Applying and Rearranging Formulas

What the formula sheet gives you

The provided sheet includes the formulas you reach for most often. A few of the ones we'll use here:

• Area of a rectangle: A = lw (area equals length times width).
• Distance: d = rt (distance equals rate times time).
• Simple interest: I = Prt (interest equals principal times rate times time).
• Area of a circle: A = πr² (area equals pi times the radius squared).

The one big idea: inverse operations

Rearranging a formula is the exact same move as solving an equation: you undo whatever is happening to the variable you want, and you do it to both sides to keep things balanced. Each operation has an opposite, its inverse:

So if a variable is being multiplied by something, you divide both sides by that something to set it free. That single idea handles nearly every rearrangement on the test.

Worked example #1: just substitute (find the area)

A rectangular garden is 8 m long and 5 m wide. Find its area using A = lw. Here the unknown (A) is already alone, so there is nothing to rearrange, you simply plug in and multiply.

• Write the formula: A = lw.
• Substitute: l = 8, w = 5, so A = 8 × 5.
• Calculate: A = 40. The unit is square metres.

  A = l w
  A = (8)(5)     (substitute l = 8, w = 5)
  A = 40 m²

Worked example #2: rearrange A = lw to solve for w

Now the twist: a rectangle has area 48 cm² and length 6 cm. Find the width. This time the unknown (w) is not alone, it is being multiplied by l. To free w, divide both sides by l.

• Start: A = lw. The w is multiplied by l.
• Undo the multiply: divide both sides by l, giving A ÷ l = w.
• Substitute and solve: w = 48 ÷ 6 = 8 cm.

  A = l w
  A / l = w        (divide both sides by l)
  w = A / l        (just flipped around to read it)
  w = 48 / 6       (substitute A = 48, l = 6)
  w = 8 cm

Worked example #3: rearrange d = rt to solve for t

A car travels 240 km at a steady 60 km/h. How long does the trip take? The distance formula is d = rt, and we want the time t. Since t is multiplied by r, we divide both sides by r.

• Start: d = rt. The t is multiplied by r.
• Undo the multiply: divide both sides by r, giving t = d ÷ r.
• Substitute and solve: t = 240 ÷ 60 = 4 hours.

  d = r t
  d / r = t        (divide both sides by r)
  t = d / r
  t = 240 / 60     (substitute d = 240, r = 60)
  t = 4 hours

Worked example #4: rearrange I = Prt to solve for r

Here is one with three things multiplied together. You invest $2,000 (the principal P) for 3 years (t) and earn $300 in simple interest (I). What was the annual interest rate r? The simple interest formula I = Prt is on your sheet.

The variable r is multiplied by both P and t. So to free r, we divide both sides by the whole product Pt at once.

• Start: I = Prt. The r is multiplied by P and by t.
• Undo the multiply: divide both sides by Pt, giving r = I ÷ (Pt).
• Work out the bottom: Pt = 2000 × 3 = 6000.
• Solve: r = 300 ÷ 6000 = 0.05, which is 5%.

  I = P r t
  I / (P t) = r          (divide both sides by P t)
  r = I / (P t)
  r = 300 / (2000 × 3)   (substitute the values)
  r = 300 / 6000
  r = 0.05  →  5%

Tips that make formula questions easy

• Rearrange first, then substitute. Get the letter you want alone before you plug in numbers. It keeps the algebra clean and the arithmetic simple.
• Ask "what is happening to my variable?" If it is multiplied, divide. If something is added, subtract. Do the opposite to both sides.
• Keep both sides balanced. Whatever you do to one side, do to the other. That is the rule that never changes.
• Plug your answer back in. A quick check against the original formula catches almost every careless error.
• Mind the units. Area is square units, distance pairs with a rate and a time, and a rate often needs a final × 100 to become a percent.

Your turn: practice problems

Use the formulas from the sheet. Rearrange where you need to, then substitute. Write out every step before you check yourself.

1. Using A = lw, a rectangle has area 72 m² and width 9 m. Find the length l.
2. Using d = rt, a runner covers 21 km in 3 hours. Find the rate r.
3. Using I = Prt, find the interest I on a $1,500 loan at r = 0.04 for t = 2 years.
4. Using I = Prt, you earned I = $90 on a principal of P = $1,200 at a rate of r = 0.05. Find the time t.