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Algebra Quiz

Algebra Quiz: Rearranging Formulas To Highlight Quantities

Practice Rearranging Formulas To Highlight Quantities in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 20

0 of 20 answered

The line equation $ax + by = c$ has constants $a$ , $b$ , and $c$ . Solve for $y$ in terms of $x$ , $a$ , $b$ , and $c$ .

What this quiz covers

This quiz focuses on Rearranging Formulas To Highlight Quantities, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The line equation $ax + by = c$ has constants $a$ , $b$ , and $c$ . Solve for $y$ in terms of $x$ , $a$ , $b$ , and $c$ .

$y = \dfrac{c - ax}{b}$ (correct answer)
$y = \dfrac{ax - c}{b}$
$y = \dfrac{b}{c - ax}$
$by = c - x$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with ax + by = c, we first subtract ax from both sides: ax + by - ax = c - ax, which gives by = c - ax. Then divide both sides by b: by/b = (c - ax)/b, which simplifies to y = (c - ax)/b. Choice A is correct because it properly isolates y using subtraction of ax followed by division by b, giving y = (c - ax)/b. Perfect! Choice B has the wrong sign—it shows ax - c instead of c - ax in the numerator, which would happen if we incorrectly subtracted c from both sides instead of ax. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal!

Question 2

In physics, Newton’s second law is $F = ma$ , where $F$ is force, $m$ is mass, and $a$ is acceleration. Solve for $m$ in terms of $F$ and $a$ .

$m = Fa$
$a = \dfrac{m}{F}$
$m = \dfrac{F}{a}$ (correct answer)
$m = \dfrac{a}{F}$

Question 3

Temperature can be converted using $C = \dfrac{5}{9}(F - 32)$ , where $C$ is degrees Celsius and $F$ is degrees Fahrenheit. Solve the formula for $F$ in terms of $C$ .

$F = \dfrac{9}{5}(C - 32)$
$F = 32 - \dfrac{9}{5}C$
$F = \dfrac{5}{9}C - 32$
$F = \dfrac{9}{5}C + 32$ (correct answer)

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with C = (5/9)(F - 32), we first multiply both sides by 9/5: (9/5)C = F - 32, then add 32 to both sides: (9/5)C + 32 = F, or F = (9/5)C + 32. Choice B is correct because it properly isolates F using multiplication by 9/5 and addition of 32, giving F = (9/5)C + 32. Perfect! Choice A incorrectly shows F = (5/9)C - 32, which mixes up the fraction and subtraction—this would give a much colder temperature than intended. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For temperature conversion, remember that multiplying by 9/5 makes numbers bigger (Fahrenheit uses a wider scale), while 5/9 makes them smaller.

Question 4

The perimeter of a rectangle is given by $P = 2l + 2w$ , where $P$ is perimeter, $l$ is length, and $w$ is width. Solve for $l$ in terms of $P$ and $w$ .

$l = \dfrac{P}{2} - 2w$
$l = \dfrac{P - 2w}{2}$ (correct answer)
$l = \dfrac{P + 2w}{2}$
$l = \dfrac{2w - P}{2}$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with P = 2l + 2w, we first subtract 2w from both sides: P - 2w = 2l, then divide both sides by 2: (P - 2w)/2 = l, or l = (P - 2w)/2. Choice A is correct because it properly isolates l using subtraction of 2w and division by 2, giving l = (P - 2w)/2. Perfect! Choice B incorrectly shows l = P/2 - 2w, which would mean l = P/2 - 2w, but if we substitute back: P = 2(P/2 - 2w) + 2w = P - 4w + 2w = P - 2w, which doesn't equal P. Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 5

Temperature conversion can be written as the literal equation $C = \frac{5}{9}(F - 32)$ , where $C$ is degrees Celsius and $F$ is degrees Fahrenheit. Solve $C = \frac{5}{9}(F - 32)$ for $F$ .

$F = \dfrac{9}{5}(C - 32)$
$F = 32 - \dfrac{9}{5}C$
$F = \dfrac{5}{9}C + 32$
$F = \dfrac{9}{5}C + 32$ (correct answer)

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! To solve C = (5/9)(F - 32) for F, we first multiply both sides by 9/5 to clear the fraction: (9/5)C = F - 32. Then we add 32 to both sides: (9/5)C + 32 = F, or F = (9/5)C + 32. Choice C is correct because it properly isolates F using multiplication by 9/5 and addition of 32, giving F = (9/5)C + 32. Perfect! Choice A incorrectly keeps the 5/9 fraction, while choice B has the wrong order of operations—we need to multiply C by 9/5 first, then add 32, not subtract 32 from C before multiplying. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving C = (5/9)(F - 32) for F is like solving 20 = (5/9)(x - 32) for x: multiply by 9/5, then add 32. Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 6

In physics, distance is modeled by $d = rt$ , where $d$ is distance, $r$ is rate (speed), and $t$ is time. Solve $d = rt$ for $t$ .

