Fractions — Questions and Answers
Welcome to HSLC Guru. This page gives complete, step-by-step solutions to every Work it Out exercise (7.1 to 7.16) of the ASSEB Class 6 New Mathematics chapter Fractions, along with a set of extra practice questions in simple English.
Summary
When one whole (a unit) is divided into several equal parts, each part is a fraction. When the unit is split into equal parts and we take just one part, it is a unit fraction or fractional unit — such as $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$. The number above the fraction bar is the numerator and the number below it is the denominator.
This chapter shows how to represent fractions on the number line; how a fraction is called a proper fraction when the numerator is smaller than the denominator and an improper fraction when the numerator is greater; how an improper fraction can be written as a mixed fraction (a whole number plus a fractional part); how equivalent fractions represent equal parts; how to compare fractions; and how to reduce a fraction to its simplest (lowest) form.
To compare, add or subtract fractions with the same denominator we work with the numerators only; for different denominators we first find a common multiple of the denominators and convert each fraction to an equivalent fraction with that common denominator. The chapter ends with Brahmagupta’s rules for addition and subtraction and a note on fractions in ancient Indian mathematics.
Summary: This page gives complete, step-by-step answers to every Work it Out exercise (7.1 to 7.16) of ASSEB Class 6 New Mathematics Chapter 7, Fractions (ভগ্নাংশ), covering unit fractions and fractional units, representing fractions on the number line, proper, improper and mixed fractions, comparing fractions with same and different denominators, equivalent fractions and the simplest (lowest) form, and the addition and subtraction of fractions, along with Brahmagupta’s rules and fractions in ancient Indian mathematics, plus extra MCQs, fill in the blanks, true or false and short answer questions.
Textbook Questions and Answers
Work it Out 7.1
1. A cake is divided into 8 pieces as shown in the figure. Are each pieces of same size? Explain why or why not.
Answer: The pieces are of the same size only if the cake is divided into 8 equal parts. Only then does each piece represent the fractional unit $\frac{1}{8}$. If the cuts do not make equal parts, the pieces are not the same size and none of them can be called $\frac{1}{8}$. So, to get a fractional unit, all 8 pieces must be exactly equal.
2. Try to draw different figures that divide the above cake into 8 equal pieces.
Answer: There are several ways to get 8 equal pieces — (a) cut a square cake into 2 equal rows and 4 equal columns to make 2 × 4 = 8 equal blocks; (b) first cut it into 4 equal strips with parallel lines and then cut each strip in half to get 8; (c) cut a circular cake from the centre into 8 equal sectors (like slices of a pizza). In every case, if the 8 pieces are equal, each piece is $\frac{1}{8}$.
Work it Out 7.2
1. Fill the appropriate measurement of the sugarcane using the fractional unit. (fractional unit $\frac{1}{8}$)
Answer: Each piece is $\frac{1}{8}$ unit long, so—
- 2 times $\frac{1}{8}$ = $\frac{2}{8}$ unit
- 3 times $\frac{1}{8}$ = $\frac{3}{8}$ unit
- 4 times $\frac{1}{8}$ = $\frac{4}{8}$ unit
- 5 times $\frac{1}{8}$ = $\frac{5}{8}$ unit
- 6 times $\frac{1}{8}$ = $\frac{6}{8}$ unit
- 7 times $\frac{1}{8}$ = $\frac{7}{8}$ unit
- 8 times $\frac{1}{8}$ = $\frac{8}{8}$ = 1 unit
Work it Out 7.3
1. Determine the numerator and denominator of the fractions $\frac{7}{9}, \frac{3}{8}, \frac{11}{15}$ and $\frac{4}{13}$.
| Fraction | Numerator | Denominator |
|---|---|---|
| $\frac{7}{9}$ | 7 | 9 |
| $\frac{3}{8}$ | 3 | 8 |
| $\frac{11}{15}$ | 11 | 15 |
| $\frac{4}{13}$ | 4 | 13 |
2. Write the above fractions in terms of fractional units.
Answer: $\frac{7}{9} = 7 \times \frac{1}{9}$, $\frac{3}{8} = 3 \times \frac{1}{8}$, $\frac{11}{15} = 11 \times \frac{1}{15}$, $\frac{4}{13} = 4 \times \frac{1}{13}$.
