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One-Step Equations - ISEE Lower Level Quantitative Reasoning

  1. A rectangular garden has a length of 10 feet. Its perimeter is 34 feet. What is the area of the garden in square feet? (Perimeter = 2 × length + 2 × width)
    Answer: 70 First, use the perimeter formula to find the width. 34 = (2 × 10) + (2 × width). This simplifies to 34 = 20 + (2 × width). Let this part of the equation be X = 2 x width. Then 20 + X = 34. Subtract 20 from both sides: X = 14. So, 2 × width = 14. Divide by 2 to find the width: width = 7 feet. The area of a rectangle is length × width. Area = 10 feet × 7 feet = 70 square feet.
  2. A recipe uses 6 cups of flour for one batch of muffins. You make 3 batches, so 3f=183f = 18 cups total. Divide both sides by 3 to find ff. Solve for ff in 3f=183f = 18.
    Answer: 6 This question tests ISEE Lower Level quantitative reasoning skills: solving one-step equations for an unknown. Solving a one-step equation involves performing the inverse operation to isolate the variable. For example, if the equation is 3f = 18, divide both sides by 3 to find f = 6. In this specific problem, students encounter a recipe context where 3 batches use 18 cups total, requiring them to divide both sides by 3 to find the flour per batch. Choice A is correct because 18 ÷ 3 = 6, which accurately represents the solution after applying the correct inverse operation. Choice B (54) results from multiplying 3 × 18, while choice C (21) and D (15) represent other arithmetic errors or misunderstandings. Teaching strategies include using familiar contexts like cooking to make math problems relatable and emphasizing the importance of understanding what the variable represents in word problems.
  3. Ella has xx stickers and gives away 5 stickers today. She still has 11 stickers, so x−5=11x - 5 = 11. Add 5 to both sides to find xx. If x−5=11x - 5 = 11, what is xx?
    Answer: 16 This question tests ISEE Lower Level quantitative reasoning skills: solving one-step equations for an unknown. Solving a one-step equation involves performing the inverse operation to isolate the variable. For example, if the equation is x - 5 = 11, add 5 to both sides to find x = 16. In this specific problem, students encounter Ella's sticker collection where she has 11 stickers left after giving away 5, requiring addition to find her original amount. Choice C is correct because 11 + 5 = 16, which accurately represents the solution after applying the correct inverse operation. Choice A (6) results from subtracting 11 - 5 instead of adding, while choices B (55) and D (11) represent multiplication errors or misunderstanding. Teaching strategies include using manipulatives like counters to model giving away and finding the original amount, reinforcing that addition undoes subtraction in equation solving.
  4. The perimeter of a square is 36 inches. An equilateral triangle has a side length that is 2 inches shorter than the side length of the square. What is the perimeter of the equilateral triangle?
    Answer: 21 inches First, find the side length of the square. A square has 4 equal sides. Let S be the side length. 4 × S = 36. To find S, divide 36 by 4: S = 9 inches. The side of the triangle is 2 inches shorter: 9 - 2 = 7 inches. An equilateral triangle has 3 equal sides. Its perimeter is 3 times its side length: 3 × 7 = 21 inches.
  5. Maya and Liam collect stamps. Before her birthday, Maya had a certain number of stamps. For her birthday, she received 24 stamps, bringing her total to 70. Liam has half as many stamps as Maya had before her birthday. How many stamps does Liam have?
    Answer: 23 First, find the number of stamps Maya had before her birthday. Let S be this number. S + 24 = 70. To find S, subtract 24 from 70: S = 70 - 24 = 46. Maya had 46 stamps. Liam has half this amount, so divide 46 by 2: 46 ÷ 2 = 23. Liam has 23 stamps.
  6. A recipe for a batch of muffins requires 3 cups of flour. A baker has a 21-cup bag of flour. After he makes several batches of muffins, he has 6 cups of flour left. How many batches of muffins did he make?
    Answer: 5 First, determine how much flour the baker used. He started with 21 cups and had 6 cups left. Let F be the flour used. 21 - F = 6. To find F, subtract 6 from 21: F = 21 - 6 = 15 cups. The baker used 15 cups of flour. Since each batch requires 3 cups, divide the total flour used by the amount per batch: 15 ÷ 3 = 5. He made 5 batches of muffins.