$d = \dfrac{r}{t}$
$t = \dfrac{d}{r}$ (correct answer)
$t = \dfrac{r}{d}$
$t = dr$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! To solve d = rt for t, we need to isolate t on one side. Since t is being multiplied by r, we do the opposite operation—divide both sides by r: d/r = rt/r, which simplifies to d/r = t, or t = d/r. Choice A is correct because it properly isolates t using division by r, giving t = d/r. Perfect! Choice B incorrectly shows t = dr (multiplying instead of dividing), while choice C has the fraction flipped as t = r/d—remember, we divide distance by rate to get time, not the other way around. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving d = rt for t is just like solving 20 = 5x for x: divide both sides by r (or 5), giving t = d/r. Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 7

In the rectangle area formula $A = lw$ (where $A$ is area, $l$ is length, and $w$ is width), solve for $w$ . Treat the other variables like numbers and use inverse operations as you would in a numeric equation.

$w = \dfrac{l}{A}$
$w = \dfrac{A}{l}$ (correct answer)
$A = \dfrac{w}{l}$
$w = Al$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with A = lw, we want to isolate w, so we divide both sides by l: A/l = lw/l, which simplifies to A/l = w, or w = A/l. Choice B is correct because it properly isolates w using division by l, giving w = A/l. Perfect! Choice A incorrectly shows w = l/A, which would mean width equals length divided by area—this reverses the fraction and doesn't match our algebraic steps. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving A = lw for w is just like solving 12 = 3x for x: divide both sides by 3 (or l), giving x = 12/3 (or w = A/l).

Question 8

Simple interest is modeled by $I = Prt$ (where $I$ is interest, $P$ is principal, $r$ is annual rate, and $t$ is time). Rearrange to solve for $r$ .

$r = \dfrac{I}{Pt}$ (correct answer)
$r = \dfrac{Pt}{I}$
$r = IPt$
$r = \dfrac{I}{P} - t$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with I = Prt, we want to isolate r, so we divide both sides by Pt (the product of P and t): I/(Pt) = Prt/(Pt), which simplifies to I/(Pt) = r, or r = I/(Pt). Choice A is correct because it properly isolates r using division by the product Pt, giving r = I/(Pt). Perfect! Choice B incorrectly shows r = Pt/I, which inverts the fraction—this would mean rate equals principal times time divided by interest, which doesn't match our algebraic steps. When checking your work, substitute back: if you rearranged I = Prt to get r = I/(Pt), multiply both sides by Pt: Pt · r = Pt · [I/(Pt)] = I, which gives Prt = I—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 9

Temperature conversion can be written as $C = \dfrac{5}{9}(F - 32)$ , where $C$ is degrees Celsius and $F$ is degrees Fahrenheit. Solve for $F$ in terms of $C$ .

$F = \dfrac{5}{9}C + 32$
$F = \dfrac{9}{5}(C - 32)$
$F = \dfrac{9}{5}C + 32$ (correct answer)
$F = 32 - \dfrac{9}{5}C$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with C = (5/9)(F - 32), we first multiply both sides by 9/5: (9/5)C = (9/5) · (5/9)(F - 32), which simplifies to (9/5)C = F - 32. Then add 32 to both sides: (9/5)C + 32 = F - 32 + 32, which gives (9/5)C + 32 = F, or F = (9/5)C + 32. Choice C is correct because it properly isolates F using multiplication by 9/5 followed by addition of 32, giving F = (9/5)C + 32. Perfect! Choice B incorrectly applies the 9/5 to (C - 32) instead of just C—the parentheses placement matters! Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 10

Ohm’s law is $V = IR$ (where $V$ is voltage, $I$ is current, and $R$ is resistance). Rearrange the formula to solve for $R$ .

$R = \dfrac{I}{V}$
$R = VI$
$R = \dfrac{V}{I}$ (correct answer)
$V = \dfrac{R}{I}$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with V = IR, we want to isolate R, so we divide both sides by I: V/I = IR/I, which simplifies to V/I = R, or R = V/I. Choice C is correct because it properly isolates R using division by I, giving R = V/I. Perfect! Choice A incorrectly shows R = I/V, which reverses the fraction—this would mean resistance equals current divided by voltage, which doesn't match our algebraic steps or the physics (higher voltage with same current means higher resistance, not lower). The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving V = IR for R is just like solving 12 = 3x for x: divide both sides by 3 (or I), giving x = 12/3 (or R = V/I).

Question 11

The perimeter of a rectangle is given by $P = 2l + 2w$ , where $P$ is perimeter, $l$ is length, and $w$ is width. Solve $P = 2l + 2w$ for $l$ .

