A store offers a discount where customers pay for an item originally priced at dollars. Which expression shows the relationship between the discount amount and the sale price?
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7th Grade Math Quiz
Practice Rewrite Expressions In Different Forms in 7th Grade Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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A store offers a discount where customers pay 0.85p for an item originally priced at p dollars. Which expression shows the relationship between the discount amount and the sale price?
This quiz focuses on Rewrite Expressions In Different Forms, giving you a quick way to practice the rules, question types, and explanations that matter most for 7th Grade Math.
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.
A store offers a discount where customers pay 0.85p for an item originally priced at p dollars. Which expression shows the relationship between the discount amount and the sale price?
Explanation: Since customers pay 0.85p, they receive a 15% discount. The discount amount is p−0.85p=0.15p. The sale price can be written as 0.85p or equivalently as p−0.15p. Choice B incorrectly identifies the sale price as the discount. Choice C shows an increase instead of a discount. Choice D uses an impossible discount percentage greater than 100%.
The cost to rent a car is 25+0.15m dollars, where m is miles driven. A customer wants to spend exactly $40. Which form of the equation 25+0.15m=40 makes it easiest to see how many miles they can drive?
Explanation: The form m=0.1540−25 directly shows how to calculate the miles: take the budget (40),subtractthebasefee(25), then divide by the per-mile rate ($0.15). This makes the solution process transparent. Choice A requires another step to solve for m. Choice C doesn't directly show the miles. Choice D creates unnecessary complexity with ratios.
A rectangular garden has area 6x2+9x square feet. Which factored form reveals the most useful information about possible dimensions?
Explanation: Factoring out the greatest common factor: 6x2+9x=3x(2x+3). This form shows the dimensions as 3x feet by (2x+3) feet, both of which are reasonable expressions for side lengths. Choice B factors out only x, not the complete GCF. Choice C factors out 3 but doesn't reveal dimensional information. Choice D uses a decimal coefficient, which is less clean than the integer form.
The temperature in degrees Fahrenheit can be converted to Celsius using C=95(F−32). Which equivalent expression shows how much the Celsius temperature changes when Fahrenheit increases by 9 degrees?
Explanation: The form 9C=5F−160 clearly shows that when F increases by 9, the term 5F increases by 5×9=45, so 9C increases by 45, meaning C increases by 5. This directly reveals the 9:5 ratio. Choice A shows the slope but not the specific 9-degree relationship. Choice B doesn't highlight the ratio. Choice D shows the inverse but doesn't emphasize the change relationship.
A class is collecting cans. Last week they collected p cans, but this week they collected 20% fewer. The amount this week is p−0.20p. Which equivalent expression makes it easiest to find the new amount?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: cans p with 20% fewer, calculate p-0.20p=p(1-0.20)=p(0.80) (factoring p shows multiply by 0.80); or sales 50with200.80pshowsequivalencebycombiningliketermsp−0.20p=0.80p,makesiteasiesttocomputeas801.20pconfusingdecreasewithincrease,0.20pwhichisonlythereduction,orp - 20$ subtracting a flat 20 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A student writes 3x+12 on the board and wants to factor it to show a common factor. Which expression is equivalent to 3x+12 and shows the greatest common factor?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, the expression 3x + 12 can be factored as 3(x + 4), pulling out the greatest common factor of 3 from both terms. The correct rewriting is 3(x + 4), which shows equivalence by reverse distributive property and highlights the common factor clearly. A common error is choosing 3(x) + 12 which doesn't fully factor, or x + 4 which ignores the 3, or 3x(12) which multiplies incorrectly to 36x. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in x=2: 3(2+4)=18, 3(2)+12=6+12=18✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).
A library charges c dollars per bookmark, and a student buys 4 bookmarks. The total cost is written as c+c+c+c. Which expression is an equivalent rewrite that shows the total as “4 times c”?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, buying 4 bookmarks at c dollars each, the total c + c + c + c can be combined as 4c, showing it's 4 times the cost per bookmark. The correct rewriting is 4c, which demonstrates equivalence by combining like terms and clearly shows the multiplication relationship. A common error is selecting c^4 which exponents instead of multiplies, or 4 + c which adds instead, or c/4 which divides wrongly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in c=2: 4(2)=8, 2+2+2+2=8✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).
A snack costs p dollars. The store adds 8% sales tax, so the total cost is p+0.08p. Which equivalent expression shows the total as one multiplication?
