- A student buys notebooks that cost $1.75 each and also pays $2.50 for a binder. If the total cost was $16.50, how many notebooks did the student buy? Write an equation and solve.
Answer: 8 notebooks
Each notebook costs $1.75, so buying n notebooks costs 1.75n, and adding the $2.50 binder gives the equation 1.75n + 2.50 = 16.50. Subtracting 2.50 from both sides gives 1.75n = 14.00, and dividing both sides by 1.75 gives n = 8. Choice A comes from a small arithmetic slip when dividing. Choice B comes from a larger miscalculation in isolating n. Choice D comes from dividing the total by 1.75 before subtracting the binder cost, which mixes up the order of operations.
- A streaming app charges $7.50 per month plus a one-time sign-up fee of $4.50. If Maya paid a total of $49.50, how many months did she pay for? Write and solve an equation of the form px+q=r.
Answer: m=6
The equation is 7.5m + 4.5 = 49.5. Subtracting 4.5 from both sides gives 7.5m = 45, and dividing by 7.5 gives m = 6, matching choice C. Checking: 7.5 x 6 + 4.5 = 45 + 4.5 = 49.5, which fits the context of 6 months of subscription. Choice A (6.6) comes from dividing 49.5 by 7.5 first instead of subtracting the fee, and choices B and D don't satisfy the equation when substituted back in.
- The equation 4(2x−3)=6x+14 represents a real-world situation where costs must be equal. After solving algebraically to get x=13, which arithmetic verification correctly confirms this solution?
Answer: Left side: 4(2⋅13−3)=4(23)=92; Right side: 6⋅13+14=92
When you have an algebraic equation and need to verify a solution, you substitute the proposed value back into the original equation and check that both sides are equal. This confirmation step is crucial because it catches any algebraic errors you might have made while solving.
To verify x=13 in the equation 4(2x−3)=6x+14, you need to carefully substitute 13 for x and evaluate each side separately.
For the left side: 4(2x−3)=4(2⋅13−3)=4(26−3)=4(23)=92
For the right side: 6x+14=6⋅13+14=78+14=92
Since both sides equal 92, the solution x=13 is correct. This matches answer choice D.
Looking at the wrong answers: Choice A makes an order of operations error on the left side, calculating 2⋅13−3 as 20 instead of 23, leading to 80 instead of 92. Choice B correctly calculates the left side as 92 but makes an arithmetic error on the right side, getting 86 instead of 92. Choice C makes a different error on the left side, somehow getting 26 instead of 23 inside the parentheses.
Remember: verification problems test your arithmetic precision more than your algebra skills. Work slowly through the order of operations, double-check your basic arithmetic, and make sure both sides of the equation give you the same final value. - Compare methods: A streaming service charges $9.75 per month plus a one-time $4.50 fee. The total paid was $53.25. How many months were paid for? (You could solve algebraically or by working backward.)
Answer: 5 months
This problem tests solving two-step equations from word problems in the form px+q=r, where we multiply the variable then add a constant, using inverse operations. The streaming service costs $9.75 per month (variable cost) plus a $4.50 fee (fixed cost), totaling $53.25, so we write 9.75m+4.50=53.25 where m is the number of months. To solve algebraically: (1) subtract 4.50 from both sides (9.75m=48.75), then (2) divide by 9.75 (m=48.75÷9.75=5). Working backward gives the same result: $53.25−$4.50=$48.75, then $48.75÷$9.75=5 months. Common errors include: stopping after one step, reversing operations, or arithmetic mistakes with decimals. Both methods—algebraic equation solving and working backward—yield 5 months, demonstrating that algebra generalizes the arithmetic approach of undoing operations in reverse order. - A school club charges a $4.50 sign-up fee plus $2.25 per week. If a student paid $24.75 total, how many weeks did they pay for?
Answer: 9 weeks
This problem tests solving two-step equations from word problems in the form px+q=r, where we multiply the variable then add a constant, using inverse operations. The club charges $2.25 per week (variable cost) plus a $4.50 sign-up fee (fixed cost), totaling $24.75, so we write 2.25w + 4.50 = 24.75 where w is the number of weeks. To solve: (1) subtract 4.50 from both sides to isolate the variable term (2.25w = 20.25), then (2) divide by 2.25 to isolate w (w = 20.25 ÷ 2.25 = 9). The student paid for 9 weeks. Common errors include: stopping after one step (getting 20.25 as the answer), reversing operations (dividing first: 24.75÷2.25=11, then subtracting 4.50=6.50), or arithmetic mistakes (20.25÷2.25=8). Strategy: identify the equation form from context (weekly rate × weeks + fixed fee = total), apply inverse operations in reverse order (subtract then divide), and verify by substituting back (2.25×9+4.50=20.25+4.50=24.75✓).
- A school fundraiser sells tubs of cookie dough for $8.75 each. The club also pays a one-time booth rental fee of $10.00. If the total cost was $80.00, how many tubs did they buy? Solve 8.75x+10=80.
Answer: x=8
This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 8.75x + 10 = 80 becomes 8.75x = 70, then divide both sides by p to isolate the variable, so 8.75x ÷ 8.75 = 70 ÷ 8.75 gives x = 8; for the other form p(x + q) = r, divide by p first then subtract q, and always verify by substituting back, like 8.75 × 8 + 10 = 70 + 10 = 80, which checks out in the context of cookie dough tubs at $8.75 each plus $10 fee totaling $80 for x tubs. For this specific problem, the equation is given as 8.75x + 10 = 80, subtract 10 to get 8.75x = 70, divide by 8.75 to find x = 8, verify 8.75 × 8 + 10 = 80, and interpret as 8 tubs bought. The correct two-step process yields x = 8, which is choice A. Common errors include subtracting only and forgetting to divide (x = 70, choice C), reversing order by dividing first (80 ÷ 8.75 ≈ 9.14 then subtract 10, nonsensical), sign errors like adding instead (8.75x = 90), arithmetic mistakes (70 ÷ 8.75 = 9 or 6), or wrong setup though it's provided. Strategy tips include identifying the px + q = r form from per-tub cost plus fixed fee, applying inverse operations in reverse (subtract then divide), maintaining equality by doing the same to both sides, verifying by substitution and context sense (8 tubs at $8.75 is $70 plus $10 totals $80, reasonable), and checking reasonableness. Compared to arithmetic working backwards (80 - 10 = 70, 70 ÷ 8.75 = 8), algebra generalizes for any total; avoid mistakes like stopping after one step, wrong order, sign errors, or mismatched setup. - A music app charges $3 to start an account and then $8 each month. Your total cost was $59. How many months did you pay for?
Answer: 7 months
This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 8m + 3 = 59 becomes 8m = 56, then divide both sides by p to isolate the variable, so 8m / 8 = 56 / 8 gives m = 7; always verify by substituting back, like 8 × 7 + 3 = 56 + 3 = 59, which checks out. For this specific problem, set up the equation from the context: 8m + 3 = 59, subtract 3 to get 8m = 56, divide by 8 to find m = 7, verify 8 × 7 + 3 = 59, and interpret as 7 months of subscription. The correct two-step process yields m = 7, which is choice A. Common errors include one-step only (59 - 3 = 56, forgetting to divide so m = 56), operation order reversed (59 / 8 = 7.375, then subtract 3 ≈ 4.375 nonsense), sign error (adding 3 instead: 8m = 62), arithmetic mistake (56 / 8 = 8 wrongly), or setup wrong (3m + 8 = 59 reversing). Strategy: identify the equation form px + q = r from the context, apply inverse operations in reverse order (subtract then divide to undo add then multiply), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation and makes sense (7 months at $8 is $56 plus $3 fee totals $59), and check reasonableness (total around 7 × 8 + small fee). Comparing to arithmetic, work backwards: 59 - 3 = 56, 56 / 8 = 7, same answer; algebra generalizes for any total; avoid mistakes like stopping after one step or wrong order. - The perimeter of a rectangle is 84 inches. The length is 6 inches more than twice the width. What are the dimensions of the rectangle?
Answer: Width = 12 inches, Length = 30 inches
Let w be the width. Then length = 2w+6. Perimeter formula: 2w+2(2w+6)=84. Simplifying: 2w+4w+12=84, so 6w=72 and w=12. Therefore length = 2(12)+6=30. Check: 2(12)+2(30)=24+60=84 ✓. Choice A results from solving 2w+2w+6=84 (forgetting to double the length). Choice C uses incorrect coefficient setup. Choice D comes from arithmetic errors in solving the equation. - Solve the equation 15x+6=96.
Answer: x=6
Subtracting 6 from both sides of 15x + 6 = 96 gives 15x = 90, and dividing by 15 gives x = 6, matching choice D. Checking: 15 x 6 + 6 = 90 + 6 = 96. Choice B (5.6) and choice A (7) both come from arithmetic slips in the division step, and choice C (90) stops after subtracting but forgets to divide by 15.
- Compare methods: A student says the equation for this situation is 7x+2=44: "Each notebook costs $7, and there is a $2 fee for the whole purchase. The total was $44." What is the correct number of notebooks?
Answer: x=6
Starting with the equation 7x plus 2 equals 44, subtracting 2 from both sides gives 7x equals 42, and dividing both sides by 7 gives x equals 6, matching choice D. Choice A, x equals 7, does not satisfy the equation, since 7 times 7 plus 2 equals 51, not 44. Choice B, x equals 42, comes from stopping after subtracting 2 and forgetting to divide by 7. Choice C, x equals 5, comes from an arithmetic slip when dividing 42 by 7. Checking the answer by substitution, 7 times 6 plus 2 equals 42 plus 2, which equals 44, confirms that x equals 6 is correct. - Verify a solution: A tutoring session costs $18 per hour plus a $6 materials fee. The total bill was $78. Which value of h correctly solves 18h+6=78 and fits the situation?
Answer: h=4
Subtracting 6 from both sides of 18h + 6 = 78 gives 18h = 72, and dividing by 18 gives h = 4, matching choice A. Checking: 18 x 4 + 6 = 72 + 6 = 78, which fits the situation of 4 hours tutored. Choices B, C, and D don't satisfy the equation: substituting each back in gives totals of $69, $60, and $84 instead of the required $78.
- A movie theater has two pricing options for groups. Option A charges $12 per person. Option B charges a $36 group fee plus $8 per person. For what group size do both options cost exactly the same amount?
Answer: 9 people, with both options costing $108 total
Let n be the number of people. Option A costs 12n and Option B costs 36+8n. Setting them equal: 12n=36+8n, so 4n=36 and n=9. Check: Option A = 12(9)=108 and Option B = 36+8(9)=36+72=108. Both options cost exactly $108 for 9 people. The other choices show incorrect group sizes that don't result in equal costs. - A class is making snack bags. Each bag has the same number of pretzels. After making 6 bags, there are 3 pretzels left over, and the class started with 51 pretzels. Let x be the number of pretzels in each bag. Which equation matches the situation, and what is x?
(Equation form: px+q=r)
Answer: 6x+3=51, so x=8
This problem tests solving two-step equations from word problems in the form px+q=r, where we multiply the variable then add a constant, using inverse operations. The class makes 6 bags with x pretzels each (6x total) plus 3 leftover pretzels equals 51 total pretzels, giving equation 6x + 3 = 51. To solve: (1) subtract 3 from both sides to isolate the variable term (6x = 48), then (2) divide by 6 to isolate x (x = 48 ÷ 6 = 8). Each bag has 8 pretzels, matching answer choice A. Common errors include: reversing the equation setup (51x + 3 = 6 as in choice C), swapping coefficients (3x + 6 = 51 as in choice D), or using subtraction instead of addition for leftovers (6x - 3 = 51 as in choice B). Strategy: identify what's multiplied (bags × pretzels per bag) and what's added (leftovers), set up equation matching the context, apply inverse operations, and verify (6×8+3=48+3=51✓). - A hobby store sells refill packs for $5 each. A science kit comes with some number of paid refill packs, plus 2 bonus packs added for free. The kit's total advertised value, treating all packs (paid and bonus) at $5 each, is $45. If p represents the number of paid refill packs, this can be modeled by 5(p+2)=45. Solve for p.
Answer: p=7
To solve 5(p+2)=45, first divide both sides by 5: p+2=9. Then subtract 2 from both sides: p=7, matching choice B. Verify: 5(7+2)=5x9=45, which checks out. Choice A stops after dividing but forgets to subtract 2, leaving p+2's value (9) as the final answer. Choice C divides 45 by 5 to get 9, then adds 2 instead of subtracting, giving 11. Choice D makes an arithmetic error, subtracting incorrectly to arrive at 5. - A movie theater sells a snack combo for $6.25 each, plus a one-time online order fee of $2.50. If the total was $40, how many combos were ordered? Write and solve an equation of the form px+q=r.
Answer: 6 combos
The equation for this situation is 6.25x + 2.5 = 40, where x is the number of combos. Subtracting 2.5 from both sides gives 6.25x = 37.5, and dividing both sides by 6.25 gives x = 6. Checking: 6.25(6) + 2.5 = 37.5 + 2.5 = 40, which matches. Choice A mistakes the price per combo for the number of combos. Choice C comes from an arithmetic slip when dividing 37.5 by 6.25. Choice D comes from dividing the total by the price per combo first, 40/6.25 = 6.4, without properly subtracting the fee first.
- A snack shop sells party bags. Each bag includes some number of paid snack packs, plus 3 bonus packs added for free. The shop values every pack, paid or bonus, at $4 each, and the total value of one bag is $36. If x represents the number of paid snack packs, this can be modeled by 4(x+3)=36. Solve for x.