$l = \dfrac{P - 2w}{2}$ (correct answer)
$l = \dfrac{P}{2} - 2w$
$l = \dfrac{P - w}{2}$
$l = \dfrac{2w - P}{2}$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! To solve P = 2l + 2w for l, we first subtract 2w from both sides: P - 2w = 2l. Then we divide both sides by 2: (P - 2w)/2 = l, or l = (P - 2w)/2. Choice A is correct because it properly isolates l using subtraction of 2w and division by 2, giving l = (P - 2w)/2. Perfect! Choice B incorrectly divides P by 2 before subtracting 2w, while choice C forgets to multiply w by 2 when subtracting—remember, we need to subtract the entire term 2w, not just w. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving P = 2l + 2w for l is just like solving 20 = 2x + 6 for x: subtract 6 (or 2w), then divide by 2. When checking your work, substitute back: if you rearranged P = 2l + 2w to get l = (P - 2w)/2, multiply your answer by 2 and add 2w: 2 · [(P - 2w)/2] + 2w = P - 2w + 2w = P—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 12

The perimeter of a rectangle is given by $P = 2l + 2w$ . Solve this literal equation for $l$ (length) in terms of $P$ and $w$ .

$l = \dfrac{P - w}{2}$
$l = \dfrac{P + 2w}{2}$
$l = \dfrac{P - 2w}{2}$ (correct answer)
$l = P - 2w$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with P = 2l + 2w, we first subtract 2w from both sides: P - 2w = 2l + 2w - 2w, which gives P - 2w = 2l. Then we divide both sides by 2: (P - 2w)/2 = 2l/2, which simplifies to l = (P - 2w)/2. Choice C is correct because it properly isolates l using subtraction of 2w followed by division by 2, giving l = (P - 2w)/2. Perfect! Choice A incorrectly shows l = (P - w)/2, forgetting that we need to subtract 2w (both width terms), not just w—this is a common error when dealing with coefficients. When checking your work, substitute back: if you rearranged P = 2l + 2w to get l = (P - 2w)/2, multiply both sides by 2 and add 2w: 2l + 2w = 2[(P - 2w)/2] + 2w = P - 2w + 2w = P—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 13

The average (mean) of $n$ numbers is given by $\bar{x} = \dfrac{S}{n}$ , where $S$ is the sum of the numbers. Solve for $S$ in terms of $\bar{x}$ and $n$ .

$S = \dfrac{\bar{x}}{n}$
$S = \bar{x}n$ (correct answer)
$S = \dfrac{n}{\bar{x}}$
$\bar{x} = Sn$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with x̄ = S/n (where x̄ is the average), we want to isolate S, so we multiply both sides by n: n · x̄ = n · (S/n), which simplifies to nx̄ = S, or S = x̄n. Choice B is correct because it properly isolates S using multiplication by n, giving S = x̄n. Perfect! Choice A incorrectly shows S = x̄/n, which would mean the sum equals the average divided by the count—this doesn't make sense because sums are larger than averages when we have multiple values. When checking your work, substitute back: if you rearranged x̄ = S/n to get S = x̄n, divide both sides by n: S/n = (x̄n)/n = x̄—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 14

The volume of a cylinder is $V = \pi r^2 h$ , where $V$ is volume, $r$ is radius, and $h$ is height. Solve for $h$ in terms of $V$ and $r$ .

$h = \dfrac{V}{\pi r^2}$ (correct answer)
$h = \dfrac{\pi r^2}{V}$
$h = \dfrac{V}{\pi r}$
$h = \pi r^2 V$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with V = πr²h, we want to isolate h, so we divide both sides by πr²: V/(πr²) = πr²h/(πr²), which simplifies to V/(πr²) = h, or h = V/(πr²). Choice A is correct because it properly isolates h using division by πr², giving h = V/(πr²). Perfect! Choice C incorrectly shows h = V/(πr), forgetting to square the r—this would give volume V = πr · V/(πr) = V, which seems right but doesn't include the r² term we need. Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); V = πr²h becomes h = V/(πr²) (divide by base area). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 15

Rearrange the formula $y = \dfrac{x - a}{b}$ (assume $b \ne 0$ ) to solve for $x$ in terms of $y$ , $a$ , and $b$ .

$x = by - a$
$x = \dfrac{y - a}{b}$
$x = by + a$ (correct answer)
$x = \dfrac{b}{y} + a$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with y = (x - a)/b, we first multiply both sides by b: by = b · (x - a)/b, which simplifies to by = x - a. Then we add a to both sides: by + a = x - a + a, which gives us x = by + a. Choice C is correct because it properly isolates x using multiplication by b followed by addition of a, giving x = by + a. Perfect! Choice B incorrectly shows x = (y - a)/b, which seems to just swap x and y in the original formula without proper algebraic steps—remember, we need to undo the operations around x systematically. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. When you see y = (x - a)/b, think of solving 3 = (x - 2)/5: multiply by 5 to get 15 = x - 2, then add 2 to get x = 17, just like we get x = by + a!

