] where 5x represents the white gumballs and 4x represents the red gumballs.

Step 3: Solve for x

Combine like terms in the equation: \[ 9x = 180 \] Divide both sides by 9 to solve for x: \[ x = \frac{180}{9} = 20 \]

Step 4: Find the Number of Red Gumballs

Now that we have x = 20, substitute it back into the expression for red gumballs, which is 4x: \[ 4x = 4 \times 20 = 80 \] Thus, there are 80 red gumballs in the jar.

Key Concepts

Solving Equations

When approaching problem-solving, equations are like mathematical puzzles. The goal is to find the value of the unknown variable, which, in this case, is represented by \( x \). Understanding how to manipulate equations is a key skill in math. Here's a simplified approach to solving an equation:

• First, identify the quantities in the problem and represent them with variables. Here, we use 5x for white gumballs and 4x for red ones.
• Set up an equation based on the problem. We established \(5x + 4x = 180\). This is because the combined total of the gumballs is 180.
• Combine like terms to simplify the equation. Adding \(5x\) and \(4x\) gives us \(9x = 180\).
• Finally, isolate the variable by dividing both sides by the coefficient. Thus, \(x = \frac{180}{9} = 20\).

Once we find \(x\), we can use it to determine the specifics of the problem, such as the number of red gumballs. Solving equations often involves these straightforward steps: set up, simplify, and isolate.

Ratio Word Problems

Ratio word problems are a type of math problem that compare parts of a whole in terms of ratios. In this exercise, we're dealing with a ratio of 5:4, which tells us that for every 5 white gumballs, there are 4 red ones. Ratios express the relative size of quantities and are solved by setting up equations like we did here. The key steps include:

• Understanding the given ratio. Here, it's crucial to see the relationship between the quantities involved, indicated by 5:4.
• Assigning variables to these quantities, such as using 5x for white and 4x for red, representing the multiplier in the relationship.
• Setting up equations based on this understanding to find the total number of items associated with each part of the ratio.
• Solving these equations by combining terms and isolating the variable is essential for finding the specific number of gumballs.

By grasping how ratios work, you can tackle many similar word problems and extract meaningful information to solve them effectively.

SAT Math Problems

SAT math problems assess your ability to apply mathematical concepts in various scenarios, and they often include word problems like this gumball one. To excel, focus on understanding what is being asked and identify the best approach to solving it. Here's how to tackle SAT math problems effectively:

• Read the problem carefully. Understand all the details and what is being asked. In our gumball problem, the key was to figure out how many are red.
• Translate the words into mathematical expressions or equations. This usually involves identifying and defining variables like \(x\), setting up equations, and determining the relationships between quantities.
• Solve the equations or expressions using your foundational math skills. This might involve arithmetic, algebra, geometry, or even trigonometry, depending on the specific problem.
• Finally, double-check your answers. It's crucial to ensure that your solution makes sense in the context of the question. Here, after finding \(x\), we confirmed that there are 80 red gumballs based on our calculations.

Practicing a variety of SAT math problems will help in developing your ability to think critically and solve problems efficiently on the test day.