GSEB Class 7 Maths Solutions Chapter 2 અપૂર્ણાંક અને દશાંશ સંખ્યાઓ Exercise 2.2

Question 1. Match the given expressions with the appropriate diagrams from (a) to (d):
(i) \( 2 \times \frac {1}{5} \)
(ii) \( 2 \times \frac {1}{2} \)
(iii) \( 3 \times \frac {2}{3} \)
(iv) \( 3 \times \frac {1}{4} \)
Answer:
(i) \( 2 \times \frac {1}{5} = \frac{1}{5}+\frac{1}{5} \) can be shown as two one-fifth parts. Thus, this information is represented by figure (d). Therefore, (i) → (d).
(ii) \( 2 \times \frac {1}{2} = \frac{1}{2}+\frac{1}{2} \) can be shown as two one-half parts. Thus, this information is represented by figure (b). Therefore, (ii) → (b).
(iii) \( 3 \times \frac {2}{3} = \frac{2}{3}+\frac{2}{3}+\frac{2}{3} \) can be shown as three two-thirds parts. Thus, this information is represented by figure (a). Therefore, (iii) → (a).
(iv) \( 3 \times \frac {1}{4} = \frac{1}{4}+\frac{1}{4}+\frac{1}{4} \) can be shown as three one-fourth parts. Thus, this information is represented by figure (c). Therefore, (iv) → (c).
In simple words: We need to match each multiplication expression with the drawing that shows that many fractional parts. For example, two one-fifths matches diagram (d) because it shows two groups, each with one-fifth shaded.

Exam Tip: Remember that \( n \times \frac{a}{b} \) means adding the fraction \( \frac{a}{b} \) to itself \( n \) times, and diagrams illustrate this by showing \( n \) identical fractional parts.

Question 2. Some figures (a) to (c) are given below. Select the corresponding answer from (i), (ii), (iii):
(i) \( 3 \times \frac {1}{5} = \frac{3}{5} \)
(ii) \( 2 \times \frac {1}{3} = \frac{2}{3} \)
(iii) \( 3 \times \frac {3}{4} = 2\frac{1}{4} \)
Answer:
(i) \( 3 \times \frac {1}{5} = \frac{1}{5}+\frac{1}{5}+\frac{1}{5} \) can be written this way. This information is shown by figure (c). Therefore, (i) → (c).
(ii) \( 2 \times \frac {1}{3} = \frac{1}{3}+\frac{1}{3} \) can be written this way. This information is shown by figure (a). Therefore, (ii) → (a).
(iii) \( 3 \times \frac {3}{4}=\frac{3}{4}+\frac{3}{4}+\frac{3}{4}=\frac{9}{4}=2\frac{1}{4} \) can be written this way. This information is shown by figure (b). Therefore, (iii) → (b).
In simple words: We match each multiplication calculation to the visual drawing that shows the same fraction. For instance, three groups of one-fifth shaded in total make three-fifths.

Exam Tip: Always analyze the figures first to understand what fractional multiplication they represent, then compare with the given expressions.

Question 3. Multiply and convert to the simplest form and express as a mixed fraction:
(i) \( 7 \times \frac {3}{5} \)
Answer:
(i) \( 7 \times \frac {3}{5} \)
\( = \frac{7 \times 3}{5} \)
\( = \frac {21}{5} \)
\( = 4\frac {1}{5} \)
In simple words: To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same. Then, we change the improper fraction into a mixed number.

Exam Tip: Remember to always simplify the fraction to its lowest terms and convert it to a mixed number if it's an improper fraction.

Question 3. (ii) \( 4 \times \frac {1}{3} \)
Answer:
(ii) \( 4 \times \frac {1}{3} \)
\( = \frac{4 \times 1}{3} \)
\( = \frac {4}{3} \)
\( = 1\frac {1}{3} \)
In simple words: We multiply the whole number 4 by the numerator 1 to get 4, keeping the denominator 3. Then, we convert the improper fraction \( \frac{4}{3} \) to the mixed number \( 1\frac{1}{3} \).

Exam Tip: Always check if the final answer can be simplified further or converted into a mixed fraction if the numerator is larger than the denominator.

