The simple interest earned on an investment is given by where is the interest earned, is the principal, is the annual interest rate (as a decimal), and is the time in years. Which of the following equations expresses in terms of , , and ?

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SAT Equivalent Expressions Question 178 | Free SAT Prep | Aniko

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Question 178·Easy·Equivalent Expressions

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The simple interest earned on an investment is given by

I = P r t

where $I$ is the interest earned, $P$ is the principal, $r$ is the annual interest rate (as a decimal), and $t$ is the time in years.

Which of the following equations expresses $r$ in terms of $I$ , $P$ , and $t$ ?

For formula-rearrangement questions, treat the variables just like numbers and focus on isolating the target variable step by step. Identify which operations are being done to that variable (such as multiplication by other symbols) and then use inverse operations (like division) on both sides to undo them. Keep the target variable on one side of the equation by itself, and group all other symbols on the other side, checking that your final expression uses only the given variables.

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Look at how $r$ appears in the equation

In the equation $I = Prt$ , notice what operations are being done to $r$ . Is it being added, subtracted, multiplied, or divided by $P$ and $t$ ?

Think about the inverse operation

Once you see how $r$ is combined with $P$ and $t$ , ask: what operation would undo that and help you get $r$ alone?

Apply the same operation to both sides

To keep the equation balanced, whatever you do to one side you must do to the other. What could you divide both sides by so that only $r$ remains on one side?

Rewrite the equation after simplifying

After you divide and cancel common factors, rewrite the equation with $r$ alone on one side, and the expression in terms of $I$ , $P$ , and $t$ on the other side.

Desmos Guide

Pick specific values that satisfy the original formula

In Desmos, define simple values such as P = 100, r = 0.05, and t = 3. Then enter I = P*r*t to compute the corresponding interest $I$ ; Desmos will give you a numerical value for $I$ .

Compute each option’s expression for r using those values

Using the $I$ , $P$ , and $t$ you just defined, type each choice as a separate expression in Desmos: r1 = I*P*t, r2 = P/(I*t), r3 = t/(I*P), and r4 = I/(P*t). Desmos will show four numerical values.

Compare with the original r value

Look back at the value you originally set for r (for example, 0.05) and see which of r1, r2, r3, or r4 matches that value. The matching expression corresponds to the equation that correctly solves for $r$ .

Step-by-step Explanation

Understand the given formula and the goal

You are given the simple interest formula

I = Prt

where $I$ is interest, $P$ is principal, $r$ is the annual interest rate, and $t$ is time in years. The question asks you to rewrite this equation so that $r$ is by itself on one side ("solve for $r$ ") using $I$ , $P$ , and $t$ .

Identify how $r$ is combined with other variables

In the equation $I = Prt$ , the variable $r$ is being multiplied by both $P$ and $t$ .

So the right-hand side is the product $P \cdot r \cdot t$ .

Use inverse operations to isolate $r$

To isolate $r$ , you need to undo the multiplication by $P$ and $t$ . The inverse of multiplying by $P$ and $t$ is dividing by $P$ and $t$ .

So divide both sides of the equation by $Pt$ :

\frac{I}{Pt} = \frac{Prt}{Pt}

On the right-hand side, $P$ cancels with $P$ , and $t$ cancels with $t$ , leaving only $r$ .

Write the final equation for $r$

After the cancellation on the right-hand side, you are left with

\frac{I}{Pt} = r.

Writing $r$ on the left gives the final expression:

r = \frac{I}{Pt}.

So the correct choice is $r = I/(Pt)$ .

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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