Work it Out 7.4
On the number line, divide the unit length between 0 and 1 into equal parts, count how many parts the marked section covers, and write the fraction. For example, in the number line below the unit length is divided into three equal parts and $\frac{2}{3}$ is marked—
Fill the box with appropriate fractions.
Answer: Reading each marked section from the figures—
- 1. The unit length is divided into 4 equal parts and the marked section reaches the 3rd mark. So the marked fraction = $\frac{3}{4}$.
- 2. The unit length is divided into 6 equal parts; the upper mark reaches the 2nd division ($\frac{2}{6} = \frac{1}{3}$) and the lower mark reaches the 5th division ($\frac{5}{6}$).
- 3. The unit length is divided into 8 equal parts; the upper mark reaches the 3rd division ($\frac{3}{8}$) and the lower mark reaches the 4th division ($\frac{4}{8} = \frac{1}{2}$).
4. Suppose a unit on a number line is divided into 9 equal parts. Figure out the different possible fractions and mark them in your notebook.
Answer: If the unit is divided into 9 equal parts, each part is $\frac{1}{9}$. So the marks between 0 and 1 give $\frac{1}{9}, \frac{2}{9}, \frac{3}{9}, \frac{4}{9}, \frac{5}{9}, \frac{6}{9}, \frac{7}{9}, \frac{8}{9}, \frac{9}{9}(=1)$. Going beyond 1 we get $\frac{10}{9}, \frac{11}{9}, \ldots, \frac{18}{9}(=2)$.
Work it Out 7.5
1. Write some proper and improper fractions.
Answer: When the numerator is smaller than the denominator the fraction is proper; when the numerator is greater than (or equal to) the denominator it is improper.
- Proper fractions: $\frac{1}{2}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{8}$
- Improper fractions: $\frac{5}{3}, \frac{7}{4}, \frac{9}{5}, \frac{11}{6}, \frac{8}{8}$
Work it Out 7.6
1. Find out the number of whole units in the improper fractions — (a) $\frac{10}{3}$ (b) $\frac{13}{4}$ (c) $\frac{17}{5}$ (d) $\frac{12}{7}$ (e) $\frac{14}{11}$
Answer: Divide the numerator by the denominator; the quotient is the number of whole units—
- (a) $\frac{10}{3}$: 10 ÷ 3 = 3 remainder 1 → 3 whole units
- (b) $\frac{13}{4}$: 13 ÷ 4 = 3 remainder 1 → 3 whole units
- (c) $\frac{17}{5}$: 17 ÷ 5 = 3 remainder 2 → 3 whole units
- (d) $\frac{12}{7}$: 12 ÷ 7 = 1 remainder 5 → 1 whole unit
- (e) $\frac{14}{11}$: 14 ÷ 11 = 1 remainder 3 → 1 whole unit
2. Write the following fractions as a mixed fraction (e.g. $\frac{10}{3} = 3\frac{1}{3}$) — (a) $\frac{11}{5}$ (b) $\frac{19}{7}$ (c) $\frac{46}{11}$ (d) $\frac{14}{13}$
- (a) $\frac{11}{5}$: 11 = 5 × 2 + 1, so $\frac{11}{5} = 2\frac{1}{5}$
- (b) $\frac{19}{7}$: 19 = 7 × 2 + 5, so $\frac{19}{7} = 2\frac{5}{7}$
- (c) $\frac{46}{11}$: 46 = 11 × 4 + 2, so $\frac{46}{11} = 4\frac{2}{11}$
- (d) $\frac{14}{13}$: 14 = 13 × 1 + 1, so $\frac{14}{13} = 1\frac{1}{13}$
Work it Out 7.7
1. Convert the following mixed fractions into improper fractions — $7\frac{1}{6}, 4\frac{3}{5}, 7\frac{1}{4}, 8\frac{3}{10}$ and $6\frac{4}{7}$
Answer: Rule: (whole number × denominator) + numerator, over the denominator—
- $7\frac{1}{6} = \frac{7 \times 6 + 1}{6} = \frac{43}{6}$
- $4\frac{3}{5} = \frac{4 \times 5 + 3}{5} = \frac{23}{5}$
- $7\frac{1}{4} = \frac{7 \times 4 + 1}{4} = \frac{29}{4}$
- $8\frac{3}{10} = \frac{8 \times 10 + 3}{10} = \frac{83}{10}$
- $6\frac{4}{7} = \frac{6 \times 7 + 4}{7} = \frac{46}{7}$
Work it Out 7.8
1. Compare the fractions (same denominator, so compare only the numerators) — (i) $\frac{7}{6}, \frac{5}{6}$ (ii) $\frac{4}{8}, \frac{5}{8}$ (iii) $\frac{2}{9}, \frac{1}{9}$
- (i) 7 > 5, so $\frac{7}{6} > \frac{5}{6}$
- (ii) 4 < 5, so $\frac{4}{8} < \frac{5}{8}$
- (iii) 2 > 1, so $\frac{2}{9} > \frac{1}{9}$
Work it Out 7.9
1. Compare the following pairs of fractions — (i) $\frac{7}{3}, \frac{5}{8}$ (ii) $\frac{4}{9}, \frac{7}{10}$ (iii) $\frac{6}{5}, \frac{8}{13}$ (iv) $3\frac{4}{7}, 4\frac{2}{5}$
- (i) Common multiple of 3 and 8 is 24; $\frac{7}{3} = \frac{56}{24}$, $\frac{5}{8} = \frac{15}{24}$. 56 > 15, so $\frac{7}{3} > \frac{5}{8}$.
- (ii) Common multiple of 9 and 10 is 90; $\frac{4}{9} = \frac{40}{90}$, $\frac{7}{10} = \frac{63}{90}$. 40 < 63, so $\frac{4}{9} < \frac{7}{10}$.
- (iii) Common multiple of 5 and 13 is 65; $\frac{6}{5} = \frac{78}{65}$, $\frac{8}{13} = \frac{40}{65}$. 78 > 40, so $\frac{6}{5} > \frac{8}{13}$.
- (iv) Comparing the whole parts, 3 < 4, so $3\frac{4}{7} < 4\frac{2}{5}$ (since $\frac{25}{7} \approx 3.57$ and $\frac{22}{5} = 4.4$).
2. Arrange the following fractions in ascending order — (i) $\frac{6}{4}, \frac{17}{12}, \frac{11}{8}, 1\frac{6}{7}$ (ii) $\frac{8}{9}, \frac{5}{6}, \frac{14}{15}, 1\frac{4}{11}$
- (i) Taking common denominator 168: $\frac{6}{4} = \frac{252}{168}$, $\frac{17}{12} = \frac{238}{168}$, $\frac{11}{8} = \frac{231}{168}$, $1\frac{6}{7} = \frac{13}{7} = \frac{312}{168}$. Ascending: $\frac{11}{8} < \frac{17}{12} < \frac{6}{4} < 1\frac{6}{7}$.
- (ii) Values: $\frac{5}{6} \approx 0.83$, $\frac{8}{9} \approx 0.89$, $\frac{14}{15} \approx 0.93$, $1\frac{4}{11} = \frac{15}{11} \approx 1.36$. Ascending: $\frac{5}{6} < \frac{8}{9} < \frac{14}{15} < 1\frac{4}{11}$.
