Rearranging equations · RiskAtoms Academy

By the end you will be able to

1Solve an equation for one unknown by doing the same thing to both sides.
2Undo operations in the right order, last one first.
3Rearrange a finance formula to make any of its letters the subject.
4Check an answer by putting it back into the equation.

Why you need it

A formula tells you the end amount from the start amount. Very often you know the end and need the start, or the rate, or the time. Rearranging turns one formula into all of those.

L1.3.03 Present value and discounting
Turning a growth formula around to find what a future amount is worth today.
L1.4.04 Yield to maturity
Finding the rate that makes a formula match a known price.

Any lesson that asks you to solve for one unknown uses the same few moves.

The idea

An equation is a balance

An equation says two things are equal. Picture a balance scale with the left side in one pan and the right side in the other. The scale is level.

The letter $x$ is the unknown: a number we don't know yet. To solve the equation, we get $x$ on its own in one pan. The one golden rule: whatever you do to one side, do to the other. Take 4 from the left pan and you must take 4 from the right, or the scale tips.

Each move uses an inverse operation: subtract to undo adding, divide to undo multiplying. You probably do this in your head already:

Think firstWhat is x?

$x + 3 = 10$
$2x = 18$
$x \div 5 = 4$

The rules

Rule 1: do the same to both sides, and undo the last step first. When $x$ has been through several steps, unwrap it like a parcel: the outer layer comes off first.

Rule 2: a formula is an equation with more letters. The same moves work. Here is a formula for money growing at simple interest:

$FV$: future value: the amount at the end
$PV$: present value: the amount at the start
$r$: the rate per year, as a decimal (5% is 0.05)
$t$: the time in years

All four versions, side by side. Each tab checks itself with the same numbers: €2,000 at 5% for 1.5 years grows to €2,150.

Solve for

FV = PV \times (1 + r\,t)

Check: $2{,}000 \times (1 + 0.05 \times 1.5) = 2{,}150.00$

The moves on one card

Whatever you do to one side, do to the other.
Undo + with −, × with ÷, and the other way round.
Unwrap from the outside in: undo the last operation first.
Treat a bracket as one block until it is on its own.
Check by putting the answer back in.

Try it

Pick an equation, choose an operation and a number, and apply it to both sides. Try “Left side only” once to see the scale tip. Stuck? Press Hint.

Try itBalance the scales

Open the equation solver →

Pick an equation

3x + 4 = 19

Operation

Number

Worked examples

1. €2,000 grows to €2,150 in 18 months at simple interest. What is the rate?

Step 1 of 4

1
Write down what you know
$FV = 2{,}150,\ PV = 2{,}000,\ t = 1.5$ . 18 months is 1.5 years. The unknown is $r$ .

2. How much to set aside today to have $10,800 in 2 years at 4% simple interest?

Step 1 of 3

1
Choose the right version
The unknown is the start amount, so use $PV = FV \div (1 + r\,t)$ .

3. A share has a P/E ratio of 15 and earnings of 2.40 per share. What is its price?

Step 1 of 2

1
Write the ratio as an equation
$\text{P/E} = \frac{\text{price}}{\text{earnings}} \quad\Rightarrow\quad 15 = \frac{\text{price}}{2.40}$

Common mistakes

Each of these is tempting. Decide what is wrong before you reveal the fix.

“x + 5 = 12, so move the 5 across: x = 12 + 5 = 17.”
“3x + 4 = 19. Divide by 3: x + 4 = 19 ÷ 3.”
“FV/PV = 1 + r t, so divide by t: FV/(PV × t) = 1 + r.”
“r is 5%, so put r = 5 into the formula.”
“I got an answer, so I'm done.”

Reaction

Check what stuck

Question 1 · objective 1

Solve 5x − 7 = 18. What is x?

Question 2 · objective 2

To solve x ÷ 2 + 3 = 11, what is the best first step?

AMultiply the left side by 2BSubtract 3 from both sidesCDivide both sides by 2DSubtract 11 from both sides

Question 3 · objective 3

FV = PV × (1 + r t). Which is the formula for t?

At = (FV − PV) × rBt = (FV ÷ PV − 1) ÷ rCt = FV ÷ (PV × r) − 1Dt = (FV ÷ PV) ÷ r − 1

Question 4 · objective 3

£5,000 grows to £5,300 at simple interest of 4% a year. Using FV = PV × (1 + r t), how many years did it take?

years

Question 5 · objective 4

A price-to-earnings ratio is P/E = price ÷ earnings. A share costs 54 and its P/E is 12. What are its earnings?

Glossary

Equation: A statement that two things are equal, written with an = sign between them.
Unknown: A letter standing for a number we do not know yet, often x.
Solve: Find the value of the unknown that makes the equation true.
Inverse operation: The operation that undoes another: subtracting undoes adding, dividing undoes multiplying.
Rearrange: Rewrite a formula so that a different letter is on its own on one side.
Subject: The letter on its own on one side of a formula, as PV is in PV = FV ÷ (1 + r t).