Question 16

In finance, simple interest is modeled by $I = Prt$ , where $I$ is interest, $P$ is principal, $r$ is annual interest rate, and $t$ is time. Rearrange the formula to solve for $r$ (in terms of $I$ , $P$ , and $t$ ).

$r = \dfrac{IP}{t}$
$r = \dfrac{I}{Pt}$ (correct answer)
$r = \dfrac{Pt}{I}$
$r = I - Pt$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! For I = P r t, isolate r by dividing both sides by (P t), since r is multiplied by both P and t, giving r = I / (P t). Choice B is correct because it properly isolates r using division by the product P t, giving r = I / (P t). Perfect! Choice C flips the fraction, but remember, to undo multiplication by P t, we divide I by P t—it's easy to mix up, but verifying with numbers helps! The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving V = I R for R is just like solving 12 = 3x for x: divide both sides by I, giving R = V / I. When checking your work, substitute back: plug in values like I=10, P=100, t=1, r=0.05 into original (I=5) and your formula to confirm it matches.

Question 17

In physics, distance traveled is modeled by $d = rt$ , where $d$ is distance, $r$ is speed (rate), and $t$ is time. Solve for $t$ in terms of $d$ and $r$ using the same steps you would use to isolate a variable in a numeric equation.

$t = d - r$
$t = \dfrac{r}{d}$
$t = dr$
$t = \dfrac{d}{r}$ (correct answer)

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with d = r t, to isolate t, divide both sides by r (treating d and r like constants), resulting in t = d / r. Choice C is correct because it properly isolates t using division, giving t = d / r. Perfect! Something like choice A multiplies instead, but that's the opposite of what we need—since r and t are multiplied, division is the inverse to undo it, so double-check those operations! Common formula rearrangements to practice: d = r t becomes t = d / r (divide by rate) and r = d / t (divide by time); A = l w becomes l = A / w (divide by width). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times! When checking your work, substitute back: if you rearranged d = r t to get t = d / r, multiply both sides by r: r · t = r · (d / r) = d, matching the original.

Question 18

Ohm's law is $V = IR$ , where $V$ is voltage, $I$ is current, and $R$ is resistance. Rearrange the formula to solve for $I$ in terms of $V$ and $R$ .

$I = \dfrac{R}{V}$
$I = VR$
$I = \dfrac{V}{R}$ (correct answer)
$R = \dfrac{V}{I}$

Question 19

Temperature conversion is given by $C = \dfrac{5}{9}(F - 32)$ , where $C$ is degrees Celsius and $F$ is degrees Fahrenheit. Solve for $F$ in terms of $C$ .

$F = \dfrac{5}{9}C + 32$
$F = \dfrac{9}{5}C - 32$
$F = \dfrac{9}{5}C + 32$ (correct answer)
$F = \dfrac{5}{9}(C - 32)$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with C = (5/9)(F - 32), we first multiply both sides by 9/5: (9/5)C = (9/5) · (5/9)(F - 32), which simplifies to (9/5)C = F - 32. Then we add 32 to both sides: (9/5)C + 32 = F - 32 + 32, giving us F = (9/5)C + 32. Choice C is correct because it properly isolates F using multiplication by 9/5 followed by addition of 32, giving F = (9/5)C + 32. Perfect! Choice A incorrectly shows F = (5/9)C + 32, keeping the original fraction 5/9 instead of using its reciprocal 9/5—when we multiply both sides by 9/5, we're undoing the original multiplication by 5/9. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For temperature conversion, think of it as solving 20 = (5/9)(x - 32) for x: multiply by 9/5, then add 32.

Question 20

In geometry, the volume of a rectangular prism is $V = lwh$ , where $V$ is volume, $l$ is length, $w$ is width, and $h$ is height. Solve $V = lwh$ for $h$ .

$h = \dfrac{V}{lw}$ (correct answer)
$h = Vlw$
$h = \dfrac{lw}{V}$
$V = \dfrac{h}{lw}$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! To solve V = lwh for h, we need to isolate h on one side. Since h is being multiplied by both l and w (giving lwh), we divide both sides by lw: V/(lw) = lwh/(lw), which simplifies to V/(lw) = h, or h = V/(lw). Choice A is correct because it properly isolates h using division by lw, giving h = V/(lw). Perfect! Choice B incorrectly shows h = lw/V (the reciprocal), while choice C multiplies all the variables together—remember, we need to divide volume by length times width to get height. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving V = lwh for h is just like solving 60 = 12x for x: divide both sides by lw (or 12), giving h = V/(lw). Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

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Algebra Quiz

Algebra Quiz: Rearranging Formulas To Highlight Quantities

Practice Rearranging Formulas To Highlight Quantities in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 20

0 of 20 answered

The line equation $ax + by = c$ has constants $a$ , $b$ , and $c$ . Solve for $y$ in terms of $x$ , $a$ , $b$ , and $c$ .

What this quiz covers

This quiz focuses on Rearranging Formulas To Highlight Quantities, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The line equation $ax + by = c$ has constants $a$ , $b$ , and $c$ . Solve for $y$ in terms of $x$ , $a$ , $b$ , and $c$ .

$y = \dfrac{c - ax}{b}$ (correct answer)
$y = \dfrac{ax - c}{b}$
$y = \dfrac{b}{c - ax}$
$by = c - x$

Question 2

In physics, Newton’s second law is $F = ma$ , where $F$ is force, $m$ is mass, and $a$ is acceleration. Solve for $m$ in terms of $F$ and $a$ .

$m = Fa$
$a = \dfrac{m}{F}$
$m = \dfrac{F}{a}$ (correct answer)
$m = \dfrac{a}{F}$

Question 3

Temperature can be converted using $C = \dfrac{5}{9}(F - 32)$ , where $C$ is degrees Celsius and $F$ is degrees Fahrenheit. Solve the formula for $F$ in terms of $C$ .

$F = \dfrac{9}{5}(C - 32)$
$F = 32 - \dfrac{9}{5}C$
$F = \dfrac{5}{9}C - 32$
$F = \dfrac{9}{5}C + 32$ (correct answer)

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with C = (5/9)(F - 32), we first multiply both sides by 9/5: (9/5)C = F - 32, then add 32 to both sides: (9/5)C + 32 = F, or F = (9/5)C + 32. Choice B is correct because it properly isolates F using multiplication by 9/5 and addition of 32, giving F = (9/5)C + 32. Perfect! Choice A incorrectly shows F = (5/9)C - 32, which mixes up the fraction and subtraction—this would give a much colder temperature than intended. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For temperature conversion, remember that multiplying by 9/5 makes numbers bigger (Fahrenheit uses a wider scale), while 5/9 makes them smaller.

Question 4

The perimeter of a rectangle is given by $P = 2l + 2w$ , where $P$ is perimeter, $l$ is length, and $w$ is width. Solve for $l$ in terms of $P$ and $w$ .

$l = \dfrac{P}{2} - 2w$
$l = \dfrac{P - 2w}{2}$ (correct answer)
$l = \dfrac{P + 2w}{2}$
$l = \dfrac{2w - P}{2}$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with P = 2l + 2w, we first subtract 2w from both sides: P - 2w = 2l, then divide both sides by 2: (P - 2w)/2 = l, or l = (P - 2w)/2. Choice A is correct because it properly isolates l using subtraction of 2w and division by 2, giving l = (P - 2w)/2. Perfect! Choice B incorrectly shows l = P/2 - 2w, which would mean l = P/2 - 2w, but if we substitute back: P = 2(P/2 - 2w) + 2w = P - 4w + 2w = P - 2w, which doesn't equal P. Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 5

Temperature conversion can be written as the literal equation $C = \frac{5}{9}(F - 32)$ , where $C$ is degrees Celsius and $F$ is degrees Fahrenheit. Solve $C = \frac{5}{9}(F - 32)$ for $F$ .

$F = \dfrac{9}{5}(C - 32)$
$F = 32 - \dfrac{9}{5}C$
$F = \dfrac{5}{9}C + 32$
$F = \dfrac{9}{5}C + 32$ (correct answer)

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! To solve C = (5/9)(F - 32) for F, we first multiply both sides by 9/5 to clear the fraction: (9/5)C = F - 32. Then we add 32 to both sides: (9/5)C + 32 = F, or F = (9/5)C + 32. Choice C is correct because it properly isolates F using multiplication by 9/5 and addition of 32, giving F = (9/5)C + 32. Perfect! Choice A incorrectly keeps the 5/9 fraction, while choice B has the wrong order of operations—we need to multiply C by 9/5 first, then add 32, not subtract 32 from C before multiplying. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving C = (5/9)(F - 32) for F is like solving 20 = (5/9)(x - 32) for x: multiply by 9/5, then add 32. Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 6

In physics, distance is modeled by $d = rt$ , where $d$ is distance, $r$ is rate (speed), and $t$ is time. Solve $d = rt$ for $t$ .