Question 3. (iii) \( 2 \times \frac {6}{7} \)
Answer:
(iii) \( 2 \times \frac {6}{7} \)
\( = \frac{2 \times 6}{7} \)
\( = \frac {12}{7} \)
\( = 1\frac {5}{7} \)
In simple words: Multiply 2 by 6 to get 12, keeping the denominator 7. Then, change the fraction \( \frac{12}{7} \) into a mixed number, which is \( 1\frac{5}{7} \).

Exam Tip: Make sure to perform the multiplication correctly and then divide to find the whole number and remainder for the mixed fraction.

Question 3. (iv) \( 5 \times \frac {2}{9} \)
Answer:
(iv) \( 5 \times \frac {2}{9} \)
\( = \frac{5 \times 2}{9} \)
\( = \frac {10}{9} \)
\( = 1\frac {1}{9} \)
In simple words: Multiply 5 by the numerator 2 to get 10, keeping the denominator 9. Convert the improper fraction \( \frac{10}{9} \) to the mixed number \( 1\frac{1}{9} \).

Exam Tip: Always show each step of your calculation clearly: multiplication of numerator, result as an improper fraction, and finally the mixed fraction.

Question 3. (v) \( \frac {2}{3} \times 4 \)
Answer:
(v) \( \frac {2}{3} \times 4 \)
\( = \frac{2 \times 4}{3} \)
\( = \frac {8}{3} \)
\( = 2\frac {2}{3} \)
In simple words: Multiply the numerator 2 by the whole number 4, which gives 8, while keeping the denominator 3. Then, change the improper fraction \( \frac{8}{3} \) into a mixed number, which is \( 2\frac{2}{3} \).

Exam Tip: The order of multiplication does not matter; \( \frac{a}{b} \times c \) is the same as \( c \times \frac{a}{b} \).

Question 3. (vi) \( \frac {5}{2} \times 6 \)
Answer:
(vi) \( \frac {5}{2} \times 6 \)
\( = \frac{5 \times 6}{2} \)
\( = \frac {30}{2} \)
\( = 15 \)
In simple words: Multiply the numerator 5 by the whole number 6 to get 30. Keep the denominator as 2. Then, divide 30 by 2, which equals 15.

Exam Tip: Look for opportunities to simplify before multiplying (e.g., dividing 6 by 2 before multiplying by 5) to make calculations easier.

Question 3. (vii) \( 11 \times \frac {4}{7} \)
Answer:
(vii) \( 11 \times \frac {4}{7} \)
\( = \frac{11 \times 4}{7} \)
\( = \frac {44}{7} \)
\( = 6\frac {2}{7} \)
In simple words: Multiply the whole number 11 by the numerator 4 to get 44. Keep the denominator 7. Convert the improper fraction \( \frac{44}{7} \) into a mixed number, which is \( 6\frac{2}{7} \).

Exam Tip: Remember to write the remainder over the original denominator when forming the mixed fraction.

Question 3. (viii) \( 20 \times \frac {4}{5} \)
Answer:
(viii) \( 20 \times \frac {4}{5} \)
\( = \frac{20 \times 4}{5} \)
\( = \frac {80}{5} \)
\( = 16 \)
In simple words: Multiply the whole number 20 by the numerator 4 to get 80. Keep the denominator 5. Then, divide 80 by 5, which results in 16.

Exam Tip: Always look to simplify by canceling common factors before multiplying, for example, \( \frac{20}{5} = 4 \), so \( 4 \times 4 = 16 \).

Question 3. (ix) \( 13 \times \frac {1}{3} \)
Answer:
(ix) \( 13 \times \frac {1}{3} \)
\( = \frac{13 \times 1}{3} \)
\( = \frac {13}{3} \)
\( = 4\frac {1}{3} \)
In simple words: Multiply 13 by the numerator 1 to get 13, keeping the denominator 3. Then, convert the improper fraction \( \frac{13}{3} \) into the mixed number \( 4\frac{1}{3} \).

Exam Tip: Be mindful of how you handle the numerator and denominator when converting an improper fraction to a mixed number.

