Exercises: Rearrange formulas - Common Core Math - HS Algebra

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

Solve $2x + 6 = 14$ for $x$ .

A.

$x = 10$

B.

$x = 4$

C.

$x = 7$

D.

$x = 3$

2.

The equation $ax = b$ is solved for $x$ . Which expression gives $x$ ?

A.

$x = b - a$

B.

$x = a - b$

C.

$x = \frac{b}{a}$

D.

$x = ab$

3.

In the formula $A = \frac{1}{2}bh$ , you want to solve for $h$ . Which statement best describes how to treat $b$ ?

A.

Set $b = 0$ so it disappears from the equation.

B.

Replace $b$ with a number before rearranging.

C.

Treat $b$ as a constant — apply inverse operations around it, just as you would with a number.

D.

Ignore $b$ because the goal is to isolate $h$ .

B

Fluency Practice

Rearrange each formula to solve for the indicated variable. Show your steps.

1.

Ohm's Law: $V = IR$ . Rearrange to solve for $R$ . Then find $R$ when $V = 24$ volts and $I = 3$ amps. Give your answer in ohms.

2.

Perimeter of a rectangle: $P = 2l + 2w$ . Rearrange to solve for $w$ . Then find $w$ when $P = 36$ cm and $l = 11$ cm. Give your answer in centimetres.

3.

Distance formula: $d = rt$ . Rearrange to solve for $t$ . Then find $t$ when $d = 180$ miles and $r = 60$ miles per hour. Give your answer in hours.

$Side-by-side comparison showing incomplete rearrangement (stopping at r = A/pi) versus correct rearrangement (taking square root to get r = sqrt(A/pi)).$

4.

Area of a circle: $A = \pi r^2$ . Rearrange to solve for $r$ (take the positive root). Then find $r$ when $A = 100\pi$ cm². Give your answer in centimetres.

5.

Simple interest total: $A = P + Prt$ , where $P$ is the principal, $r$ is the annual rate, and $t$ is time in years. Which expression correctly gives $P$ in terms of $A$ , $r$ , and $t$ ?

A.

$P = \frac{A}{1 + rt}$

B.

$P = A - rt$

C.

$P = \frac{A}{1} + rt$

D.

$P = \frac{A - P}{rt}$

C

Varied Practice

These problems present formula rearrangement in different ways.

D

Application Problems

For each problem, identify the formula, rearrange it to solve for the unknown, then substitute and compute. Include units in your final answer.

1.

A car travels 270 miles in 4.5 hours at a constant speed.

Use the formula $d = rt$ to find the car's average speed $r$ in miles per hour.

2.

An investment earns $90 in simple interest over 3 years. The principal is $500.

Use the formula $I = Prt$ to find the annual interest rate $r$ . Express your answer as a decimal.

$A cylinder with radius 5 cm and unknown height h.$

3.

A cylindrical water tank has a volume of $500\pi$ cubic centimetres and a radius of 5 cm.

1.

Use the formula $V = \pi r^2 h$ . Rearrange the formula to solve for $h$ . Write the rearranged formula.

Enter the coefficient of $V$ in the rearranged formula $h = \frac{V}{\square}$ (give the expression in the denominator as a plain expression, e.g., "pi r^2").

2.

Using $h = \frac{V}{\pi r^2}$ , find the height $h$ in centimetres when $V = 500\pi$ cm³ and $r = 5$ cm.

E

Error Analysis

Each problem shows a student's work that contains an error. Identify the mistake.

1.

Jordan solved $A = P + Prt$ for $P$ :

$A = P + Prt$
Divide both sides by $P$ : $\frac{A}{P} = 1 + rt$
Therefore $P = \frac{A}{1 + rt}$ ✓

Jordan concludes Step 3 is correct.

Is Jordan's work valid? If not, what is the error in Step 2?

A.

Step 2 is wrong — Jordan cannot divide both sides by $P$ here because $P$ has not been factored out of all terms first. Dividing $A$ by $P$ is not equivalent to the original equation.

B.

Step 2 is correct, but Step 3 should be $P = A - (1 + rt)$ instead.

C.

The error is in Step 1 — the formula $A = P + Prt$ is incorrect.

D.

Jordan's work is entirely correct.

2.

Sam solved $A = \pi r^2$ for $r$ :

$A = \pi r^2$
Divide both sides by $\pi$ : $r = \frac{A}{\pi}$
Answer: $r = \frac{A}{\pi}$

What error did Sam make?

A.

Sam should have multiplied both sides by $\pi$ , not divided.

B.

Sam divided correctly but forgot to take the square root of both sides. The correct result is $r = \sqrt{\frac{A}{\pi}}$ .

C.

Sam made no error — $r = \frac{A}{\pi}$ is correct.

D.

Sam should have subtracted $\pi$ from both sides in Step 2.

F

Challenge / Extension

These problems go beyond the core skill. Try them if you finish early or want an extra challenge.

1.

Kinematics formula: $v^2 = u^2 + 2as$ , where $v$ is final velocity, $u$ is initial velocity, $a$ is acceleration, and $s$ is distance.

Rearrange to solve for $u$ (take the non-negative root). Then find $u$ when $v = 10$ m/s, $a = 3$ m/s², and $s = 8$ m.

Give your answer in m/s (exact value, or simplify if possible).

2.