Rearranging Simple Formulas

A formula is an equation that shows the relationship between two or more quantities.

For example:

v = d/t

connects:

• v = speed
• d = distance
• t = time

If we know distance and time, we can calculate speed directly. But sometimes we need to find distance or time instead.

We can rearrange the formula so the variable we want is isolated.

From:

v = d/t

we can rearrange to:

d = vt

or:

t = d/v

These are not new formulas. They are equivalent versions of the same relationship.

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Variables in Formulas

A variable is a symbol representing a quantity that can have different values.

Consider:

A = lw

The variables are:

• A = area
• l = length
• w = width

The formula tells us that the area of a rectangle equals its length multiplied by its width.

In formulas, letters often have specific meanings rather than simply representing an unknown number.

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Identifying Variables

Consider:

F = ma

The variables are:

• F = force
• m = mass
• a = acceleration

Consider:

P = E/t

The variables are:

• P = power
• E = energy transferred
• t = time

Before rearranging a formula, make sure you understand what each variable represents.

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Constants and Variables

Not every symbol in a formula necessarily changes.

For example:

C = 2πr

The quantities C and r are variables.

The number 2 is a constant.

π is also a constant:

π ≈ 3.14159

Recognizing the difference between variables and constants helps us understand formulas correctly.

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What Does Rearranging Mean?

To rearrange a formula means to rewrite it so that a different variable is isolated.

For example:

F = ma

already has F isolated.

If we want m instead:

m = F/a

If we want a:

a = F/m

The relationship between force, mass, and acceleration has not changed.

Only the form of the equation has changed.

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The Goal: Isolate the Required Variable

When rearranging a formula, ask:

Which variable do I need by itself?

For:

P = E/t

if the question asks for E, we want:

E = ?

If the question asks for t, we want:

t = ?

This helps us decide which operations need to be undone.

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The Balance Principle

A formula is an equation, so both sides must remain equal.

If we perform an operation on one side, we must perform the same operation on the other side.

For example:

x + 5 = 12

Subtract 5 from both sides:

x = 7

The same principle applies when rearranging formulas.

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Inverse Operations

Inverse operations undo each other.

For example, if a variable is multiplied by 5:

y = 5x

divide by 5 to isolate x:

x = y/5

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Rearranging Addition Formulas

Consider:

y = x + 4

Make x the subject.

Subtract 4 from both sides:

y − 4 = x

Rewrite:

x = y − 4

Therefore:

y = x + 4

can be rearranged as:

x = y − 4

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Rearranging Subtraction Formulas

Consider:

y = x − 7

Make x the subject.

Add 7 to both sides:

y + 7 = x

Therefore:

x = y + 7

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Be Careful with Subtraction Order

Consider:

y = 10 − x

Make x the subject.

Add x to both sides:

y + x = 10

Subtract y:

x = 10 − y

The answer is:

x = 10 − y

not:

x = y − 10

Subtraction order matters.

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Rearranging Multiplication Formulas

Consider:

y = 4x

Make x the subject.

Divide both sides by 4:

y/4 = x

Therefore:

x = y/4

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Rearranging Division Formulas

Consider:

y = x/5

Make x the subject.

Multiply both sides by 5:

5y = x

Therefore:

x = 5y

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A Reliable Strategy

For simple formulas, use:

Identify → Target → Undo → Simplify → Check

Identify

Identify all variables and what they represent.

Target

Decide which variable needs to be isolated.

Undo

Use inverse operations to remove everything attached to that variable.

Simplify

Write the rearranged formula clearly.

Check

Confirm that the new formula is equivalent to the original.

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Distance, Speed, and Time

One of the most useful examples is:

v = d/t

where:

• v = speed
• d = distance
• t = time

To make d the subject, multiply both sides by t:

vt = d

Rewrite:

d = vt

To make t the subject:

Start with:

v = d/t

Multiply by t:

vt = d

Divide by v:

t = d/v

Therefore:

v = d/t

d = vt

t = d/v

All three represent the same relationship.

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Applying the Rearranged Formula

Suppose:

v = 20 m/s

and:

t = 15 s

Find d.

