SAT Linear Equations in One Variable Question 120 | Free SAT Prep | Aniko

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Question 120·Hard·Linear Equations in One Variable

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A jar contained some identical marbles. Evan removed one-third of the marbles plus 4 more from the jar. Later, Maya removed half of the marbles that remained, leaving 20 marbles in the jar. How many marbles were originally in the jar?

Your answer

(Express the answer as an integer)

For word problems with sequential actions, first define a variable for the original amount, then carefully translate each step into algebra, keeping track of what is removed versus what remains. Use parentheses to group expressions like “one-third of the marbles plus 4 more” so subtraction is applied to the entire quantity, and simplify after each person’s action. Finally, set the expression for the final amount equal to the given number and solve the resulting one-variable linear equation using inverse operations.

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Start by choosing a variable

Let $x$ represent the number of marbles originally in the jar. Every other quantity should be written in terms of $x$ .

Express the marbles after Evan’s turn

Evan removed $\tfrac{1}{3}$ of the marbles plus 4 more. How can you write the number of marbles removed and then the number remaining in terms of $x$ ?

Express the marbles after Maya’s turn

Take the amount left after Evan and find half of that. That half is what remains in the jar. Set this expression equal to 20.

Solve the linear equation

You should now have an equation of the form something like $\dfrac{x}{3} - 2 = 20$ . Isolate $x$ using inverse operations (undo subtraction, then undo division).

Desmos Guide

Enter the equation for the remaining marbles

In Desmos, type the equation y = x/3 - 2. This represents the number of marbles left after both Evan and Maya in terms of the original number $x$ .

Graph the final amount and find the intersection

On a new line, type y = 20. Then, look for the point where the graph of y = x/3 - 2 intersects the horizontal line y = 20. The x-coordinate of this intersection gives the original number of marbles.

Step-by-step Explanation

Define the variable

Let $x$ be the number of marbles originally in the jar.

Write how many marbles remain after Evan

Evan removed one-third of the marbles plus 4 more, so he removed $x/3 + 4$ marbles.

The number of marbles left after Evan is

x - \bigg(\frac{x}{3} + 4\bigg).

Simplify this expression:

x - \frac{x}{3} - 4 = \frac{3x}{3} - \frac{x}{3} - 4 = \frac{2x}{3} - 4.

So after Evan, there are $\frac{2x}{3} - 4$ marbles left in the jar.

Write how many marbles remain after Maya and set up the equation

Maya removed half of the marbles that remained after Evan.

Half of $\frac{2x}{3} - 4$ is

\frac{1}{2}\bigg(\frac{2x}{3} - 4\bigg) = \frac{2x}{3} \cdot \frac{1}{2} - 4 \cdot \frac{1}{2} = \frac{x}{3} - 2.

If Maya removes half, the other half is left in the jar. That remaining half is also $\frac{x}{3} - 2$ marbles, and we are told this equals 20:

\frac{x}{3} - 2 = 20.

This is the linear equation we need to solve for $x$ .

Solve the equation for the original number of marbles

Solve

\frac{x}{3} - 2 = 20.

Add 2 to both sides:

\frac{x}{3} = 22.

Multiply both sides by 3:

x = 66.

So, there were originally 66 marbles in the jar.

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation