Algebra Quiz: Rearranging Formulas To Highlight Quantities
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Rearranging Formulas To Highlight QuantitiesQuestion 1 of 20

In physics, distance is modeled by d=rtd = rt, where dd is distance, rr is rate (speed), and tt is time. Solve d=rtd = rt for tt.

d=rtd = \dfrac{r}{t}
t=drt = \dfrac{d}{r}
t=rdt = \dfrac{r}{d}
t=drt = dr
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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.

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

In physics, distance is modeled by d=rtd = rt, where dd is distance, rr is rate (speed), and tt is time. Solve d=rtd = rt for tt.

  1. d=rtd = \dfrac{r}{t}
  2. t=drt = \dfrac{d}{r} (correct answer)
  3. t=rdt = \dfrac{r}{d}
  4. t=drt = 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 2

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

  1. w=lAw = \dfrac{l}{A}
  2. w=Alw = \dfrac{A}{l} (correct answer)
  3. A=wlA = \dfrac{w}{l}
  4. w=Alw = 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 3

In finance, simple interest is modeled by I=PrtI = Prt, where II is interest, PP is principal, rr is annual interest rate, and tt is time. Rearrange the formula to solve for rr (in terms of II, PP, and tt).

  1. r=IPtr = \dfrac{IP}{t}
  2. r=IPtr = \dfrac{I}{Pt} (correct answer)
  3. r=PtIr = \dfrac{Pt}{I}
  4. r=I−Ptr = 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 4

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

  1. t=d−rt = d - r
  2. t=rdt = \dfrac{r}{d}
  3. t=drt = dr
  4. t=drt = \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 5

Temperature conversion is given by C=59(F−32)C = \dfrac{5}{9}(F - 32), where CC is degrees Celsius and FF is degrees Fahrenheit. Solve for FF in terms of CC.

  1. F=59C+32F = \dfrac{5}{9}C + 32
  2. F=95C−32F = \dfrac{9}{5}C - 32
  3. F=95C+32F = \dfrac{9}{5}C + 32 (correct answer)
  4. F=59(C−32)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 6

Solve for yy in the literal equation ax+by=cax + by = c (treat aa, bb, and cc as constants).

  1. y=c−axby = \dfrac{c - ax}{b} (correct answer)
  2. y=cab−xy = \dfrac{c}{ab} - x
  3. by=c−aby = c - a
  4. y=ax−cby = \dfrac{ax - c}{b}
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 ax + by = c for y, we first subtract ax from both sides: by = c - ax. Then we divide both sides by b: y = (c - ax)/b. Choice A is correct because it properly isolates y using subtraction of ax and division by b, giving y = (c - ax)/b. Perfect! Choice B incorrectly shows y = (ax - c)/b, which has the wrong sign—we subtract ax from c, not c from ax. Choice C tries to separate the fraction incorrectly, forgetting that we need to divide the entire expression (c - ax) by b. 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 ax + by = c for y is like solving 3 + 2y = 11 for y: subtract 3 (or ax), then divide by 2 (or b). 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

The line equation ax+by=cax + by = c has constants aa, bb, and cc. Solve for yy in terms of xx, aa, bb, and cc.

  1. y=c−axby = \dfrac{c - ax}{b} (correct answer)
  2. y=ax−cby = \dfrac{ax - c}{b}
  3. y=bc−axy = \dfrac{b}{c - ax}
  4. by=c−xby = 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 8

Temperature conversion can be written as the literal equation C=59(F−32)C = \frac{5}{9}(F - 32), where CC is degrees Celsius and FF is degrees Fahrenheit. Solve C=59(F−32)C = \frac{5}{9}(F - 32) for FF.

  1. F=95(C−32)F = \dfrac{9}{5}(C - 32)
  2. F=32−95CF = 32 - \dfrac{9}{5}C
  3. F=59C+32F = \dfrac{5}{9}C + 32
  4. F=95C+32F = \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 9

In physics, Newton's second law is F=maF = ma, where FF is force, mm is mass, and aa is acceleration. Solve for mm in terms of FF and aa.

  1. m=Fam = Fa
  2. a=mFa = \dfrac{m}{F}
  3. m=Fam = \dfrac{F}{a} (correct answer)
  4. m=aFm = \dfrac{a}{F}
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 F = ma, we want to isolate m, so we divide both sides by a: F/a = ma/a, which simplifies to F/a = m, or m = F/a. Choice C is correct because it properly isolates m using division by a, giving m = F/a. Perfect! Choice A incorrectly multiplies F by a—remember, to undo multiplication by a, we divide by a, not multiply more. When checking your work, substitute back: if you rearranged F = ma to get m = F/a, multiply both sides of your answer by a: a · m = a · (F/a) = F, which gives ma = F—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 10

In algebra, consider the literal equation ax+by=cax + by = c, where aa, bb, and cc are constants. Solve for yy in terms of xx, aa, bb, and cc using the same steps as solving a numeric equation.

  1. y=bc−axy = \dfrac{b}{c - ax}
  2. y=c−axby = \dfrac{c - ax}{b} (correct answer)
  3. y=ax−cby = \dfrac{ax - c}{b}
  4. y=c−ax−by = c - ax - b
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 x + b y = c, subtract a x from both sides to get b y = c - a x, then divide by b to isolate y, giving y = (c - a x)/b. Choice A is correct because it properly isolates y using subtraction and division, giving y = (c - a x)/b. Perfect! Choice B switches the signs inside, but subtracting a x correctly moves it to the other side positively—it's tricky, but treating it like numbers clarifies it! 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! When checking your work, substitute back: multiply your y expression by b and add a x to see if you get c.

Question 11

Simple interest is modeled by I=PrtI = Prt (where II is interest, PP is principal, rr is annual rate, and tt is time). Rearrange to solve for rr.

  1. r=IPtr = \dfrac{I}{Pt} (correct answer)
  2. r=PtIr = \dfrac{Pt}{I}
  3. r=IPtr = IPt
  4. r=IP−tr = \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 12

In the distance formula d=rtd = rt (where dd is distance, rr is rate, and tt is time), solve for tt in terms of dd and rr.

  1. t=drt = dr
  2. t=rdt = \dfrac{r}{d}
  3. r=dtr = \dfrac{d}{t}
  4. t=drt = \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 = rt, we want to isolate t, so we divide both sides by r: d/r = rt/r, which simplifies to d/r = t, or t = d/r. Choice C is correct because it properly isolates t using division by r, giving t = d/r. Perfect! Choice B incorrectly shows t = dr, which would mean distance equals rate times dr—but that gives d = r(dr) = dr², not our original formula. 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 13

In physics, kinetic energy is given by K=12mv2K = \dfrac{1}{2}mv^2, where KK is kinetic energy, mm is mass, and vv is speed. Solve for mm in terms of KK and vv.

  1. m=v22Km = \dfrac{v^2}{2K}
  2. m=2Kv2m = 2Kv^2
  3. m=2Kv2m = \dfrac{2K}{v^2} (correct answer)
  4. m=K2v2m = \dfrac{K}{2v^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 K = (1/2)mv², we first multiply both sides by 2: 2K = mv², then divide both sides by v²: 2K/v² = m, or m = 2K/v². Choice B is correct because it properly isolates m using multiplication by 2 and division by v², giving m = 2K/v². Perfect! Choice A incorrectly shows m = K/(2v²), which forgets to multiply by 2 first—this would give only half the correct mass. When checking your work, substitute back: if you rearranged K = (1/2)mv² to get m = 2K/v², multiply your answer by (1/2)v²: (1/2)v² · m = (1/2)v² · (2K/v²) = K—same as the original! This 'does it work backward?' check confirms you rearranged correctly.

Question 14

A discounted price can be modeled by S=P(1−r)S = P(1 - r), where PP is the original price, rr is the discount rate (as a decimal), and SS is the sale price. Solve for rr in terms of PP and SS.

  1. r=1−PSr = 1 - \dfrac{P}{S}
  2. r=SP−1r = \dfrac{S}{P} - 1
  3. r=1−SPr = 1 - \dfrac{S}{P} (correct answer)
  4. r=PS−1r = \dfrac{P}{S} - 1
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! Start with S = P (1 - r); divide both sides by P to get S / P = 1 - r, then subtract 1 from both sides (or rearrange) to isolate r as r = 1 - S / P. Choice C is correct because it properly isolates r using division and subtraction, giving r = 1 - S / P. Perfect! Choice D flips the fraction and subtracts 1 incorrectly, but since we divide S by P and subtract from 1, the order matters—great effort, and plugging in numbers like P=100, S=80, r=0.2 checks it! When checking your work, substitute back: if r = 1 - S / P, plug into original: P (1 - (1 - S / P)) = P (S / P) = S, perfect match. Common formula rearrangements to practice: d = r t becomes t = d / r (divide by rate) and r = d / t (divide by time). The same formulas show up repeatedly in math and science, so learning these rearrangements once helps you many times!

Question 15

In geometry, the volume of a rectangular prism is V=lwhV = lwh, where VV is volume, ll is length, ww is width, and hh is height. Solve V=lwhV = lwh for hh.

  1. h=Vlwh = \dfrac{V}{lw} (correct answer)
  2. h=Vlwh = Vlw
  3. h=lwVh = \dfrac{lw}{V}
  4. V=hlwV = \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!

Question 16

Temperature conversion can be written as C=59(F−32)C = \dfrac{5}{9}(F - 32), where CC is degrees Celsius and FF is degrees Fahrenheit. Solve for FF in terms of CC.

  1. F=59C+32F = \dfrac{5}{9}C + 32
  2. F=95(C−32)F = \dfrac{9}{5}(C - 32)
  3. F=95C+32F = \dfrac{9}{5}C + 32 (correct answer)
  4. F=32−95CF = 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 17

The volume of a rectangular prism is V=lwhV = lwh, where VV is volume, ll is length, ww is width, and hh is height. Solve for hh in terms of VV, ll, and ww.

  1. h=V−lwh = V - lw
  2. h=Vlwh = \dfrac{V}{lw} (correct answer)
  3. h=lwVh = \dfrac{lw}{V}
  4. h=Vlwh = Vlw
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 V = l w h, divide both sides by (l w) to isolate h, as h is multiplied by l and w, giving h = V / (l w). Choice A is correct because it properly isolates h using division by the product l w, giving h = V / (l w). Perfect! Choice D multiplies everything, but remember, to undo the multiplication, division is key—it's a common mix-up, but verifying with numbers like V=24, l=3, w=2, h=4 helps confirm! 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.

Question 18

Solve for yy in the literal equation ax+by=cax + by = c (assume b≠0b \ne 0). Use the same steps you would use to solve a numeric equation for yy.

  1. y=c−axby = \dfrac{c - ax}{b} (correct answer)
  2. y=c+axby = \dfrac{c + ax}{b}
  3. y=bc−axy = \dfrac{b}{c - ax}
  4. by=c−axby = c - ax (already solved for yy)
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 we 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 incorrectly shows y = (c + ax)/b, adding ax instead of subtracting it—remember, when we move a term to the other side, we use the opposite operation. Common formula rearrangements to practice: ax + by = c becomes y = (c - ax)/b (subtract ax, divide by b) or x = (c - by)/a (subtract by, divide by a). These linear equation rearrangements are fundamental for graphing lines and solving systems!

Question 19

Ohm's law is V=IRV = IR (where VV is voltage, II is current, and RR is resistance). Rearrange the formula to solve for RR.

  1. R=IVR = \dfrac{I}{V}
  2. R=VIR = VI
  3. R=VIR = \dfrac{V}{I} (correct answer)
  4. V=RIV = \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 20

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

  1. l=P−2w2l = \dfrac{P - 2w}{2} (correct answer)
  2. l=P2−2wl = \dfrac{P}{2} - 2w
  3. l=P−w2l = \dfrac{P - w}{2}
  4. l=2w−P2l = \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.