Use:

d = vt

Substitute:

d = (20)(15)

Therefore:

d = 300 m

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Finding Time

Suppose:

d = 500 km

and:

v = 100 km/h

Use:

t = d/v

Substitute:

t = 500/100

Therefore:

t = 5 h

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Newton's Second Law

Newton's Second Law is:

F = ma

where:

• F = net force
• m = mass
• a = acceleration

To make m the subject:

Divide by a:

m = F/a

To make a the subject:

Divide by m:

a = F/m

Therefore:

F = ma

can also be written:

m = F/a

or:

a = F/m

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Applying F = ma

Suppose:

F = 60 N

m = 12 kg

Find acceleration.

Use the rearranged formula:

a = F/m

Substitute:

a = 60/12

Therefore:

a = 5 m/s²

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Density

Density is:

ρ = m/V

where:

• ρ = density
• m = mass
• V = volume

To make m the subject:

Multiply by V:

ρV = m

Therefore:

m = ρV

To make V the subject:

Start with:

ρ = m/V

Multiply by V:

ρV = m

Divide by ρ:

V = m/ρ

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Density Example

A material has:

ρ = 8 g/cm³

and:

m = 200 g

Find its volume.

Use:

V = m/ρ

Substitute:

V = 200/8

Therefore:

V = 25 cm³

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Weight

Weight is calculated using:

W = mg

where:

• W = weight
• m = mass
• g = gravitational field strength

Make m the subject:

m = W/g

Make g the subject:

g = W/m

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Weight Example

An object weighs:

196 N

where:

g = 9.8 N/kg

Find its mass.

Use:

m = W/g

Substitute:

m = 196/9.8

Therefore:

m = 20 kg

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Pressure

Pressure is:

p = F/A

where:

• p = pressure
• F = force
• A = area

Make F the subject:

F = pA

Make A the subject:

A = F/p

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Pressure Example

A pressure of:

500 Pa

acts over an area of:

0.20 m²

Find the force.

Use:

F = pA

Substitute:

F = (500)(0.20)

Therefore:

F = 100 N

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Electrical Relationships

Electrical formulas also require rearrangement.

For example:

V = IR

where:

• V = voltage
• I = current
• R = resistance

Make I the subject:

I = V/R

Make R the subject:

R = V/I

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Ohm's Law Example

Suppose:

V = 12 V

and:

R = 4 Ω

Find current.

Use:

I = V/R

Substitute:

I = 12/4

Therefore:

I = 3 A

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Electrical Power

Electrical power can be calculated using:

P = VI

Make V the subject:

V = P/I

Make I the subject:

I = P/V

Example:

A device uses:

P = 120 W

at:

V = 12 V

Find current:

I = P/V

I = 120/12

I = 10 A

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Energy, Power, and Time

Power is:

P = E/t

Make E the subject:

E = Pt

Make t the subject:

t = E/P

Example:

A 100 W device operates for 30 s.

E = Pt

E = (100)(30)

Therefore:

E = 3000 J

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Rectangle Area

The area of a rectangle is:

A = lw

Make l the subject:

l = A/w

Make w the subject:

w = A/l

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Rectangle Example

A rectangle has:

A = 60 cm²

and:

w = 5 cm

Find its length.

Use:

l = A/w

Substitute:

l = 60/5

Therefore:

l = 12 cm

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Triangle Area

The area of a triangle is:

A = 1/2 bh

Make b the subject.

Start with:

A = 1/2 bh

Multiply both sides by 2:

2A = bh

Divide by h:

b = 2A/h

To make h the subject:

h = 2A/b

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Triangle Example

A triangle has:

A = 36 cm²

and:

b = 9 cm

Find its height.

Use:

h = 2A/b

Substitute:

h = 2(36)/9

h = 72/9

Therefore:

h = 8 cm

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Circle Circumference

Circumference is:

C = 2πr

Make r the subject.

Divide both sides by 2π:

r = C/(2π)

If:

C = 31.4 cm

then:

r = 31.4/(2π)

Using π ≈ 3.14:

r = 5 cm

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Temperature Conversion

The Celsius-to-Fahrenheit formula is:

F = (9/5)C + 32

Suppose we want C.

Start:

F = (9/5)C + 32

Subtract 32:

F − 32 = (9/5)C

Multiply by 5/9:

C = (5/9)(F − 32)

This is a multi-step rearrangement.

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Temperature Example

Convert:

86°F

to Celsius.

Use:

C = (5/9)(F − 32)

Substitute:

C = (5/9)(86 − 32)

C = (5/9)(54)

Therefore:

C = 30°C

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Undo Operations in Reverse Order

Consider:

y = 3x + 5

To isolate x, ask:

What has happened to x?

First:

x was multiplied by 3

Then:

5 was added

Undo these operations in reverse order.

Subtract 5:

y − 5 = 3x

Divide by 3:

x = (y − 5)/3

This reverse-order idea is extremely useful.

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Another Two-Step Formula

Consider:

y = 4x − 7

Make x the subject.

Add 7:

y + 7 = 4x

Divide by 4:

x = (y + 7)/4

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Formula with a Divided Expression

Consider:

y = (x + 3)/5

Make x the subject.

Multiply by 5:

5y = x + 3

Subtract 3:

x = 5y − 3

Notice that we undo the outermost operation first.

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Formula with Subtraction in the Numerator

Consider:

y = (x − 4)/3

Multiply by 3:

3y = x − 4

Add 4:

x = 3y + 4

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Variables on Both Sides

Some formulas contain the required variable more than once or on both sides.

For example:

y = ax + bx

Factor x:

y = x(a + b)

Divide by a + b:

x = y/(a + b)

Factoring can sometimes be an important part of rearranging formulas.

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Factoring to Isolate a Variable

Consider:

P = 2l + 2w

Suppose we want l.

Subtract 2w:

P − 2w = 2l

Divide by 2:

l = (P − 2w)/2

This can also be written:

l = P/2 − w

Both forms are equivalent.

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Checking a Rearranged Formula

One way to check rearrangement is to reverse the algebra.

Suppose:

v = d/t

was rearranged to:

d = vt

Starting from:

d = vt

divide by t:

d/t = v

Therefore:

v = d/t

We have returned to the original formula.

This supports that the rearrangement is correct.

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Checking with Numerical Values

We can also test a rearranged formula using numbers.

Original:

F = ma

Let:

m = 4 kg

a = 5 m/s²

Then:

F = 20 N

Now test:

m = F/a

m = 20/5

m = 4 kg

The rearranged formula gives the expected value.

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Checking by Units

Units can also help detect mistakes.

For:

v = d/t

speed has units:

m/s

Rearrange:

d = vt

Units:

(m/s)(s) = m

Correct.

For:

t = d/v

units:

m ÷ (m/s) = s

Correct.

This is called dimensional checking or dimensional analysis.

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Formula Triangles

Some simple formulas are sometimes remembered using formula triangles.

For example, for:

v = d/t

a triangle may show d above v and t.

However, formula triangles are memory aids rather than a replacement for algebra.

Learning to rearrange equations using inverse operations works for much more complicated formulas.

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Practical Example: Travel

A car travels:

360 km

at an average speed of:

90 km/h

Find the time.

Start with:

v = d/t

Make t the subject:

t = d/v

Substitute:

t = 360/90

Therefore:

t = 4 h

Interpretation:

The journey takes 4 hours at an average speed of 90 km/h.

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Practical Example: Density

A liquid has:

density = 0.8 g/cm³

and:

volume = 250 cm³

Find its mass.

Start:

ρ = m/V

Rearrange:

m = ρV

Substitute:

m = (0.8)(250)

Therefore:

m = 200 g

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Practical Example: Electricity

A resistor has:

R = 15 Ω

and carries:

I = 2 A

Find voltage.

Use:

V = IR

Substitute:

V = (2)(15)

Therefore:

V = 30 V

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Practical Example: Geometry

A triangle has:

area = 54 cm²

and:

height = 9 cm

Find the base.

Start:

A = 1/2 bh

Rearrange:

b = 2A/h

Substitute:

b = 2(54)/9

Therefore:

b = 12 cm

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Practical Example: Cost

Suppose the total cost of renting equipment is:

C = 15h + 25

where:

• C = total cost
• h = number of hours
• $25 = fixed fee
• $15 = hourly charge

Make h the subject.

Subtract 25:

C − 25 = 15h

Divide by 15:

h = (C − 25)/15

If the total bill is $100:

h = (100 − 25)/15

h = 75/15

Therefore:

h = 5 hours

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Choosing Whether to Rearrange First

Sometimes we can substitute values before rearranging.

