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### Course: Digital SAT Math>Unit 8

Lesson 6: Isolating quantities: medium

# Isolating quantities | Lesson

A guide to isolating quantities on the digital SAT

## What are isolating quantities problems?

Many real-world scenarios can be described using equations and formulas with multiple variables. For example, the formula for the area, $A$, of a rectangle with length $\ell$ and width $w$ is $A=\ell w$. In this form, area ($A$) is isolated: it is alone on one side of the equation.
If we want to find the equivalent equation, but with $\ell$ isolated, we can divide both sides of the equation by $w$, giving us $\ell =\frac{A}{w}$.
You can learn anything. Let's do this!

## How do I isolate quantities?

### Manipulating formulas: temperature

Manipulating formulas: temperatureSee video transcript

### Like solving equations, but with more variables

Isolating quantities is similar to solving equations. However, instead of ending up with a variable equal to a constant, we end up with a variable equal to an expression containing other variables. In fact, if you've rewritten a linear equation like $x+y=1$ in slope-intercept form, then you've already isolated $y$ by rearranging a linear equation!
The most important thing to remember is that the rearranged equation will remain equivalent to the original equation only if we always treat both sides equally: whenever we do something to one side, we must do the exact same thing to the other side.
To isolate a quantity in an equation or formula:
1. Write down the original equation. If needed, translate the word problem or given context into an equation.
2. Perform the same operation on both sides of the equation to begin isolating the desired quantity.
3. Repeat Step 2 until the desired quantity is isolated.

#### Let's look at some examples!

A physics student uses the formula ${E}_{\text{k}}=\frac{1}{2}m{v}^{2}$ to calculate the kinetic energy in joules, ${E}_{\text{k}}$, of an object with a mass of $m$ kilograms traveling at a speed of $v$ meters per second. What is $v$ in terms of ${E}_{\text{k}}$ and $m$ ?

The distance $d$ traveled by an object moving at constant velocity is found by multiplying the velocity $v$ of the subject by time $t$. Write an equation that gives the time $t$ in terms of $d$ and $v$.

### Try it!

try: identify the steps to isolate a variable
$V=\pi {r}^{2}h$
The equation above shows the volume $V$ of a right cylinder with radius $r$ and height $h$.
To rewrite the equation to isolate $r$, we must first isolate ${r}^{2}$. To do so, we
both sides of the equation by $\pi h$.
Next, to isolate $r$, we
both sides of the equation.
Which of the following correctly expresses $r$ ?

practice: isolate a quantity in one step
Giselle uses the formula $I=Prt$ to calculate $I$, the simple interest in dollars, she collects from a loan of $P$ dollars at an interest rate of $r$ per month over a period of $t$ months. Which of the following expresses $P$ in terms of the other variables?

practice: write an equation, then isolate a quantity
The perimeter $P$ of a rectangle is found by multiplying the sum of the rectangle's length $l$ and width $w$ by $2$. Which of the following equations gives the width $w$ in terms of $P$ and $l$ ?

practice: isolate a quantity with exponents
$F=\frac{G{m}_{1}{m}_{2}}{{d}^{2}}$
The equation above shows the $F$, the gravitational force between two objects, in terms of $G$, the gravitational constant, ${m}_{1}$ and ${m}_{2}$, the masses of the two objects, and $d$, the distance between the two objects. Which of the following correctly expresses $d$ in terms of $F$, $G$, ${m}_{1}$, and ${m}_{2}$ ?

## Want to join the conversation?

• Finally something easy after so many hard topics
• my first thought
• I can't believe that this is a separate unit
• There are a few reasons its separate. First of all, it makes it easier for people to find it and bookmark it. Second, it helps Khans reputation and gets them more customers if they have a unit for SAT prep so that they can point it out.
• why math section is classified into three level: foundation, Medium and advance ? although there is a same question and same explanation?
• the practice questions are different for each level, you gotta check them out
• This was very basic and covered in previous lessons. I dont see the point of adding this
• will they give us the formulas in the questions? Like should I memorize the C to F formula?
• Of course not, this is a math exam, not physics, so you only have to memorize math formulas
• I thought this was far easier but also taught very well.