Question 3. (x) \( 15 \times \frac {3}{5} \)
Answer:
(x) \( 15 \times \frac {3}{5} \)
\( = \frac{15 \times 3}{5} \)
\( = \frac {45}{5} \)
\( = 9 \)
In simple words: Multiply 15 by 3 to get 45, keeping the denominator 5. Then, divide 45 by 5, which gives a whole number, 9.

Exam Tip: Always simplify fractions to their lowest terms. If the result is a whole number, write it as such without a denominator.

Question 4. Shade the following:
(i) In \( \frac {1}{2} \) part of the circles in figure (a).
(ii) In \( \frac {2}{3} \) part of the triangles in figure (b).
(iii) In \( \frac {3}{5} \) part of the squares in figure (c).
Answer:
(i) \( \frac {1}{2} \) part of the 12 circles in figure (a)
\( = 12 \times \frac {1}{2} \)
\( = 6 \)
We shall shade 6 circles.
(ii) \( \frac {2}{3} \) part of the 9 triangles in figure (b)
\( = 9 \times \frac {2}{3} \)
\( = 3 \times 2 \)
\( = 6 \)
We shall shade 6 triangles.
(iii) \( \frac {3}{5} \) part of the 15 squares in figure (c)
\( = 15 \times \frac {3}{5} \)
\( = 3 \times 3 \)
\( = 9 \)
We shall shade 9 squares.
In simple words: To shade a fraction of a total number of shapes, you multiply the total shapes by the fraction. For example, for 12 circles, shading half means you shade 6 circles.

Exam Tip: When asked to shade a fraction of a group of items, calculate the exact number of items to be shaded by multiplying the total number by the given fraction.

Question 5. Find:
(a) (i) \( \frac {1}{2} \) of 24 and (ii) \( \frac {1}{2} \) of 46 each.
Answer:
(a) (i) \( \frac {1}{2} \) of 24
\( = 24 \times \frac {1}{2} \)
\( = 12 \times 1 \)
\( = 12 \)
(ii) \( \frac {1}{2} \) of 46
\( = 46 \times \frac {1}{2} \)
\( = 23 \times 1 \)
\( = 23 \)
In simple words: To find a fraction of a number, we multiply the fraction by that number. Half of 24 is 12, and half of 46 is 23.

Exam Tip: The word 'of' in mathematics usually means multiplication. Simplify before multiplying for easier calculations.

Question 5. (b) (i) \( \frac {2}{3} \) of 18 and (ii) \( \frac {2}{3} \) of 27 each.
Answer:
(b) (i) \( \frac {2}{3} \) of 18
\( = 18 \times \frac {2}{3} \)
\( = 6 \times 2 \)
\( = 12 \)
(ii) \( \frac {2}{3} \) of 27
\( = 27 \times \frac {2}{3} \)
\( = 9 \times 2 \)
\( = 18 \)
In simple words: To calculate two-thirds of a number, first divide the number by 3, and then multiply the result by 2. For instance, two-thirds of 18 is 12.

Exam Tip: When finding a fraction of a whole number, divide the whole number by the denominator first, then multiply by the numerator to avoid larger numbers.

Question 5. (c) (i) \( \frac {3}{4} \) of 16 and (ii) \( \frac {3}{4} \) of 36 each.
Answer:
(c) (i) \( \frac {3}{4} \) of 16
\( = 16 \times \frac {3}{4} \)
\( = 4 \times 3 \)
\( = 12 \)
(ii) \( \frac {3}{4} \) of 36
\( = 36 \times \frac {3}{4} \)
\( = 9 \times 3 \)
\( = 27 \)
In simple words: To find three-fourths of a number, divide the number by 4 and then multiply the result by 3. For instance, three-fourths of 16 is 12.

Exam Tip: Be accurate with your division and multiplication steps to get the correct fractional part of the given numbers.

Question 5. (d) (i) \( \frac {4}{5} \) of 20 and (ii) \( \frac {4}{5} \) of 35 each.
Answer:
(d) (i) \( \frac {4}{5} \) of 20
\( = 20 \times \frac {4}{5} \)
\( = 4 \times 4 \)
\( = 16 \)
(ii) \( \frac {4}{5} \) of 35
\( = 35 \times \frac {4}{5} \)
\( = 7 \times 4 \)
\( = 28 \)
In simple words: To find four-fifths of a number, divide the number by 5, then multiply the answer by 4. For instance, four-fifths of 20 is 16.

