Q1: Which of the following shows \(\frac{5}{8}\) decomposed into unit fractions? (a) \(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\) (b) \(\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}\) (c) \(\frac{2}{8} + \frac{3}{8}\) (d) \(\frac{5}{1} + \frac{5}{1} + \frac{5}{1} + \frac{5}{1} + \frac{5}{1}\)
Solution:
Ans: (a) Explanation: A unit fraction has 1 as the numerator. To decompose \(\frac{5}{8}\), we write it as five copies of \(\frac{1}{8}\). Option (b) uses the wrong denominator. Option (c) shows a sum but not unit fractions. Option (d) uses improper fractions with denominator 1, not 8.
Q2: What is another way to write \(\frac{3}{4}\)? (a) \(\frac{1}{4} + \frac{1}{4}\) (b) \(\frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3}\) (c) \(\frac{2}{4} + \frac{1}{4}\) (d) \(\frac{1}{2} + \frac{1}{3}\)
Solution:
Ans: (c) Explanation: We can decompose \(\frac{3}{4}\) into fractions with the same denominator that add up to \(\frac{3}{4}\). Here, \(\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\). Option (a) only equals \(\frac{2}{4}\). Option (b) uses the wrong denominator. Option (d) uses different denominators that don't add to \(\frac{3}{4}\).
Q3: Maria decomposed a fraction as \(\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\). What fraction did she start with? (a) \(\frac{1}{6}\) (b) \(\frac{4}{6}\) (c) \(\frac{6}{4}\) (d) \(\frac{1}{4}\)
Solution:
Ans: (b) Explanation: When we add four copies of \(\frac{1}{6}\), we get \(\frac{4}{6}\). The numerator tells us how many unit fractions we have, which is 4. Option (a) is just one unit fraction. Option (c) has the numerator and denominator reversed. Option (d) has the wrong denominator.
Ans: (b) Explanation: When decomposing a fraction, the parts must add up to the original fraction. Here, \(\frac{5}{10} + \frac{2}{10} = \frac{7}{10}\). Option (a) only adds to \(\frac{6}{10}\). Option (c) uses the wrong denominator and adds to more than 1. Option (d) uses different denominators that don't work.
Q5: Sam says \(\frac{6}{8} = \frac{3}{8} + \frac{3}{8}\). Is he correct? (a) Yes, because \(3 + 3 = 6\) and the denominators stay the same (b) No, because you cannot add fractions (c) No, because the denominators are different (d) Yes, but only if you change both fractions to sixths
Solution:
Ans: (a) Explanation: Sam is correct. When fractions have the same denominator, we add the numerators: \(3 + 3 = 6\), so \(\frac{3}{8} + \frac{3}{8} = \frac{6}{8}\). Option (b) is incorrect because we can add fractions. Option (c) is wrong because the denominators are the same. Option (d) is unnecessary because the fractions already work as eighths.
Q6: Which shows a correct way to decompose \(\frac{5}{6}\) into three parts? (a) \(\frac{1}{6} + \frac{2}{6} + \frac{2}{6}\) (b) \(\frac{2}{6} + \frac{2}{6} + \frac{2}{6}\) (c) \(\frac{1}{3} + \frac{1}{3} + \frac{1}{3}\) (d) \(\frac{1}{6} + \frac{1}{6} + \frac{1}{6}\)
Solution:
Ans: (a) Explanation: The three parts must add to \(\frac{5}{6}\). Here, \(\frac{1}{6} + \frac{2}{6} + \frac{2}{6} = \frac{5}{6}\). Option (b) adds to \(\frac{6}{6}\), which is too much. Option (c) equals 1 whole, not \(\frac{5}{6}\). Option (d) only adds to \(\frac{3}{6}\).
Q7: A fraction is decomposed as \(\frac{2}{5} + \frac{3}{5}\). What is the original fraction? (a) \(\frac{5}{5}\) (b) \(\frac{5}{10}\) (c) \(\frac{2}{3}\) (d) \(\frac{6}{5}\)
Solution:
Ans: (a) Explanation: When we add fractions with the same denominator, we add the numerators: \(2 + 3 = 5\), so \(\frac{2}{5} + \frac{3}{5} = \frac{5}{5}\), which equals 1 whole. Option (b) has the wrong denominator. Option (c) is incorrect arithmetic. Option (d) multiplies instead of adding numerators.
Q8: Which statement about decomposing fractions is true? (a) You can only decompose fractions into two parts (b) When decomposing, all the parts must have the same denominator as the original fraction (c) The numerator must always stay the same when you decompose (d) You can only decompose unit fractions
Solution:
Ans: (b) Explanation: When decomposing a fraction, the parts must have the same denominator as the original so they can add correctly. Option (a) is wrong because you can decompose into any number of parts. Option (c) is incorrect because the numerator is split up. Option (d) is false because any fraction can be decomposed.
Section B: Fill in the Blanks
Q9: A fraction that has 1 as its numerator is called a __________ fraction.
Solution:
Ans: unit Explanation: A unit fraction is a fraction with a numerator of 1, such as \(\frac{1}{2}\), \(\frac{1}{3}\), or \(\frac{1}{5}\). These are the building blocks for decomposing other fractions.
