Fraction Benchmarks

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Answer Key & Explanations

Fraction Benchmarks · Grade 3 · Worksheet 2

1. Charlotte draws a number line from 0 to 1 and divides it into 17 equal parts. She places a point at the mark that is 14 parts away from 0. Is this point closer to 0, 1/2, or 1? Answer: 1 Solution: The point is 14 parts from 0, so it is at 14/17. The benchmark 1/2 is halfway between 0 and 1. Since 1/2 = 8.5/17, it lies between the 8th and 9th marks.
   Full step-by-step solution

   Step 1: The number line is divided into 17 equal parts, so each part is 1/17. Step 2: The point is 14 parts from 0, so it is at 14/17. Step 3: The benchmark 1/2 is halfway between 0 and 1. Since 1/2 = 8.5/17, it lies between the 8th and 9th marks. Step 4: Distance from 14/17 to 0: 14/17 - 0 = 14/17. Step 5: Distance from 14/17 to 1/2: 14/17 - 8.5/17 = 5.5/17. Step 6: Distance from 14/17 to 1: 1 - 14/17 = 17/17 - 14/17 = 3/17. Step 7: Compare the distances: 3/17 (to 1) is smaller than 5.5/17 (to 1/2) and much smaller than 14/17 (to 0). So 14/17 is closest to 1. The answer is 1.

2. Liam baked a large pan of brownies and cut it into 8 equal pieces. His family ate 3 pieces after dinner. What fraction of the brownies did they eat? Answer: 3/8 Solution: Liam cut the pan of brownies into 8 equal pieces. This means the whole pan is represented by 8 pieces. Identify how many pieces were eaten.
   Full step-by-step solution

   Step 1: Understand the total number of equal pieces. Liam cut the pan of brownies into 8 equal pieces. This means the whole pan is represented by 8 pieces. So, the total is 8 out of 8 pieces, or the whole, which is represented by the fraction 8/8. Step 2: Identify how many pieces were eaten. The problem states that the family ate 3 pieces. Step 3: Determine the fraction that represents the eaten part. The fraction is calculated as the number of pieces eaten divided by the total number of pieces. So, the fraction is (pieces eaten) / (total pieces) = 3 / 8. Step 4: State the final answer. Therefore, the fraction of the brownies that the family ate is 3/8.

3. Emma is saving up to buy a new bike that costs $80. She has saved $35 so far. Is the fraction of the total cost that Emma has saved closer to 0, 1/2, or 1? Answer: closer to 1/2 Solution: The total cost of the bike is $80. The fraction saved is 35/80. Find half of the total: 80 divided by 2 equals 40.
   Full step-by-step solution

   Step 1: The total cost of the bike is $80. The fraction saved is 35/80. Step 2: Find half of the total: 80 divided by 2 equals 40. So 1/2 of the cost is $40, or 40/80. Step 3: Compare 35/80 to the benchmarks 0 (0/80), 1/2 (40/80), and 1 (80/80). Step 4: Calculate the distance from 35 to each benchmark: - Distance to 0: 35 - 0 = 35 - Distance to 1/2 (40): 40 - 35 = 5 - Distance to 1 (80): 80 - 35 = 45 Step 5: The smallest distance is 5, which is the distance to 40 (1/2). Step 6: Since $35 is only $5 away from $40, the fraction 35/80 is closest to 1/2. Answer: The fraction of the total cost Emma has saved is closer to 1/2.

4. Mason is filling a water bottle that holds 12 cups of water. He has already poured in 7 cups. Is the amount of water in the bottle closer to 0, 1/2, or 1 full bottle? Answer: closer to 1/2 Solution: The bottle holds 12 cups total. Mason has poured 7 cups, so the fraction filled is 7/12. Compare 7/12 to the benchmark 1/2.
   Full step-by-step solution

   Step 1: The bottle holds 12 cups total. Mason has poured 7 cups, so the fraction filled is 7/12. Step 2: Compare 7/12 to the benchmark 1/2. 1/2 of 12 is 6 cups. Since 7 is greater than 6, 7/12 is more than 1/2. Step 3: Compare 7/12 to the benchmark 1. 1 full bottle is 12/12. Since 7 is less than 12, 7/12 is less than 1. Step 4: Compare 7/12 to 0. 7/12 is much greater than 0. Step 5: The difference from 1/2 is 7/12 - 6/12 = 1/12. The difference from 1 is 12/12 - 7/12 = 5/12. The difference from 0 is 7/12 - 0 = 7/12. Step 6: The smallest difference is 1/12, so 7/12 is closest to 1/2. The answer is: closer to 1/2.

5. 2/3 + 1/6 = ? Answer: 5/6 Solution: We need to add the fractions 2/3 and 1/6. The denominators are 3 and 6. The least common multiple of 3 and 6 is 6.
   Full step-by-step solution

   Step 1: Identify the problem We need to add the fractions 2/3 and 1/6. Step 2: Find a common denominator The denominators are 3 and 6. The least common multiple of 3 and 6 is 6. So the common denominator is 6. Step 3: Convert 2/3 to have denominator 6 Multiply the numerator and denominator of 2/3 by 2: 2/3 = (2 × 2)/(3 × 2) = 4/6. Step 4: Rewrite the addition with common denominator 2/3 + 1/6 = 4/6 + 1/6. Step 5: Add the numerators Since the denominators are the same, add the numerators: 4 + 1 = 5. So, 4/6 + 1/6 = 5/6. Step 6: Simplify if possible 5/6 is already in simplest form because 5 and 6 have no common factors other than 1. Final Answer: 5/6

6. Mason draws a number line from 0 to 1 and divides it into 9 equal parts. He places a point at the mark that is 2 parts away from 0. Is this point closer to 0, 1/2, or 1? Answer: 0 Solution: The benchmark 1/2 is halfway. Since 1/2 = 4.5/9, it is between the 4th and 5th marks from 0. Compare the distance from 2/9 to 0: 2/9 - 0 = 2/9.
   Full step-by-step solution

   Step 1: The number line is divided into 9 equal parts, so each part is 1/9 of the whole. Step 2: The point is placed 2 parts from 0, so it is at 2/9 on the number line. Step 3: The benchmark 1/2 is halfway. Since 1/2 = 4.5/9, it is between the 4th and 5th marks from 0. Step 4: Compare the distance from 2/9 to 0: 2/9 - 0 = 2/9. Step 5: Compare the distance from 2/9 to 1/2: 1/2 - 2/9 = 9/18 - 4/18 = 5/18, which is about 0.278. Also 2/9 is about 0.222. Step 6: Since 2/9 (0.222) is closer to 0 than to 1/2 (0.5) or to 1 (1.0), the point is closer to 0. The answer is 0.

7. A rectangular vegetable garden is divided into 12 equal sections. The gardener plants corn in 5 sections, beans in 4 sections, and squash in the remaining sections. If you drew a diagram showing all 12 sections with different colors for each vegetable, what fraction of the garden is planted with squash? Answer: 1/4 Solution: Find the total number of sections used for corn and beans: 5 + 4 = 9 sections Find the number of sections left for squash: 12 - 9 = 3 sections Write the fraction of garden with squash: 3/12 Simplify the fraction: 3/12 = 1/4 The answer is 1/4.
   Full step-by-step solution

   Step 1: Find the total number of sections used for corn and beans: 5 + 4 = 9 sections Step 2: Find the number of sections left for squash: 12 - 9 = 3 sections Step 3: Write the fraction of garden with squash: 3/12 Step 4: Simplify the fraction: 3/12 = 1/4 The answer is 1/4.

8. A rectangular chocolate bar is divided into 16 equal squares arranged in 4 rows and 4 columns. If 3 squares are dark chocolate and 5 squares are milk chocolate, what fraction of the chocolate bar is white chocolate? Answer: 1/2 Solution: Find the total number of squares in the chocolate bar. There are 16 equal squares. Find the number of squares that are NOT white chocolate.
   Full step-by-step solution

   Step 1: Find the total number of squares in the chocolate bar. There are 16 equal squares. Step 2: Find the number of squares that are NOT white chocolate. Dark chocolate squares: 3. Milk chocolate squares: 5. Total non-white squares: 3 + 5 = 8. Step 3: Find the number of white chocolate squares. Total squares - non-white squares = 16 - 8 = 8. Step 4: Write the fraction of white chocolate squares. Number of white squares / total squares = 8/16. Step 5: Simplify the fraction. 8/16 = 1/2. The answer is 1/2.

9. A rectangular garden is divided into 8 equal sections. If 5 sections are planted with vegetables and the rest are planted with flowers, what fraction of the garden is planted with flowers? Answer: 3/8 Solution: The garden is divided into 8 equal sections. Determine how many sections are flowers. We are told 5 sections are planted with vegetables.
   Full step-by-step solution

   Step 1: Understand the problem. The garden is divided into 8 equal sections. So, the whole garden is represented by 8 sections. Step 2: Determine how many sections are flowers. We are told 5 sections are planted with vegetables. The rest are flowers. So, flowers = total sections - vegetable sections = 8 - 5 = 3 sections. Step 3: Write the fraction for flowers. Since there are 8 equal sections in total, and 3 of them are flowers, the fraction is: flowers / total sections = 3/8. Step 4: Conclusion. The fraction of the garden planted with flowers is 3/8.