$d = \dfrac{r}{t}$
$t = \dfrac{d}{r}$ (correct answer)
$t = \dfrac{r}{d}$
$t = dr$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! To solve d = rt for t, we need to isolate t on one side. Since t is being multiplied by r, we do the opposite operation—divide both sides by r: d/r = rt/r, which simplifies to d/r = t, or t = d/r. Choice A is correct because it properly isolates t using division by r, giving t = d/r. Perfect! Choice B incorrectly shows t = dr (multiplying instead of dividing), while choice C has the fraction flipped as t = r/d—remember, we divide distance by rate to get time, not the other way around. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving d = rt for t is just like solving 20 = 5x for x: divide both sides by r (or 5), giving t = d/r. Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 7

$w = \dfrac{l}{A}$
$w = \dfrac{A}{l}$ (correct answer)
$A = \dfrac{w}{l}$
$w = Al$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with A = lw, we want to isolate w, so we divide both sides by l: A/l = lw/l, which simplifies to A/l = w, or w = A/l. Choice B is correct because it properly isolates w using division by l, giving w = A/l. Perfect! Choice A incorrectly shows w = l/A, which would mean width equals length divided by area—this reverses the fraction and doesn't match our algebraic steps. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving A = lw for w is just like solving 12 = 3x for x: divide both sides by 3 (or l), giving x = 12/3 (or w = A/l).

Question 8

Simple interest is modeled by $I = Prt$ (where $I$ is interest, $P$ is principal, $r$ is annual rate, and $t$ is time). Rearrange to solve for $r$ .

$r = \dfrac{I}{Pt}$ (correct answer)
$r = \dfrac{Pt}{I}$
$r = IPt$
$r = \dfrac{I}{P} - t$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with I = Prt, we want to isolate r, so we divide both sides by Pt (the product of P and t): I/(Pt) = Prt/(Pt), which simplifies to I/(Pt) = r, or r = I/(Pt). Choice A is correct because it properly isolates r using division by the product Pt, giving r = I/(Pt). Perfect! Choice B incorrectly shows r = Pt/I, which inverts the fraction—this would mean rate equals principal times time divided by interest, which doesn't match our algebraic steps. When checking your work, substitute back: if you rearranged I = Prt to get r = I/(Pt), multiply both sides by Pt: Pt · r = Pt · [I/(Pt)] = I, which gives Prt = I—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 9

Temperature conversion can be written as $C = \dfrac{5}{9}(F - 32)$ , where $C$ is degrees Celsius and $F$ is degrees Fahrenheit. Solve for $F$ in terms of $C$ .

$F = \dfrac{5}{9}C + 32$
$F = \dfrac{9}{5}(C - 32)$
$F = \dfrac{9}{5}C + 32$ (correct answer)
$F = 32 - \dfrac{9}{5}C$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with C = (5/9)(F - 32), we first multiply both sides by 9/5: (9/5)C = (9/5) · (5/9)(F - 32), which simplifies to (9/5)C = F - 32. Then add 32 to both sides: (9/5)C + 32 = F - 32 + 32, which gives (9/5)C + 32 = F, or F = (9/5)C + 32. Choice C is correct because it properly isolates F using multiplication by 9/5 followed by addition of 32, giving F = (9/5)C + 32. Perfect! Choice B incorrectly applies the 9/5 to (C - 32) instead of just C—the parentheses placement matters! Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 10

Ohm’s law is $V = IR$ (where $V$ is voltage, $I$ is current, and $R$ is resistance). Rearrange the formula to solve for $R$ .

$R = \dfrac{I}{V}$
$R = VI$
$R = \dfrac{V}{I}$ (correct answer)
$V = \dfrac{R}{I}$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with V = IR, we want to isolate R, so we divide both sides by I: V/I = IR/I, which simplifies to V/I = R, or R = V/I. Choice C is correct because it properly isolates R using division by I, giving R = V/I. Perfect! Choice A incorrectly shows R = I/V, which reverses the fraction—this would mean resistance equals current divided by voltage, which doesn't match our algebraic steps or the physics (higher voltage with same current means higher resistance, not lower). The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving V = IR for R is just like solving 12 = 3x for x: divide both sides by 3 (or I), giving x = 12/3 (or R = V/I).

Question 11

The perimeter of a rectangle is given by $P = 2l + 2w$ , where $P$ is perimeter, $l$ is length, and $w$ is width. Solve $P = 2l + 2w$ for $l$ .

