Rewrite Expressions in Different Forms - 7th Grade Math

A student is doing mental math to find 15% of 80. Which rewrite makes the calculation easiest without a calculator?
Answer: $(10\%+5\%)\times 80=8+4$ 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, 15% of 100 is 0.15(100)=15, or rewritten as (10% + 5%)(100)=10 + 5=15, breaking it into easier percentages for mental math. The correct rewriting is (10% + 5%) × 80 = 8 + 4, which shows equivalence by decomposing 15% and makes calculation easiest by adding simple amounts without a calculator. A common error is 0.15 × 80, which is correct but not the rewritten form for ease, or 15 × 80=1200 way off, or 80 ÷ 15 confusing with division. 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 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"?
Answer: $2(l+w)$ 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).
To quickly find the total, a student wants to rewrite $7\times 23+7\times 77$ using factoring. Which expression is equivalent and makes the calculation easiest?
Answer: $7(23+77)$ 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, to compute 5×12 + 5×18, rewrite as 5(12+18)=5(30)=150, factoring out the common 5 for easier addition inside. The correct rewriting is 7×23 + 7×77 = 7(23+77), which shows equivalence by factoring out 7 and makes calculation easiest by simplifying to 7(100)=700. A common error is choosing 7(23×77), confusing addition with multiplication, or 7(23-77) with subtraction, or (7+23)77 rearranging incorrectly without factoring. 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 store increases the price of a notebook by 5%. The new price can be written as $p+0.05p$ . Which equivalent expression shows this as "multiply the original price by one number," making it easier to calculate?
Answer: $1.05p$ 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 notebook price p increased by 5%, the new price p + 0.05p can be factored as p(1 + 0.05) = 1.05p, showing it's equivalent to multiplying the original price by 1.05 for quick calculation. The correct rewriting is 1.05p, which shows equivalence by combining like terms and reveals the utility of a single multiplication for efficiency. A common error is choosing 0.05p or p(0.05), which only represents the increase amount, forgetting to include the original p (should be 1.05p), or mistaking it for 5p which multiplies by 5 instead of 1.05. 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=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).
A store marks up a $40 board game by 30%. The new price can be written as $40 + 0.30(40)$ . Which equivalent expression shows the markup as one multiplication and is easiest to compute?
Answer: $40(1.30)$ Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation ( $7 \times 23 + 7 \times 77$ 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 $40 \times 1.3 = 52$ . 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 $1.05(10) = 10 + 0.05(10)$ ? $10.5 = 10.5\checkmark$ ), (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 ( $a + 0.1a \neq 0.1a$ , $= 1.1a$ ), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A gym charges a monthly fee that can be expressed as $45 + 0.1(m-100)$ dollars, where $m$ is the number of minutes used. A second expression for the same plan is $45+0.1m-10$ . For which values of $m$ are these two expressions equal?
Answer: For all values of $m$ , since the two expressions simplify to the same expression Expand the first expression: 45+0.1(m-100)=45+0.1m-10=35+0.1m. The second expression, 45+0.1m-10, also simplifies to 35+0.1m. Since both expressions simplify to the exact same expression, they are equal for every value of m, not just one specific value, matching choice C. Choice A picks out one specific value where the parentheses happen to equal zero, but the expressions are equal for every value of m, not just that one. Choice B and choice D each pick another single value where the expressions happen to match numerically, which is true for every value of m, not just those specific ones.
A rectangle has length $l$ and width $w$ . Its perimeter is written as $2l+2w$ . Which rewrite shows the structure of the perimeter most clearly (and can be useful for mental math)?
Answer: $2(l+w)$ 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 rectangle with l=10, w=5 has perimeter 2(10)+2(5)=20+10=30, or rewritten as 2(10+5)=2(15)=30, showing the structure of twice the sum of sides. The correct rewriting is 2l + 2w = 2(l + w), which shows equivalence by factoring out 2 and clearly reveals the perimeter as twice the sum of length and width, useful for mental math. A common error is choosing $(l+w)^2$ , confusing perimeter with area, or 2lw which is area, or l+2w incomplete factoring. 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 rectangular garden has area $6x^2 + 9x$ square feet. Which factored form reveals the most useful information about possible dimensions?
Answer: $3x(2x + 3)$ because it shows one side is $3x$ and the other is $2x + 3$ Factoring out the greatest common factor: $6x^2 + 9x = 3x(2x + 3)$ . 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.
To quickly compute $7\times 23 + 7\times 77$ , a student wants to factor using the distributive property. Which expression is an equivalent factored form that makes mental math easiest?
Answer: $7(23+77)$ 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 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 $c$ "?
Answer: $4c$ 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 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?
Answer: $3(x+4)$ 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 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?
Answer: $0.80p$ 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 $50 with 20% discount: 50-0.20(50)=50(0.80)=40 (rewrite reveals multiply by 0.80 for 20% decrease). The correct rewriting $0.80p$ shows equivalence by combining like terms p - 0.20p = 0.80p, makes it easiest to compute as 80% of p, and reveals the decrease relationship clearly. Common errors include $1.20p$ confusing decrease with increase, $0.20p$ which is only the reduction, or $p - 20$ 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 hoodie costs $60. A coupon takes 20% off, so the amount you pay is $60 - 0.2(60)$ . Which equivalent expression is the simplest way to calculate the final price?