Answer: x=6
To solve 4(x+3)=36, first divide both sides by 4: x+3=9. Then subtract 3 from both sides: x=6, matching choice C. Verify: 4(6+3)=4x9=36, which checks out. Choice A comes from an arithmetic slip. Choice B stops after dividing but forgets to subtract 3, leaving the value of x+3 (9) plus an extra step gone wrong. Choice D divides 36 by 4 to get 9, then adds 3 instead of subtracting, giving 12 mistakenly recorded, or represents a similar addition error. - A science museum sells tickets for 9 each. There is also a one-time processing fee of 4.50 for the whole order. If the total cost was 58.50, how many tickets were bought?
Answer: 6 tickets
This question tests solving two-step equations from word problems in the px+q=r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px+q=r, first subtract q from both sides to isolate the variable term, for example, 9t+4.5=58.5 becomes 9t=54, then divide both sides by p to isolate the variable, so 9t/9=54/9 gives t=6; always verify by substituting back, like 9×6+4.5=54+4.5=58.5, which checks out. For this specific problem, set up the equation from the context: 9t+4.5=58.5, subtract 4.5 to get 9t=54, divide by 9 to find t=6, verify 9×6+4.5=58.5, and interpret as 6 tickets bought. The correct two-step process yields t=6, which is choice A. Common errors include dividing first (58.5/9=6.5, subtract 4.5=2 nonsense), arithmetic mistakes (54/9=5 wrongly), one-step only (58.5−4.5=54, forget divide), sign error (adding 4.5 instead), or setup wrong (4.5t+9=58.5). Strategy: identify the equation form px+q=r from the context, apply inverse operations in reverse order (subtract then divide to undo add then multiply), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation and makes sense (6 tickets at 9 is 54 plus 4.50 fee totals 58.50), and check reasonableness (total near 6×9+ small fee). Comparing to arithmetic, work backwards: 58.5−4.5=54, 54/9=6, same answer; algebra generalizes for any total; avoid mistakes like wrong order or stopping after one step. - At a sports store, each item costs $4. You are buying one identical item for each of your friends, plus 2 extra items for yourself and a backup. If the total cost was $36, how many friends were you shopping for?
Answer: 7 friends
Since you're buying one item for each friend plus 2 extra items for yourself and a backup, the total number of items is x + 2, where x is the number of friends. The equation for the total cost is 4(x + 2) = 36. Dividing both sides by 4 gives x + 2 = 9, and subtracting 2 gives x = 7. Checking: 4(7 + 2) = 4(9) = 36, which matches. Choice B comes from stopping after finding the total number of items, 9, without subtracting the 2 extra. Choice C and choice D come from errors in the order of the division and subtraction steps. - Solve the equation 3(x+4)=27.
Answer: x=5
This question tests solving two-step equations in the p(x + q) = r form, where you multiply a grouped expression by a coefficient, using inverse operations to isolate the variable. To solve p(x + q) = r, first divide both sides by p to isolate the grouped term, for example, 3(x + 4) = 27 becomes x + 4 = 9, then subtract q from both sides to isolate the variable, so x = 9 - 4 = 5; always verify by substituting back, like 3(5 + 4) = 3 × 9 = 27, which checks out. For this specific problem, start with 3(x + 4) = 27, divide by 3 to get x + 4 = 9, subtract 4 to find x = 5, and verify 3(5 + 4) = 27. The correct two-step process yields x = 5, which is choice A. Common errors include distributing wrongly (3x + 4 = 27, then mistakes), order reversed (subtract 4 from 27 = 23, divide by 3 ≈ 7.67 nonsense), arithmetic error (27 / 3 = 8, 8 - 4 = 4), or expanding incorrectly (3x + 12 = 27). Strategy: identify the equation form p(x + q) = r, apply inverse operations in reverse order (divide then subtract to undo multiply after add), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation, and check reasonableness (5 + 4 = 9, times 3 is 27). Comparing to arithmetic, work backwards: 27 / 3 = 9, 9 - 4 = 5, same answer; algebra generalizes for any right side; avoid mistakes like wrong order or distribution errors. - Solve the equation 5(x+3)=40.
Answer: x=5
Dividing both sides by 5 first gives x + 3 = 8. Subtracting 3 from both sides gives x = 5. Checking: 5(5+3) = 5(8) = 40, which matches. Choice A stops after dividing by 5 and forgets to subtract 3 to finish isolating x. Choice B comes from incorrectly expanding the expression as 5x + 3 = 40, leading to 5x = 37, then stopping without dividing by 5. Choice D comes from adding 3 instead of subtracting it after dividing by 5.