Exam Tip: Always make sure to perform all arithmetic operations correctly, especially when dealing with fractions of larger numbers.

Question 6. Multiply and express as a mixed fraction:
(a) \( 3 \times 5\frac {1}{5} \)
Answer:
(a) \( 3 \times 5\frac {1}{5} \)
\( = 3 \times \frac {26}{5} \)
\( = \frac{3 \times 26}{5} \)
\( = \frac {78}{5} \)
\( = 15\frac {3}{5} \)
In simple words: First, convert the mixed number to an improper fraction. Then, multiply the whole number by the numerator and keep the denominator. Finally, convert the improper fraction back to a mixed number.

Exam Tip: It is crucial to convert mixed fractions to improper fractions before performing multiplication to simplify the process.

Question 6. (b) \( 5 \times 6\frac {3}{4} \)
Answer:
(b) \( 5 \times 6\frac {3}{4} \)
\( = 5 \times \frac {27}{4} \)
\( = \frac{5 \times 27}{4} \)
\( = \frac {135}{4} \)
\( = 33\frac {3}{4} \)
In simple words: Change the mixed number \( 6\frac{3}{4} \) to an improper fraction \( \frac{27}{4} \). Then, multiply 5 by 27 to get 135, keeping the denominator 4. Finally, convert \( \frac{135}{4} \) back to the mixed number \( 33\frac{3}{4} \).

Exam Tip: Always double-check your conversion from mixed to improper fractions and vice versa, as errors here will affect the final answer.

Question 6. (c) \( 7 \times 2\frac {1}{4} \)
Answer:
(c) \( 7 \times 2\frac {1}{4} \)
\( = 7 \times \frac {9}{4} \)
\( = \frac{7 \times 9}{4} \)
\( = \frac {63}{4} \)
\( = 15\frac {3}{4} \)
In simple words: Convert \( 2\frac{1}{4} \) to the improper fraction \( \frac{9}{4} \). Multiply 7 by 9 to get 63, keeping the denominator 4. Change \( \frac{63}{4} \) back to the mixed number \( 15\frac{3}{4} \).

Exam Tip: Practice your multiplication tables to speed up calculations when multiplying numerators by whole numbers.

Question 6. (d) \( 4 \times 6\frac {1}{3} \)
Answer:
(d) \( 4 \times 6\frac {1}{3} \)
\( = 4 \times \frac {19}{3} \)
\( = \frac{4 \times 19}{3} \)
\( = \frac {76}{3} \)
\( = 25\frac {1}{3} \)
In simple words: Change \( 6\frac{1}{3} \) to \( \frac{19}{3} \). Then, multiply 4 by 19 to get 76, keeping the denominator 3. Finally, convert \( \frac{76}{3} \) to the mixed number \( 25\frac{1}{3} \).

Exam Tip: Be careful with multiplication, especially with larger numbers, to avoid calculation errors.

Question 6. (e) \( 3\frac {1}{4} \times 6 \)
Answer:
(e) \( 3\frac {1}{4} \times 6 \)
\( = \frac{13}{4} \times 6 \)
\( = \frac{13 \times 6}{4} \)
\( = \frac {39}{2} \)
\( = 19\frac {1}{2} \)
In simple words: Convert \( 3\frac{1}{4} \) to \( \frac{13}{4} \). Then, multiply 13 by 6 to get 78, keeping the denominator 4. Simplify \( \frac{78}{4} \) to \( \frac{39}{2} \) and convert it to the mixed number \( 19\frac{1}{2} \).

Exam Tip: Always simplify fractions to their lowest terms before converting them to mixed numbers, if possible, as it makes the division easier.