  7. A dog weighs 42 pounds. The dog weighs 6 times as much as a cat. What is the combined weight of the dog and the cat?
    Answer: 49 pounds First, find the weight of the cat. Let C be the cat's weight. 6 × C = 42. To find C, divide 42 by 6: C = 7 pounds. The cat weighs 7 pounds. To find the combined weight, add the dog's weight and the cat's weight: 42 + 7 = 49 pounds.
  8. Ava saves money and already has $8 in her jar. She wants $20 total, so x+8=20x + 8 = 20. Subtract 8 from both sides to find xx. What is the value of xx in x+8=20x + 8 = 20?
    Answer: 12 This question tests ISEE Lower Level quantitative reasoning skills: solving one-step equations for an unknown. Solving a one-step equation involves performing the inverse operation to isolate the variable. For example, if the equation is x + 8 = 20, subtract 8 from both sides to find x = 12. In this specific problem, students encounter Ava's savings scenario where she has $8 and needs $20 total, requiring subtraction to find how much more she needs. Choice B is correct because 20 - 8 = 12, which accurately represents the solution after applying the correct inverse operation. Choice A (28) results from adding 8 + 20 instead of subtracting, while choices C (18) and D (8) represent other arithmetic mistakes. Teaching strategies include using visual models like number lines to show the relationship between parts and wholes, and encouraging students to think about what makes sense in the context of the problem.
  9. A factory produces toy cars in batches. Each batch contains the same number of cars. If 3 batches contain a total of 180 cars, how many cars would be in 5 batches?
    Answer: 300 First, find the number of cars in one batch. Let C be this number. 3 × C = 180. To find C, divide 180 by 3: C = 180 ÷ 3 = 60. There are 60 cars in one batch. To find the number of cars in 5 batches, multiply 60 by 5: 60 × 5 = 300. There would be 300 cars.
  10. Maya buys 4 notebooks for $3 each at school. She writes the equation 4x=124x = 12 for the total cost. Divide both sides by 4 to find xx. Solve for xx in 4x=124x = 12.
    Answer: 3 This question tests ISEE Lower Level quantitative reasoning skills: solving one-step equations for an unknown. Solving a one-step equation involves performing the inverse operation to isolate the variable. For example, if the equation is 4x = 12, divide both sides by 4 to find x = 3. In this specific problem, students encounter the equation 4x = 12, which requires dividing both sides by 4 to solve for x. Choice C is correct because 12 ÷ 4 = 3, which accurately represents the solution after applying the correct inverse operation. Choice A (48) is incorrect because it results from multiplying 4 × 12 instead of dividing, while choice B (8) and choice D (16) represent other common arithmetic errors. Teaching strategies include practicing inverse operations through varied examples and reinforcing the concept of isolating the variable by performing the same operation on both sides of the equation.
  11. At a school fair, a game costs 4 tickets to play. Maria started the day with a roll of tickets. After playing the game several times, she has 15 tickets left. She knows she used 20 tickets. How many tickets did she have at the start of the day?
    Answer: 35 The number of times Maria played the game is extra information. The core problem is about the tickets she started with. Let T be the starting number of tickets. She used 20 tickets and had 15 left. So, T - 20 = 15. To find T, add 20 and 15: T = 20 + 15 = 35. She started with 35 tickets.
  12. Liam saves $5 each week for a new game. He writes the equation 5w=355w = 35 for his goal. Divide both sides by 5 to find ww. Find the number that satisfies 5w=355w = 35.
    Answer: 7 This question tests ISEE Lower Level quantitative reasoning skills: solving one-step equations for an unknown. Solving a one-step equation involves performing the inverse operation to isolate the variable. For example, if the equation is 5w = 35, divide both sides by 5 to find w = 7. In this specific problem, students encounter Liam's savings equation 5w = 35, which requires dividing both sides by 5 to determine how many weeks (w) he needs to save. Choice B is correct because 35 ÷ 5 = 7, which accurately represents the solution after applying the correct inverse operation. Choice A (30) might result from subtracting 5 from 35, while choice D (175) comes from multiplying 5 × 35 instead of dividing. Teaching strategies include using real-world contexts like savings goals to make equations meaningful and encouraging students to verify their answers by substituting back into the original equation.
  13. A large pizza is cut into a number of equal slices. After Tom eats 3 slices, there are 5 slices remaining. If a different, identical pizza was instead cut into twice as many equal slices, how many slices would it have in total?