3. Arrange the following fractions in descending order — (i) $\frac{16}{15}, \frac{7}{10}, \frac{4}{5}, 1\frac{2}{13}$ (ii) $\frac{5}{7}, \frac{11}{14}, \frac{14}{21}, 2\frac{3}{7}$
- (i) Values: $1\frac{2}{13} = \frac{15}{13} \approx 1.15$, $\frac{16}{15} \approx 1.07$, $\frac{4}{5} = 0.8$, $\frac{7}{10} = 0.7$. Descending: $1\frac{2}{13} > \frac{16}{15} > \frac{4}{5} > \frac{7}{10}$.
- (ii) Values: $2\frac{3}{7} = \frac{17}{7} \approx 2.43$, $\frac{11}{14} \approx 0.79$, $\frac{5}{7} \approx 0.71$, $\frac{14}{21} = \frac{2}{3} \approx 0.67$. Descending: $2\frac{3}{7} > \frac{11}{14} > \frac{5}{7} > \frac{14}{21}$.
Work it Out 7.10
1. Represent the fractions $\frac{1}{2}, \frac{3}{6}$ and $\frac{6}{12}$ with the help of a circular figure.
Answer: Divide one circle into 2 parts and shade 1 part to get $\frac{1}{2}$; divide an equal circle into 6 parts and shade 3 to get $\frac{3}{6}$; divide another into 12 parts and shade 6 to get $\frac{6}{12}$. In all three figures exactly half of the circle is shaded.
2. Do the fractions $\frac{1}{2}, \frac{3}{6}$ and $\frac{6}{12}$ represent equal parts?
Answer: Yes. Each of them represents exactly half of the circle, so they represent equal parts.
3. Are $\frac{1}{2}, \frac{3}{6}$ and $\frac{6}{12}$ equivalent fractions?
Answer: Yes, they are equivalent fractions because each has the value $\frac{1}{2}$.
4. Determine whether $\frac{2}{6}, \frac{4}{12}$ and $\frac{8}{24}$ are equivalent fractions.
Answer: $\frac{2}{6} = \frac{1}{3}$, $\frac{4}{12} = \frac{1}{3}$, $\frac{8}{24} = \frac{1}{3}$. All three equal $\frac{1}{3}$, so they are equivalent fractions.
5. Write three equivalent fractions for $\frac{3}{4}$.
Answer: Multiplying numerator and denominator by the same number — $\frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{12}{16}$.
Work it Out 7.11
1. Find three equivalent fractions for each of the following — (i) $\frac{3}{2}$ (ii) $\frac{7}{9}$ (iii) $\frac{4}{7}$ (iv) $\frac{5}{8}$
- (i) $\frac{3}{2} = \frac{6}{4} = \frac{9}{6} = \frac{12}{8}$
- (ii) $\frac{7}{9} = \frac{14}{18} = \frac{21}{27} = \frac{28}{36}$
- (iii) $\frac{4}{7} = \frac{8}{14} = \frac{12}{21} = \frac{16}{28}$
- (iv) $\frac{5}{8} = \frac{10}{16} = \frac{15}{24} = \frac{20}{32}$
Work it Out 7.12
1. Reduce the following fractions to their simplest (lowest) form — (i) $\frac{45}{60}$ (ii) $\frac{90}{150}$ (iii) $\frac{48}{72}$ (iv) $\frac{32}{80}$ (v) $\frac{120}{465}$
- (i) $\frac{45}{60} = \frac{45 \div 15}{60 \div 15} = \frac{3}{4}$
- (ii) $\frac{90}{150} = \frac{90 \div 30}{150 \div 30} = \frac{3}{5}$
- (iii) $\frac{48}{72} = \frac{48 \div 24}{72 \div 24} = \frac{2}{3}$
- (iv) $\frac{32}{80} = \frac{32 \div 16}{80 \div 16} = \frac{2}{5}$
- (v) $\frac{120}{465} = \frac{120 \div 15}{465 \div 15} = \frac{8}{31}$
Work it Out 7.13
1. Consider the following figures and find out the appropriate fraction.
Answer (i): Each square is divided into 8 equal triangles. In the first square the top half, that is 4 triangles, is shaded ($\frac{4}{8}$) and in the second square 6 triangles are shaded ($\frac{6}{8}$). So—
$$\frac{4}{8} + \frac{6}{8} = \frac{10}{8} = 1\frac{2}{8} = 1\frac{1}{4}$$
Answer (ii): Each circle is divided into 12 equal sectors. In the first circle 4 sectors are shaded ($\frac{4}{12}$) and in the second circle 9 sectors are shaded ($\frac{9}{12}$). So—
$$\frac{4}{12} + \frac{9}{12} = \frac{13}{12} = 1\frac{1}{12}$$
Work it Out 7.14
1. Find the value —
- (i) $\frac{3}{4} + \frac{1}{3} = \frac{9}{12} + \frac{4}{12} = \frac{13}{12} = 1\frac{1}{12}$
- (ii) $\frac{2}{3} + \frac{5}{6} = \frac{4}{6} + \frac{5}{6} = \frac{9}{6} = \frac{3}{2} = 1\frac{1}{2}$
- (iii) $\frac{2}{3} + \frac{4}{5} + \frac{3}{7} = \frac{70}{105} + \frac{84}{105} + \frac{45}{105} = \frac{199}{105} = 1\frac{94}{105}$
- (iv) $\frac{9}{2} + \frac{5}{3} + \frac{7}{6} = \frac{27}{6} + \frac{10}{6} + \frac{7}{6} = \frac{44}{6} = \frac{22}{3} = 7\frac{1}{3}$
- (v) $\frac{1}{7} + \frac{2}{7} + \frac{3}{7} = \frac{6}{7}$
- (vi) $\frac{3}{5} + \frac{4}{3} + \frac{7}{10} = \frac{18}{30} + \frac{40}{30} + \frac{21}{30} = \frac{79}{30} = 2\frac{19}{30}$
Work it Out 7.15
1. Find the value (same denominator, so subtract the numerators) —
- (i) $\frac{7}{10} – \frac{4}{10} = \frac{3}{10}$
- (ii) $\frac{6}{11} – \frac{3}{11} = \frac{3}{11}$
- (iii) $\frac{11}{13} – \frac{5}{13} = \frac{6}{13}$
- (iv) $\frac{5}{17} – \frac{2}{17} = \frac{3}{17}$
- (v) $\frac{9}{22} – \frac{2}{22} = \frac{7}{22}$
Work it Out 7.16
1. Find the value (different denominators, so use a common multiple and convert to equivalent fractions before subtracting) —
- (i) $\frac{2}{5} – \frac{3}{15} = \frac{6}{15} – \frac{3}{15} = \frac{3}{15} = \frac{1}{5}$
- (ii) $\frac{5}{6} – \frac{4}{9} = \frac{15}{18} – \frac{8}{18} = \frac{7}{18}$
- (iii) $\frac{10}{3} – \frac{13}{4} = \frac{40}{12} – \frac{39}{12} = \frac{1}{12}$
- (iv) $\frac{2}{5} – \frac{3}{11} = \frac{22}{55} – \frac{15}{55} = \frac{7}{55}$
- (v) $\frac{7}{8} – \frac{3}{5} = \frac{35}{40} – \frac{24}{40} = \frac{11}{40}$
- (vi) $\frac{11}{20} – \frac{1}{6} = \frac{33}{60} – \frac{10}{60} = \frac{23}{60}$
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. What is the denominator of the fraction $\frac{3}{7}$? (a) 3 (b) 7 (c) 10 (d) 4
Answer: (b) 7
2. Which of the following is a proper fraction? (a) $\frac{7}{4}$ (b) $\frac{5}{5}$ (c) $\frac{3}{8}$ (d) $\frac{9}{2}$
Answer: (c) $\frac{3}{8}$
3. The fraction $\frac{4}{3}$ is a — (a) proper fraction (b) improper fraction (c) equivalent fraction (d) unit fraction
Answer: (b) improper fraction
4. An equivalent fraction of $\frac{2}{3}$ is — (a) $\frac{3}{2}$ (b) $\frac{4}{6}$ (c) $\frac{2}{6}$ (d) $\frac{6}{3}$
Answer: (b) $\frac{4}{6}$