For:

F = ma

if:

F = 40

and:

m = 8

we could write:

40 = 8a

and solve:

a = 5

Or we could rearrange first:

a = F/m

then substitute:

a = 40/8 = 5

Both methods are correct.

However, rearranging first is especially useful when the same type of calculation will be repeated several times.

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Common Mistakes

Mistake: Performing an operation on only one side.

An equation must remain balanced.

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Mistake: Moving terms without understanding the operation.

Terms do not simply "jump across" an equals sign.

For:

y = x + 5

we subtract 5 from both sides to obtain:

x = y − 5

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Mistake: Reversing subtraction incorrectly.

From:

y = 10 − x

the correct rearrangement is:

x = 10 − y

not:

x = y − 10

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Mistake: Multiplying when division is required.

From:

F = ma

the correct rearrangement for mass is:

m = F/a

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Mistake: Dividing only part of an expression.

From:

y = 3x + 6

we get:

x = (y − 6)/3

The entire expression y − 6 is divided by 3.

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Mistake: Ignoring parentheses.

(y − 6)/3

is not the same as:

y − 6/3

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Mistake: Substituting too early and losing track of the algebra.

For unfamiliar formulas, rearranging symbolically first can make the relationship clearer.

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Mistake: Forgetting units.

After applying a rearranged formula to a practical problem, include the appropriate unit.

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Key Terms

Formula — equation representing a relationship between quantities.

Variable — symbol representing a quantity that may change or be unknown.

Constant — quantity with a fixed value.

Subject of a formula — variable isolated on one side of the formula.

Rearrange — rewrite a formula so that a different variable becomes the subject.

Isolate — get a variable alone on one side.

Inverse operations — operations that undo each other.

Equivalent formulas — different forms of a formula representing the same mathematical relationship.

Substitution — replacing variables with known values.

Factoring — rewriting an expression as a product of factors.

Dimensional analysis — using units to examine whether a mathematical relationship is consistent.

Practical application — use of mathematics to solve a real-world problem.

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Key Takeaways

• A formula is an equation relating two or more quantities.
• Variables in formulas usually represent specific physical or mathematical quantities.
• Rearranging changes the form of a formula but not the relationship it represents.
• The required variable should be isolated on one side.
• Use inverse operations to undo operations affecting the required variable.
• Addition is undone by subtraction.
• Subtraction is undone by addition.
• Multiplication is undone by division.
• Division is undone by multiplication.
• Operations are often undone in reverse order.
• Both sides of a formula must remain balanced.
• Parentheses are important when an entire expression is divided.
• Subtraction order must be handled carefully.
• Formulas such as v = d/t can be rearranged into several equivalent forms.
• F = ma can become m = F/a or a = F/m.
• ρ = m/V can become m = ρV or V = m/ρ.
• Factoring may help isolate a variable that appears in multiple terms.
• Rearranged formulas can be checked by reversing the algebra.
• Numerical substitution can also help check a rearrangement.
• Units provide another useful way to check formulas.
• Formula triangles may help with simple relationships, but algebraic rearrangement is more powerful.
• Formula rearrangement is widely used in mathematics, physics, chemistry, engineering, finance, and everyday problem-solving.

Check Your Understanding

1. What does it mean to make a variable the subject of a formula?
2. Identify all variables in F = ma.
3. Rearrange y = x + 7 to make x the subject.
4. Rearrange y = x − 5 to make x the subject.
5. Rearrange y = 4x to make x the subject.
6. Rearrange y = x/6 to make x the subject.
7. Rearrange v = d/t to make d the subject.
8. Rearrange v = d/t to make t the subject.
9. Rearrange F = ma to make m the subject.
10. Rearrange F = ma to make a the subject.
11. Rearrange ρ = m/V to make m the subject.
12. Rearrange ρ = m/V to make V the subject.
13. Rearrange V = IR to make R the subject.
14. Rearrange P = E/t to make t the subject.
15. Rearrange A = 1/2 bh to make h the subject.
16. Rearrange y = 5x + 3 to make x the subject.
17. A car travels 450 km at 90 km/h. Rearrange and use v = d/t to determine the travel time.
18. A material has a density of 6 g/cm³ and a volume of 20 cm³. Rearrange and use ρ = m/V to determine its mass.
19. Explain two different ways you could check whether a rearranged formula is correct.
20. Explain why learning algebraic rearrangement is more useful than relying only on formula triangles.