Explanation: This question tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, a $50 item with 8% tax is 50 + 0.08(50) = 50 + 4 = 54, or rewritten as 50(1 + 0.08) = 50(1.08) = 54, showing the total as 108% of original. The correct rewriting is p + 0.08p = 1.08p, which shows equivalence by combining like terms and reveals the utility of one multiplication for total cost including tax. A common error is p(0.08) only the tax amount, or p + 0.8 adding flat 0.8, or 8p multiplying by 8 incorrectly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A snack pack costs \40andthestoreaddsa3040+0.3(40)$. Which equivalent expression is the most efficient rewrite for calculating the selling price?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a snack pack at $40 with 30% markup, the selling price 40 + 0.3(40) can be rewritten as 40(1 + 0.3) = 40(1.3), making it efficient to calculate as a single multiplication. The correct rewriting is 40(1.3), which demonstrates equivalence by factoring out 40 and shows the utility for quick mental math on the marked-up price. A common error is choosing 40(0.3) which is only the markup amount (forgetting the original), or 0.3(80) which equals the markup but not the total, or 40+0.3 which adds a flat 0.3 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: 40(1.3)=52, 40+0.3(40)=40+12=52✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).
A rectangle has length l and width w. Its perimeter can be written as 2l+2w. Which rewrite shows the meaning more clearly as “twice the sum of length and width”?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: perimeter of rectangle with sides 5 and 3 is 2×5 + 2×3=2(5+3)=2(8)=16 (factoring 2 shows twice the sum); or length l width w: 2l + 2w=2(l+w) (rewrite clarifies the structure). The correct rewriting 2(l+w) shows equivalence by factoring out 2 from both terms, clearly reveals the meaning as twice the sum of length and width, and simplifies understanding of the perimeter formula. Common errors include l + w which omits the doubling, (2l)(2w) which multiplies instead of adding, or 2(l-w) which subtracts incorrectly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A coach buys 23 water bottles and 77 sports drinks, each costing \7.Thetotalcostis7\times 23 + 7\times 77$. Which rewrite uses factoring to make the calculation easiest?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, for buying 23 water bottles and 77 sports drinks at $7 each, the total 7×23 + 7×77 can be factored as 7(23 + 77) = 7(100), making the calculation straightforward. The correct rewriting is 7(23+77), which shows equivalence by factoring out the common 7 and simplifies to 700 easily. A common error is choosing 7(23×77) which multiplies instead of adds (wrong operation), or (7+23)77 which adds 7 to 23 incorrectly, or 7×(23-77) which subtracts leading to negative. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in values: 7(23+77)=7(100)=700, 7×23 + 7×77=161+539=700✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).
A video game has a 25% off sale. If the original price is p, the sale price is p−0.25p. Which rewrite shows the sale price as “multiply by the part that remains,” so it’s quicker to compute?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a video game price p discounted by 25%, the sale price p - 0.25p can be factored as p(1 - 0.25) = 0.75p, showing it's equivalent to multiplying by the remaining 75% for quicker computation. The correct rewriting is 0.75p, which shows equivalence through factoring and highlights the utility of multiplying by the portion that remains after the discount. A common error is selecting 1.25p, which would be for a 25% increase instead of decrease (sign error), or 0.25p which is just the discount amount, forgetting the original minus that (should be 0.75p), or p-25 which subtracts a flat 25 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if p=100, does 0.75(100)=100-0.25(100)? 75=75✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).