    Answer: 16 First, find the total number of slices in the first pizza. Let S be the original number of slices. S - 3 = 5. To find S, add 3 and 5: S = 3 + 5 = 8. The first pizza had 8 slices. The second pizza is cut into twice as many slices. Multiply the number of slices by 2: 8 × 2 = 16. The second pizza would have 16 slices.
  14. The temperature was 58°F in the afternoon. It dropped to 42°F by the evening. The next morning, the temperature had risen by half the amount it dropped the previous night. What was the temperature in the morning?
    Answer: 50°F First, find the temperature drop. Let D be the drop. 58 - D = 42. To find D, subtract 42 from 58: D = 58 - 42 = 16°F. The temperature dropped by 16 degrees. The next morning, it rose by half that amount: 16 ÷ 2 = 8°F. To find the new temperature, add the rise to the evening temperature: 42 + 8 = 50°F.
  15. When a certain number is divided by 6, the result is 8. What is the result if that same number is then decreased by 10?
    Answer: 38 First, find the original number. Let N be the number. N ÷ 6 = 8. To find N, multiply 8 by 6: N = 8 × 6 = 48. The number is 48. Next, decrease this number by 10: 48 - 10 = 38. The final result is 38.
  16. A roll of ribbon was 150 cm long. A piece was cut off to make 5 identical bows. After the piece was cut, 85 cm of ribbon remained on the roll. What was the length of ribbon used for each bow?
    Answer: 13 cm First, find the total length of the piece that was cut off. Let P be this piece. 150 - P = 85. To find P, subtract 85 from 150: P = 150 - 85 = 65 cm. This 65 cm piece was used to make 5 identical bows. To find the length for one bow, divide 65 by 5: 65 ÷ 5 = 13 cm.
  17. A smoothie recipe needs xx cups of yogurt plus 4 cups fruit. The total is 10 cups, so x+4=10x + 4 = 10. Subtract 4 from both sides to find xx. Solve for xx in x+4=10x + 4 = 10.
    Answer: 6 This question tests ISEE Lower Level quantitative reasoning skills: solving one-step equations for an unknown. Solving a one-step equation involves performing the inverse operation to isolate the variable. For example, if the equation is x + 4 = 10, subtract 4 from both sides to find x = 6. In this specific problem, students encounter a smoothie recipe where yogurt plus 4 cups of fruit equals 10 cups total, requiring subtraction to find the yogurt amount. Choice C is correct because 10 - 4 = 6, which accurately represents the solution after applying the correct inverse operation. Choice A (14) results from adding 4 + 10, while choices B (2) and D (4) might come from dividing or other calculation errors. Teaching strategies include using measuring cups or visual representations to make the problem tangible and reinforcing that subtraction undoes addition when solving equations.
  18. Kayla bought 4 identical markers and paid with a $20 bill. She received $8 in change. What was the cost of one marker?
    Answer: $3 First, find the total cost of the markers by subtracting the change from the amount paid: $20 - $8 = $12. The 4 markers together cost $12. Let M be the cost of one marker. 4 × M = $12. To find M, divide the total cost by the number of markers: $12 ÷ 4 = $3. Each marker cost $3.
  19. Noah buys a toy for $9 and has $15 total. He writes x+9=15x + 9 = 15 for his money. Subtract 9 from both sides to find xx. What is the value of xx in x+9=15x + 9 = 15?
    Answer: 6 This question tests ISEE Lower Level quantitative reasoning skills: solving one-step equations for an unknown. Solving a one-step equation involves performing the inverse operation to isolate the variable. For example, if the equation is x + 9 = 15, subtract 9 from both sides to find x = 6. In this specific problem, students encounter Noah's money situation where he spends $9 from his $15 total, requiring subtraction to find his original amount. Choice B is correct because 15 - 9 = 6, which accurately represents the solution after applying the correct inverse operation. Choice A (24) results from adding 9 + 15 instead of subtracting, while choices C (4) and D (9) represent other calculation errors. Teaching strategies include using money contexts to make equations concrete and teaching students to check their work by substituting the answer back into the original equation to verify that 6 + 9 = 15.
  20. Chen has a collection of 112 seashells. This is 14 more than his friend David has. How many seashells do they have altogether?
    Answer: 210 First, find the number of seashells David has. Let D be David's number of shells. D + 14 = 112. To find D, subtract 14 from 112: D = 112 - 14 = 98. David has 98 shells. To find the total, add Chen's shells and David's shells: 112 + 98 = 210. They have 210 seashells altogether.

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