5. The value of $\frac{5}{8} + \frac{2}{8}$ is — (a) $\frac{7}{16}$ (b) $\frac{7}{8}$ (c) $\frac{3}{8}$ (d) $\frac{10}{8}$
Answer: (b) $\frac{7}{8}$
6. The simplest form of $\frac{45}{60}$ is — (a) $\frac{3}{4}$ (b) $\frac{9}{12}$ (c) $\frac{5}{6}$ (d) $\frac{15}{20}$
Answer: (a) $\frac{3}{4}$
7. Written as a mixed fraction, $\frac{11}{5}$ is — (a) $2\frac{1}{5}$ (b) $1\frac{6}{5}$ (c) $5\frac{1}{11}$ (d) $2\frac{3}{5}$
Answer: (a) $2\frac{1}{5}$
8. To compare two fractions with the same denominator, we compare the — (a) denominators (b) numerators (c) both (d) neither
Answer: (b) numerators
9. The fractional unit of the fraction $\frac{7}{9}$ is — (a) $\frac{1}{7}$ (b) $\frac{1}{9}$ (c) $\frac{7}{1}$ (d) $\frac{9}{7}$
Answer: (b) $\frac{1}{9}$
10. The value of $\frac{6}{6}$ is — (a) 0 (b) 6 (c) 1 (d) $\frac{1}{6}$
Answer: (c) 1
Fill in the Blanks
- The number above the fraction bar is called the ______. (numerator)
- The number below the fraction bar is called the ______. (denominator)
- A fraction whose numerator is greater than its denominator is called an ______ fraction. (improper)
- $\frac{1}{2}, \frac{2}{4}, \frac{4}{8}$ are ______ fractions. (equivalent)
- When 1 is the only common factor of the numerator and denominator, the fraction is in its ______ form. (simplest / lowest)
True or False
- $\frac{3}{4}$ is a proper fraction. — True
- $\frac{9}{5} > \frac{8}{5}$. — True
- $\frac{2}{3}$ and $\frac{4}{6}$ are not equivalent. — False (both equal $\frac{2}{3}$)
- A mixed fraction contains a whole number and a fractional part. — True
- To add fractions with the same denominator, the denominators must also be added. — False (the denominator stays the same; only the numerators are added)
Short Answer Questions
1. What is the difference between a proper and an improper fraction?
Answer: In a proper fraction the numerator is smaller than the denominator (e.g. $\frac{3}{8}$). In an improper fraction the numerator is greater than (or equal to) the denominator (e.g. $\frac{9}{5}$).
2. Write three equivalent fractions of $\frac{2}{3}$.
Answer: $\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12}$.
3. Find the value of $\frac{7}{8} – \frac{3}{8}$.
Answer: $\frac{7}{8} – \frac{3}{8} = \frac{4}{8} = \frac{1}{2}$.
4. What is a unit fraction (fractional unit)?
Answer: When a unit is divided into several equal parts, each part is called a unit fraction or fractional unit. Its numerator is always 1 — for example $\frac{1}{2}, \frac{1}{5}, \frac{1}{9}$.
Key Terms
| Term | Meaning |
|---|---|
| Fraction | One or more equal parts of a whole |
| Numerator | The number above the fraction bar |
| Denominator | The number below the fraction bar |
| Unit fraction | A fraction whose numerator is 1 (e.g. $\frac{1}{5}$) |
| Number line | A straight line on which numbers are represented |
| Proper fraction | A fraction whose numerator is smaller than its denominator |
| Improper fraction | A fraction whose numerator is greater than its denominator |
| Mixed fraction | A number with a whole number part and a fractional part |
| Equivalent fractions | Different fractions that represent equal parts |
| Simplest form | The form in which the only common factor of numerator and denominator is 1 |
| Common multiple | A number that is a multiple of two or more given numbers |