Q10: To decompose \(\frac{4}{5}\) into unit fractions, you would write it as \(\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \) __________.
Solution:
Ans: \(\frac{1}{5}\) Explanation: Since \(\frac{4}{5}\) means four copies of \(\frac{1}{5}\), we need four unit fractions: \(\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{4}{5}\).
Q11: The fraction \(\frac{3}{8}\) can be decomposed as \(\frac{1}{8} + \) __________.
Solution:
Ans: \(\frac{2}{8}\) Explanation: We need parts that add to \(\frac{3}{8}\). Since \(\frac{1}{8} + \frac{2}{8} = \frac{3}{8}\), the missing part is \(\frac{2}{8}\). The numerators add up: \(1 + 2 = 3\).
Q12: When you add \(\frac{4}{10} + \frac{3}{10}\), you get __________.
Solution:
Ans: \(\frac{7}{10}\) Explanation: To add fractions with the same denominator, add the numerators and keep the denominator: \(4 + 3 = 7\), so \(\frac{4}{10} + \frac{3}{10} = \frac{7}{10}\).
Q13: The fraction \(\frac{6}{12}\) decomposed into two equal parts is \(\frac{3}{12} + \) __________.
Solution:
Ans: \(\frac{3}{12}\) Explanation: To split \(\frac{6}{12}\) into two equal parts, we divide the numerator by 2: \(6 ÷ 2 = 3\). So each part is \(\frac{3}{12}\), and \(\frac{3}{12} + \frac{3}{12} = \frac{6}{12}\).
Q14: If \(\frac{2}{7} + \frac{3}{7} + \frac{1}{7} = \frac{n}{7}\), then \(n = \) __________.
Solution:
Ans: 6 Explanation: When adding fractions with the same denominator, we add the numerators: \(2 + 3 + 1 = 6\). Therefore, \(n = 6\) and the sum is \(\frac{6}{7}\).
Section C: Word Problems
Q15: Emma ate \(\frac{2}{8}\) of a pizza and her brother ate \(\frac{3}{8}\) of the same pizza. What fraction of the pizza did they eat together?
Solution:
Ans:
To find the total, we add the two fractions:
\(\frac{2}{8} + \frac{3}{8} = \frac{2+3}{8} = \frac{5}{8}\) Final Answer: \(\frac{5}{8}\) of the pizza
Q16: A ribbon is \(\frac{7}{12}\) yard long. Jake wants to decompose this length into three pieces where two pieces are equal and one piece is \(\frac{1}{12}\) yard. What is the length of each of the two equal pieces?
Solution:
Ans:
The total length is \(\frac{7}{12}\) yard.
One piece is \(\frac{1}{12}\) yard.
Remaining length: \(\frac{7}{12} - \frac{1}{12} = \frac{6}{12}\)
This must be split into two equal pieces: \(\frac{6}{12} ÷ 2 = \frac{3}{12}\) Final Answer: Each equal piece is \(\frac{3}{12}\) yard
Q17: Marcus is filling a bottle with water. He pours \(\frac{3}{10}\) liter first, then \(\frac{4}{10}\) liter more. How much water is in the bottle now?
Solution:
Ans:
Add the two amounts of water:
\(\frac{3}{10} + \frac{4}{10} = \frac{3+4}{10} = \frac{7}{10}\) Final Answer: \(\frac{7}{10}\) liter
Q18: A recipe calls for \(\frac{5}{6}\) cup of flour. Lily wants to add the flour in three separate scoops. She uses \(\frac{2}{6}\) cup and then \(\frac{2}{6}\) cup. How much more flour does she need to add?
Solution:
Ans:
Amount already added: \(\frac{2}{6} + \frac{2}{6} = \frac{4}{6}\)
Amount still needed: \(\frac{5}{6} - \frac{4}{6} = \frac{1}{6}\) Final Answer: \(\frac{1}{6}\) cup
Q19: A garden path is divided into equal sections. Tom painted \(\frac{1}{5}\) of the path in the morning and \(\frac{3}{5}\) of the path in the afternoon. What fraction of the path did Tom paint in all? Did he finish painting the entire path?
Solution:
Ans:
Total painted: \(\frac{1}{5} + \frac{3}{5} = \frac{4}{5}\)
The entire path would be \(\frac{5}{5}\), which equals 1 whole.
Since \(\frac{4}{5} < \frac{5}{5}\),="" tom="" did="" not=""> Final Answer: Tom painted \(\frac{4}{5}\) of the path. No, he did not finish painting the entire path.
Q20: Sophia has a chocolate bar divided into 12 equal pieces. She gave \(\frac{3}{12}\) to her friend and kept \(\frac{5}{12}\) for herself. She wants to give the rest to her sister. What fraction of the chocolate bar will her sister get?
Solution:
Ans:
Total given away and kept: \(\frac{3}{12} + \frac{5}{12} = \frac{8}{12}\)
Whole chocolate bar: \(\frac{12}{12}\)
Amount left for sister: \(\frac{12}{12} - \frac{8}{12} = \frac{4}{12}\) Final Answer: \(\frac{4}{12}\) of the chocolate bar
The document Worksheet (with Solutions): Decomposing Fractions is a part of the Grade 4 Course Math Grade 4.
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