Question 6. (f) \( 3\frac {2}{5} \times 8 \)
Answer:
(f) \( 3\frac {2}{5} \times 8 \)
\( = \frac {17}{5} \times 8 \)
\( = \frac{17 \times 8}{5} \)
\( = \frac {136}{5} \)
\( = 27\frac {1}{5} \)
In simple words: First, change the mixed number \( 3\frac{2}{5} \) into the improper fraction \( \frac{17}{5} \). Then, multiply 17 by 8 to get 136, keeping the denominator 5. Finally, convert \( \frac{136}{5} \) back to the mixed number \( 27\frac{1}{5} \).

Exam Tip: Be meticulous with your calculations to avoid small errors, especially when working with mixed numbers and improper fractions.

Question 7. Find:
(a) (i) \( \frac {1}{2} \) of \( 2\frac {3}{4} \) and (ii) \( \frac {1}{2} \) of \( 4\frac {2}{9} \).
Answer:
(a) (i) \( \frac {1}{2} \) of \( 2\frac {3}{4} \)
\( = \frac {11}{4} \times \frac{1}{2} \)
\( = \frac{11 \times 1}{4 \times 2} \)
\( = \frac {11}{8} \)
\( = 1\frac {3}{8} \)
(ii) \( \frac {1}{2} \) of \( 4\frac {2}{9} \)
\( = \frac {38}{9} \times \frac{1}{2} \)
\( = \frac{38 \times 1}{9 \times 2} \)
\( = \frac {19}{9} \)
\( = 2\frac {1}{9} \)
In simple words: To find a fraction of a mixed number, first change the mixed number into an improper fraction. Then, multiply the numerators and denominators. Simplify the resulting fraction and convert it back to a mixed number if needed.

Exam Tip: Always remember to convert mixed fractions to improper fractions before performing any multiplication or division operations.

Question 7. (b) (i) \( \frac {5}{8} \) of \( 3\frac {5}{6} \) and (ii) \( \frac {5}{8} \) of \( 9\frac {2}{3} \).
Answer:
(b) (i) \( \frac {5}{8} \) of \( 3\frac {5}{6} \)
\( = \frac {23}{6} \times \frac{5}{8} \)
\( = \frac{23 \times 5}{6 \times 8} \)
\( = \frac {115}{48} \)
\( = 2\frac {19}{48} \)
(ii) \( \frac {5}{8} \) of \( 9\frac {2}{3} \)
\( = \frac {29}{3} \times \frac{5}{8} \)
\( = \frac{29 \times 5}{3 \times 8} \)
\( = \frac {145}{24} \)
\( = 6\frac {1}{24} \)
In simple words: To find a fraction of a mixed number, change the mixed number to an improper fraction. Multiply the numerators and then the denominators. Finally, simplify the resulting improper fraction back to a mixed number.

Exam Tip: Be careful when multiplying fractions, especially with larger numerators and denominators, and remember to reduce the final mixed fraction to its lowest terms.

Question 8. Vidya and Pratap went for a picnic. Their mother gave them a water bag filled with 5 liters of water. Vidya drank \( \frac {2}{5} \) part of it. Pratap drank the remaining water.
(i) How much water did Vidya drink?
(ii) What fraction of water did Pratap drink?
Answer:
(i) The total quantity of water was 5 liters.
Vidya drank \( \frac {2}{5} \) part of the 5 liters of water.
So, the amount of water Vidya drank \( = 5 \times \frac {2}{5} \) liters
\( = 1 \times 2 \) liters
\( = 2 \) liters.
(ii) Pratap drank the remaining water.
The amount of water Pratap drank \( = 5 \) liters \( - 2 \) liters \( = 3 \) liters.
The fraction of water Pratap drank \( = \frac{3 \text{ liters}}{5 \text{ liters}} = \frac{3}{5} \).
Alternatively, the fraction of water Pratap drank \( = 1 - \) the fraction Vidya drank
\( = 1 - \frac {2}{5} \)
\( = \frac{5-2}{5} \)
\( = \frac {3}{5} \).
In simple words: To find how much Vidya drank, multiply the total water by her fraction. To find Pratap's fraction, subtract Vidya's fraction from the whole (1) or subtract the amount Vidya drank from the total and then form a fraction.

Exam Tip: For word problems involving fractions, identify the total quantity, the fractional parts, and then perform calculations carefully to find the specific amounts or remaining fractions.