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 form $$m = \frac{40 - 25}{0.15}$$ directly shows how to calculate the miles: take the budget ($40), subtract the base fee ($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.

Expanding the first expression: $$45 + 0.1(m - 100) = 45 + 0.1m - 10 = 35 + 0.1m$$. The second expression is $$45 + 0.1m - 10 = 35 + 0.1m$$. The expressions are algebraically equivalent for all values of $$m$$. Choice A suggests they're equal only at $$m = 100$$. Choice B suggests they're equal only at $$m = 0$$. Choice D suggests they're equal only at $$m = 200$$.

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.

Original area: $$\ell w$$. After 20% increase: new length = $$1.2\ell$$, new width = $$1.2w$$. New area = $$(1.2\ell)(1.2w) = 1.44\ell w$$. The increase is $$1.44\ell w - \ell w = 0.44\ell w$$, which represents a 44% increase. Choice A incorrectly adds the individual percent increases (20% + 20%). Choice C gives the factor by which one dimension increases. Choice D gives the actual increase in area, not the percent.

The form $$9C = 5F - 160$$ clearly shows that when $$F$$ increases by 9, the term $$5F$$ increases by $$5 \times 9 = 45$$, 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.

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).

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).

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).

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).