$l = \dfrac{P - 2w}{2}$ (correct answer)
$l = \dfrac{P}{2} - 2w$
$l = \dfrac{P - w}{2}$
$l = \dfrac{2w - P}{2}$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! To solve P = 2l + 2w for l, we first subtract 2w from both sides: P - 2w = 2l. Then we divide both sides by 2: (P - 2w)/2 = l, or l = (P - 2w)/2. Choice A is correct because it properly isolates l using subtraction of 2w and division by 2, giving l = (P - 2w)/2. Perfect! Choice B incorrectly divides P by 2 before subtracting 2w, while choice C forgets to multiply w by 2 when subtracting—remember, we need to subtract the entire term 2w, not just w. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving P = 2l + 2w for l is just like solving 20 = 2x + 6 for x: subtract 6 (or 2w), then divide by 2. When checking your work, substitute back: if you rearranged P = 2l + 2w to get l = (P - 2w)/2, multiply your answer by 2 and add 2w: 2 · [(P - 2w)/2] + 2w = P - 2w + 2w = P—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 12

The perimeter of a rectangle is given by $P = 2l + 2w$ . Solve this literal equation for $l$ (length) in terms of $P$ and $w$ .

$l = \dfrac{P - w}{2}$
$l = \dfrac{P + 2w}{2}$
$l = \dfrac{P - 2w}{2}$ (correct answer)
$l = P - 2w$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with P = 2l + 2w, we first subtract 2w from both sides: P - 2w = 2l + 2w - 2w, which gives P - 2w = 2l. Then we divide both sides by 2: (P - 2w)/2 = 2l/2, which simplifies to l = (P - 2w)/2. Choice C is correct because it properly isolates l using subtraction of 2w followed by division by 2, giving l = (P - 2w)/2. Perfect! Choice A incorrectly shows l = (P - w)/2, forgetting that we need to subtract 2w (both width terms), not just w—this is a common error when dealing with coefficients. When checking your work, substitute back: if you rearranged P = 2l + 2w to get l = (P - 2w)/2, multiply both sides by 2 and add 2w: 2l + 2w = 2[(P - 2w)/2] + 2w = P - 2w + 2w = P—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 13

The average (mean) of $n$ numbers is given by $\bar{x} = \dfrac{S}{n}$ , where $S$ is the sum of the numbers. Solve for $S$ in terms of $\bar{x}$ and $n$ .

$S = \dfrac{\bar{x}}{n}$
$S = \bar{x}n$ (correct answer)
$S = \dfrac{n}{\bar{x}}$
$\bar{x} = Sn$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with x̄ = S/n (where x̄ is the average), we want to isolate S, so we multiply both sides by n: n · x̄ = n · (S/n), which simplifies to nx̄ = S, or S = x̄n. Choice B is correct because it properly isolates S using multiplication by n, giving S = x̄n. Perfect! Choice A incorrectly shows S = x̄/n, which would mean the sum equals the average divided by the count—this doesn't make sense because sums are larger than averages when we have multiple values. When checking your work, substitute back: if you rearranged x̄ = S/n to get S = x̄n, divide both sides by n: S/n = (x̄n)/n = x̄—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 14

The volume of a cylinder is $V = \pi r^2 h$ , where $V$ is volume, $r$ is radius, and $h$ is height. Solve for $h$ in terms of $V$ and $r$ .

$h = \dfrac{V}{\pi r^2}$ (correct answer)
$h = \dfrac{\pi r^2}{V}$
$h = \dfrac{V}{\pi r}$
$h = \pi r^2 V$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with V = πr²h, we want to isolate h, so we divide both sides by πr²: V/(πr²) = πr²h/(πr²), which simplifies to V/(πr²) = h, or h = V/(πr²). Choice A is correct because it properly isolates h using division by πr², giving h = V/(πr²). Perfect! Choice C incorrectly shows h = V/(πr), forgetting to square the r—this would give volume V = πr · V/(πr) = V, which seems right but doesn't include the r² term we need. Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); V = πr²h becomes h = V/(πr²) (divide by base area). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 15

Rearrange the formula $y = \dfrac{x - a}{b}$ (assume $b \ne 0$ ) to solve for $x$ in terms of $y$ , $a$ , and $b$ .

$x = by - a$
$x = \dfrac{y - a}{b}$
$x = by + a$ (correct answer)
$x = \dfrac{b}{y} + a$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with y = (x - a)/b, we first multiply both sides by b: by = b · (x - a)/b, which simplifies to by = x - a. Then we add a to both sides: by + a = x - a + a, which gives us x = by + a. Choice C is correct because it properly isolates x using multiplication by b followed by addition of a, giving x = by + a. Perfect! Choice B incorrectly shows x = (y - a)/b, which seems to just swap x and y in the original formula without proper algebraic steps—remember, we need to undo the operations around x systematically. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. When you see y = (x - a)/b, think of solving 3 = (x - 2)/5: multiply by 5 to get 15 = x - 2, then add 2 to get x = 17, just like we get x = by + a!