A notebook costs 6.Astoreishavinga256 - 0.25(6)$. Which equivalent expression shows the discount as “multiply by the part you pay” and makes it easier to calculate?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: notebook 6with2520 with 25% off: 20-0.25(20)=20(0.75)=15 (rewrite reveals multiply by 0.75 for 25% discount). The correct rewriting 6(0.75)showsequivalencebyfactoringoutthecommonfactor6toget6(1−0.25),simplifiestomultiplyingbytheportionpaid,andmakesmentalcalculationeasyas756(1.25) for an increase instead of decrease, 6(0.25)whichisjustthediscount,or6 - 25$ which subtracts a flat 25 incorrectly ignoring percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
To quickly compute 7×23+7×77, a student wants to factor using the distributive property. Which expression is an equivalent factored form that makes mental math easiest?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: compute 5×12 + 5×18 as 5(12+18)=5(30)=150 (factoring 5 simplifies addition inside); or 7×23 + 7×77=7(23+77)=7(100)=700 (rewrite makes mental math easy by summing to 100 first). The correct rewriting 7(23+77) shows equivalence by factoring out the common 7 using distributive property, makes mental math easiest by adding 23+77=100 then 7×100=700, and reveals the shared factor. Common errors include (7+23)77 which factors incorrectly, 7(23)+77 which doesn't factor fully, or 7(23-77) which changes addition to subtraction. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A runner’s time for a mile is t minutes. After training, their time decreases by 10%, so the new time is t−0.10t. Which equivalent expression correctly represents the new time?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: time t with 10% decrease, calculate t-0.10t=t(1-0.10)=t(0.90) (combining shows multiply by 0.90); or speed 50 mph decreased by 10%: 50-0.10(50)=50(0.90)=45 (rewrite reveals new amount directly). The correct rewriting 0.90t shows equivalence by combining like terms t - 0.10t = 0.90t, correctly represents the decreased time, and simplifies to a single multiplication. Common errors include 1.10t confusing decrease with increase, 0.10t which is just the decrease, or t+0.10 adding instead of subtracting. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A school club buys 12 identical water bottles for a trip and 8 more for new members. Each bottle costs 3.50. The total cost is 3.5×12+3.5×8. Which rewrite makes the calculation easiest?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: cost of 12 items at 3eachplus8at3: 3×12 + 3×8=3(12+8)=3(20)=60 (factoring 3 simplifies); or 3.5×12 + 3.5×8=3.5(20)=70 (rewrite makes addition easy first). The correct rewriting 3.5(12+8) shows equivalence by factoring out 3.5 using distributive property, makes calculation easiest by adding 12+8=20 then 3.5×20=70, and is useful for mental math. Common errors include 3.5(12-8) changing to subtraction, (3.5×12)×8 which associates differently and complicates, or 3.5×(12×8) which multiplies counts first incorrectly for total cost. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A video game has a score bonus that adds 5% more points to your current score s. The new score is written as s+0.05s. Which rewrite best shows the relationship as one multiplication?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: salary 25/hrwith51.05sshowsequivalencebyfactoringoutstocombinetermsass(1+0.05),revealstherelationshipasasinglemultiplicationforthetotalscoreafterbonus,andsimplifiescomputation.Commonerrorsinclude0.05sforgettingtheoriginalscore,5smisinterpretingpercentasmultiplyingby5,ors + 5$ adding a flat 5 instead of 5%. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A student buys 4 identical pens. If each pen costs c dollars, the total cost is written as c+c+c+c. Which rewrite best simplifies the expression and shows the relationship?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: total cost of 4 items at 2each:2+2+2+2=4×2=8(combiningliketermsshowsmultiplication);or4pensatc:c+c+c+c=4c(rewritesimplifiestocoefficienttimesvariable).Thecorrectrewriting4cshowsequivalencebycombiningfourliketermsintomultiplication,simplifiestheexpression,andclearlyrevealstherelationshipas4timesthecostperpen.Commonerrorsincludec^4misusingexponents,c + 4addinginsteadofmultiplying,or3c$ undercounting the number of pens. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A store marks up a \40boardgameby3040 + 0.30(40)$. Which equivalent expression shows the markup as one multiplication and is easiest to compute?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: item 25 with 30% markup, calculate 25+0.3(25)=25+7.5=32.5, or rewrite: 25+0.3(25)=25(1+0.3)=25(1.3)=32.5 (factoring 25 shows multiply by 1.3); or 40 with 30% markup: 40+0.3(40)=40(1.3)=52 (rewrite reveals multiply by 1.3 for 30% increase). The correct rewriting 40(1.30) shows equivalence by factoring out 40 to get 40(1+0.30), represents the markup as one multiplication, and is easiest to compute as 40×1.3=52. Common errors include 40(0.30) which is just the markup amount, 40(0.70) confusing with discount, or 40+0.30 adding 30 cents instead of 30%. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1−r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a=0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A science club’s supply amount increases by 20%. If the original amount is a, the new amount is a+0.2a. Which equivalent expression correctly shows this as a single multiplication and explains the relationship between adding 20% and multiplying?
Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a supply amount a increased by 20%, the new amount a + 0.2a can be factored as a(1 + 0.2) = 1.2a, explaining that adding 20% is equivalent to multiplying by 1.2. The correct rewriting is 1.2a, which shows equivalence by combining terms and reveals the multiplication relationship for percentage increase. A common error is choosing a(0.2) which is only the increase (forgetting original), or 0.8a which is for decrease, or 2a which doubles instead of adding 20%. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in a=10: 1.2(10)=12, 10+0.2(10)=10+2=12✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).