Answer: $60(0.8)$ 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 ( $7 \times 23 + 7 \times 77$ 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 $100 item with 20% off is $100 - 0.2(100) = 100 - 20 = 80$ , or rewritten as $100(1 - 0.2) = 100(0.8) = 80$ , simplifying to multiplication by 0.8. The correct rewriting is $60 - 0.2(60) = 60(0.8)$ , which shows equivalence by factoring out 60 and is the simplest for calculating the final price as 80% of original. A common error is $60(1.2)$ confusing with increase, or $60(0.2)$ only the discount, or $60 - 20$ subtracting flat 20 incorrectly for this 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 ( $a + 0.1a \neq 0.1a$ , $= 1.1a$ ), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).
A notebook costs $6. A store is having a 25% off sale. The sale price can be written as $6 - 0.25(6)$ . Which equivalent expression shows the discount as "multiply by the part you pay" and makes it easier to calculate?
Answer: $6(0.75)$ 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 $6 with 25% off, calculate 6-0.25(6)=6-1.5=4.5, or rewrite: 6-0.25(6)=6(1-0.25)=6(0.75)=4.5 (factoring 6 shows multiply by 0.75); or shirt $20 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) shows equivalence by factoring out the common factor 6 to get 6(1-0.25), simplifies to multiplying by the portion paid, and makes mental calculation easy as 75% of 6 is 4.5. Common errors include choosing $6(1.25) for an increase instead of decrease, $6(0.25) which is just the discount, or $6 - 25$ 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).
A snack pack costs $40 and the store adds a 30% markup. The selling price is $40+0.3(40)$ . Which equivalent expression is the most efficient rewrite for calculating the selling price?
Answer: $40(1.3)$ 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 video game item costs $p$ coins. During a sale, the price is reduced by 25%, so the sale price is $p-0.25p$ . Which rewrite shows the sale price as a single multiplication (and makes the meaning clearer)?
Answer: $0.75p$ 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, if an item costs 100 coins with 25% off, it's 100 - 0.25(100) = 100 - 25 = 75, or rewritten as 100(1 - 0.25) = 100(0.75) = 75, showing multiplication by 0.75 for the discount. The correct rewriting is p - 0.25p = 0.75p, which shows equivalence by factoring out p and clarifies the sale price as 75% of original, useful for quick mental math. A common error is choosing 1.25p, confusing decrease with increase, or 0.25p only the discount amount, or p - 25 subtracting 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 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 bookstore sells 4 identical notebooks that each cost $c$ dollars. The total cost can be written as $c+c+c+c$ . Which rewrite best shows the relationship and simplifies the expression?
Answer: $4c$ 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, buying 3 items at d dollars each is d + d + d = 3d, combining like terms to show total as multiple of price. The correct rewriting is c + c + c + c = 4c, which shows equivalence by combining like terms and best simplifies to reveal the relationship of total cost as 4 times the unit price. A common error is $c^4$ confusing with exponents, or c + 4 adding constant, or c/4 dividing instead of multiplying. 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?
Answer: $4c$ 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 $2 each: 2+2+2+2=4×2=8 (combining like terms shows multiplication); or 4 pens at c: c+c+c+c=4c (rewrite simplifies to coefficient times variable). The correct rewriting $4c$ shows equivalence by combining four like terms into multiplication, simplifies the expression, and clearly reveals the relationship as 4 times the cost per pen. Common errors include $c^4$ misusing exponents, $c + 4$ adding instead of multiplying, or $3c$ 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 coach buys 23 water bottles and 77 sports drinks, each costing $7. The total cost is $7\times 23 + 7\times 77$ . Which rewrite uses factoring to make the calculation easiest?
Answer: $7(23+77)$ 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 science kit costs $40. The store marks it up by 30%, so the new price is $40 + 0.3(40)$ . Which rewrite shows the price as "multiply by a single factor" and helps you compute quickly?
Answer: $40(1.3)$ 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 ( $7 \times 23 + 7 \times 77$ 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 30% markup is $50 + 0.3(50) = 50 + 15 = 65$ , or rewritten as $50(1 + 0.3) = 50(1.3) = 65$ , showing quick multiplication by 1.3. The correct rewriting is $40 + 0.3(40) = 40(1.3)$ , which shows equivalence by factoring out 40 and helps compute quickly as a single multiplication factor. A common error is $40(0.3)$ only the markup, or $0.7(40)$ confusing with discount, or $40 + 0.3$ adding flat 0.3. 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 $1.05(10)=10+0.05(10)$ ? $10.5=10.5\checkmark$ ), (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% → $\times 1.15$ ), decrease by r% means multiply by $(1-r)$ (decrease by 20% → $\times 0.80$ ). Mistakes: forgetting original amount ( $a+0.1a \neq 0.1a$ , $=1.1a$ ), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Start this quiz