Question 16

$r = \dfrac{IP}{t}$
$r = \dfrac{I}{Pt}$ (correct answer)
$r = \dfrac{Pt}{I}$
$r = I - Pt$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! For I = P r t, isolate r by dividing both sides by (P t), since r is multiplied by both P and t, giving r = I / (P t). Choice B is correct because it properly isolates r using division by the product P t, giving r = I / (P t). Perfect! Choice C flips the fraction, but remember, to undo multiplication by P t, we divide I by P t—it's easy to mix up, but verifying with numbers helps! The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving V = I R for R is just like solving 12 = 3x for x: divide both sides by I, giving R = V / I. When checking your work, substitute back: plug in values like I=10, P=100, t=1, r=0.05 into original (I=5) and your formula to confirm it matches.

Question 17

$t = d - r$
$t = \dfrac{r}{d}$
$t = dr$
$t = \dfrac{d}{r}$ (correct answer)

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with d = r t, to isolate t, divide both sides by r (treating d and r like constants), resulting in t = d / r. Choice C is correct because it properly isolates t using division, giving t = d / r. Perfect! Something like choice A multiplies instead, but that's the opposite of what we need—since r and t are multiplied, division is the inverse to undo it, so double-check those operations! Common formula rearrangements to practice: d = r t becomes t = d / r (divide by rate) and r = d / t (divide by time); A = l w becomes l = A / w (divide by width). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times! When checking your work, substitute back: if you rearranged d = r t to get t = d / r, multiply both sides by r: r · t = r · (d / r) = d, matching the original.

Question 18

Ohm's law is $V = IR$ , where $V$ is voltage, $I$ is current, and $R$ is resistance. Rearrange the formula to solve for $I$ in terms of $V$ and $R$ .

$I = \dfrac{R}{V}$
$I = VR$
$I = \dfrac{V}{R}$ (correct answer)
$R = \dfrac{V}{I}$

Question 19

Temperature conversion is given by $C = \dfrac{5}{9}(F - 32)$ , where $C$ is degrees Celsius and $F$ is degrees Fahrenheit. Solve for $F$ in terms of $C$ .

$F = \dfrac{5}{9}C + 32$
$F = \dfrac{9}{5}C - 32$
$F = \dfrac{9}{5}C + 32$ (correct answer)
$F = \dfrac{5}{9}(C - 32)$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! Starting with C = (5/9)(F - 32), we first multiply both sides by 9/5: (9/5)C = (9/5) · (5/9)(F - 32), which simplifies to (9/5)C = F - 32. Then we add 32 to both sides: (9/5)C + 32 = F - 32 + 32, giving us F = (9/5)C + 32. Choice C is correct because it properly isolates F using multiplication by 9/5 followed by addition of 32, giving F = (9/5)C + 32. Perfect! Choice A incorrectly shows F = (5/9)C + 32, keeping the original fraction 5/9 instead of using its reciprocal 9/5—when we multiply both sides by 9/5, we're undoing the original multiplication by 5/9. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For temperature conversion, think of it as solving 20 = (5/9)(x - 32) for x: multiply by 9/5, then add 32.

Question 20

In geometry, the volume of a rectangular prism is $V = lwh$ , where $V$ is volume, $l$ is length, $w$ is width, and $h$ is height. Solve $V = lwh$ for $h$ .

$h = \dfrac{V}{lw}$ (correct answer)
$h = Vlw$
$h = \dfrac{lw}{V}$
$V = \dfrac{h}{lw}$

Explanation: This question tests your ability to rearrange formulas with multiple variables—an essential skill for working with formulas in science, geometry, and real life. Rearranging formulas (sometimes called solving literal equations) works exactly like solving regular equations, with one difference: instead of finding a number, we're finding a formula that expresses one variable in terms of the others. The same algebraic moves apply—we just keep the variables as letters instead of substituting numbers! To solve V = lwh for h, we need to isolate h on one side. Since h is being multiplied by both l and w (giving lwh), we divide both sides by lw: V/(lw) = lwh/(lw), which simplifies to V/(lw) = h, or h = V/(lw). Choice A is correct because it properly isolates h using division by lw, giving h = V/(lw). Perfect! Choice B incorrectly shows h = lw/V (the reciprocal), while choice C multiplies all the variables together—remember, we need to divide volume by length times width to get height. The secret to rearranging formulas: pretend the variable you want to solve for is x (like in regular equations), and treat all the other variables like they're numbers. Use the same steps—add, subtract, multiply, divide, just like normal! For example, solving V = lwh for h is just like solving 60 = 12x for x: divide both sides by lw (or 12), giving h = V/(lw). Common formula rearrangements to practice: d = rt becomes t = d/r (divide by rate) and r = d/t (divide by time); A = lw becomes l = A/w (divide by width); P = 2l + 2w becomes l = (P - 2w)/2 (subtract 2w, divide by 2). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!