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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 and width w is A=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 isolated, we can divide both sides of the equation by w, giving us =Aw.
You can learn anything. Let's do this!

How do I isolate quantities?

Manipulating formulas: temperature

Khan Academy video wrapper
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 Ek=12mv2 to calculate the kinetic energy in joules, Ek, of an object with a mass of m kilograms traveling at a speed of v meters per second. What is v in terms of Ek 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
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 r2. To do so, we
both sides of the equation by πh.
Next, to isolate r, we
both sides of the equation.
Which of the following correctly expresses r ?
Choose 1 answer:

Your turn!

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?
Choose 1 answer:

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 ?
Choose 1 answer:

practice: isolate a quantity with exponents
The equation above shows the F, the gravitational force between two objects, in terms of G, the gravitational constant, m1 and m2, 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, m1, and m2 ?
Choose 1 answer:

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