Word Problems to Expressions
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ISEE Upper Level: Quantitative Reasoning › Word Problems to Expressions
Which algebraic expression represents the total distance? A cyclist rides for $x$ hours at $12$ miles per hour, then continues for $y$ hours at $9$ miles per hour, without stopping. Write an expression for the total miles traveled during the trip.
$12x+9y$
$12(x+y)+9$
$21xy$
$12x-9y$
Explanation
This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes a cyclist traveling at two different speeds for different durations, requiring the distance formula (distance = rate × time) for each segment. Choice A is correct because it accurately represents the total distance as 12x + 9y, where 12x is the distance for the first segment and 9y is the distance for the second segment. Choice B incorrectly groups the hours before applying rates, Choice C subtracts instead of adds, and Choice D multiplies all values together. To help students: Remind them that distance equals rate times time for each segment separately. Practice identifying when to add versus multiply components in multi-part journey problems.
What is the algebraic expression for the total cost? A student buys $x$ textbooks at $18$ dollars each and $y$ workbooks at $12$ dollars each, then pays a $7$ dollar shipping fee. No discounts apply to the order. What expression represents the total amount paid?
$7-18x-12y$
$18(x+y)+7$
$18x+12y+7$
$18x+12y$
Explanation
This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes buying x textbooks at $18 each, y workbooks at $12 each, and paying a $7 shipping fee. Choice B is correct because it accurately represents the total cost as 18x + 12y + 7, combining all purchase costs with the shipping fee. Choice A omits the shipping fee, Choice C incorrectly groups x and y, and Choice D subtracts everything from 7. To help students: Identify all cost components including fees that apply to the entire order. Practice distinguishing between per-item costs (that multiply) and fixed fees (that add once).
Which algebraic expression represents the total distance? A runner completes $x$ laps of a $0.4$ mile track and then runs $y$ additional miles on a trail. No other distances are included in the workout. Write an expression for the total miles run.
$0.4xy$
$0.4x-y$
$0.4x+y$
$0.4(x+y)$
Explanation
This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes running x laps of a 0.4-mile track (giving 0.4x miles) and then y additional miles on a trail. Choice A is correct because it accurately represents the total distance as 0.4x + y, combining the track distance with the trail distance. Choice B incorrectly groups x and y before multiplying, Choice C subtracts the trail miles, and Choice D multiplies all terms together. To help students: Clarify that the y miles are already in miles, not laps. Practice problems where units differ between parts to avoid confusion about what needs conversion.
Formulate an expression based on the described situation. A smoothie recipe requires $1.5$ cups of yogurt per serving and $0.5$ cups of fruit per serving, and you plan to make $x$ servings. You also add $2$ extra cups of ice total. What expression gives the total cups of ingredients used?
$1.5(x+0.5)+2$
$1.5x+0.5x+2$
$2x+2$
$1.5x+0.5x-2$
Explanation
This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes making x servings with 1.5 cups yogurt and 0.5 cups fruit per serving, plus 2 extra cups of ice total. Choice B is correct because it accurately represents the total as 1.5x + 0.5x + 2, which simplifies to 2x + 2, showing all ingredients including the fixed ice amount. Choice A shows the simplified form but doesn't clearly represent the components, Choice C subtracts the ice, and Choice D incorrectly groups terms. To help students: Encourage writing out each component separately before combining. Practice problems with decimal coefficients and fixed additions to build comfort with these variations.
How can the given quantities be expressed algebraically? A student saves $x$ dollars each week for $4$ weeks, then saves $y$ dollars each week for $3$ more weeks. No money is spent during this time. What expression represents total savings after seven weeks?
$7(x+y)$
$4x+3y$
$4y+3x$
$4x+3y+7$
Explanation
This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes saving x dollars per week for 4 weeks (giving 4x total) and y dollars per week for 3 weeks (giving 3y total). Choice B is correct because it accurately represents the total savings as 4x + 3y, combining both saving periods. Choice A incorrectly adds x and y before multiplying by 7, Choice C reverses the coefficients, and Choice D incorrectly adds 7 as a separate term. To help students: Emphasize calculating each saving period separately before combining. Practice problems where different rates apply to different time periods to build pattern recognition.
Translate the given scenario into an algebraic expression. A student buys $x$ notebooks at $3$ dollars each and $y$ pens at $2$ dollars each, then adds a $5$ dollar folder. What expression represents the total cost for all items purchased?
$3x+2y$
$3(x+y)+5$
$3x+2y+5$
$5-3x-2y$
Explanation
This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes purchasing x notebooks at $3 each (giving 3x), y pens at $2 each (giving 2y), and adding a $5 folder. Choice B is correct because it accurately represents the total cost as 3x + 2y + 5, combining all three components. Choice A omits the folder cost, Choice C incorrectly groups x and y together before multiplying by 3, and Choice D incorrectly subtracts the costs from 5. To help students: Encourage them to identify each cost component separately before combining. Practice breaking down multi-step problems into individual parts to avoid missing elements or misapplying operations.
What is the algebraic expression for the total cost? A student buys $x$ packs of index cards for $4$ dollars each and $y$ binders for $6$ dollars each, then uses a $10$ dollar coupon. Write an expression for the amount paid after the coupon is applied.
$10-4x-6y$
$4(x+y)-10$
$4x+6y+10$
$4x+6y-10$
Explanation
This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes buying x packs at $4 each and y binders at $6 each, then applying a $10 coupon which reduces the total. Choice C is correct because it accurately represents the final cost as 4x + 6y - 10, where the coupon is subtracted from the total purchase price. Choice A incorrectly adds the coupon value, Choice B subtracts everything from 10, and Choice D incorrectly groups x and y. To help students: Clarify that coupons reduce the total, requiring subtraction. Practice distinguishing between discounts (subtraction) and additional fees (addition) in word problems.
Translate the given scenario into an algebraic expression. A car travels $x$ hours on the highway at $65$ miles per hour and then $y$ hours in city traffic at $30$ miles per hour. Assume the speeds stay constant within each segment. What expression represents the total distance traveled?
$65x+30y$
$65x-30y$
$65(x+y)+30$
$95(x+y)$
Explanation
This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes a car traveling at 65 mph for x hours and 30 mph for y hours, requiring the distance formula for each segment. Choice A is correct because it accurately represents the total distance as 65x + 30y, where each term is the product of speed and time for that segment. Choice B incorrectly adds the times before multiplying, Choice C incorrectly groups variables, and Choice D subtracts the city distance. To help students: Reinforce that distance = rate × time must be calculated separately for each segment with different speeds. Practice multi-segment journey problems to build confidence with this common pattern.