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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 $ $w$ , giving us $\ell ={\displaystyle \frac{A}{w}}$ .

**isolated**, we can divide both sides of the equation by**You can learn anything. Let's do this!**

## How do I isolate quantities?

### Manipulating formulas: temperature

### 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:

- Write down the original equation. If needed, translate the word problem or given context into an equation.
- Perform the same operation on both sides of the equation to begin isolating the desired quantity.
- Repeat Step 2 until the desired quantity is isolated.

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

A physics student uses the formula ${E}_{\text{k}}={\displaystyle \frac{1}{2}}m{v}^{2}$ 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!

## Your turn!

## Want to join the conversation?

- Finally something easy after so many hard topics(208 votes)
- my first thought(23 votes)

- I can't believe that this is a separate unit(76 votes)
- 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.(3 votes)

- why math section is classified into three level: foundation, Medium and advance ? although there is a same question and same explanation?(32 votes)
- the practice questions are different for each level, you gotta check them out(58 votes)

- This was very basic and covered in previous lessons. I dont see the point of adding this(16 votes)
- will they give us the formulas in the questions? Like should I memorize the C to F formula?(8 votes)
- Of course not, this is a math exam, not physics, so you only have to memorize math formulas(13 votes)

- I thought this was far easier but also taught very well.(7 votes)
- lo único que aprendi de física fue a despejar(5 votes)
- This is part of the
math section??**advanced**(3 votes)- Yes, its with advanced math because people usually think of SATs as hard. Also because its normally only found in higher grade levels.(2 votes)

- In the last question, why do you have to multiply both sides by d^2 in the first step instead of by Gm1m2? What I got as the answer was the the square root of d=the square root of F(Gm1m2), which I understand is incorrect. I just don't understand why you have to multiply by the denominator instead of the numerator. Thanks! :)(2 votes)
- if you multiply it by the numerator, d^2 will still be in the denominator (it would be F/Gm1m2 = 1/d^2) so then you would have to take the recipricals of both sides to get Gm1m2/F = d^2. it would just be easier to multiply both sides by d^2 and divide by F. you don't have to its just faster. hope this helps!(2 votes)

- Why don't they just use transposition for these questions?(2 votes)