GSEB Class 7 Maths Solutions Chapter 2 Fractions and Decimals Exercise 2.2

Question 1. Which of the drawings (a) to (d) show:
(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) For \( 2 \times \frac { 1 }{ 5 } \), this means we add \( \frac { 1 }{ 5 } \) two times, so \( \frac { 1 }{ 5 } + \frac { 1 }{ 5 } \). This is represented by drawing (d). Each shape in (d) has 5 equal parts, and one part is shaded. There are two such shapes, showing two one-fifths.
Thus, (i) matches drawing (d).

(ii) For \( 2 \times \frac { 1 }{ 2 } \), this means we add \( \frac { 1 }{ 2 } \) two times, so \( \frac { 1 }{ 2 } + \frac { 1 }{ 2 } \). This is represented by drawing (b). Each shape in (b) has 2 equal parts, and one part is shaded. There are two such shapes, showing two one-halves.
Thus, (ii) matches drawing (b).

(iii) For \( 3 \times \frac { 2 }{ 3 } \), this means we add \( \frac { 2 }{ 3 } \) three times, so \( \frac { 2 }{ 3 } + \frac { 2 }{ 3 } + \frac { 2 }{ 3 } \). This is represented by drawing (a). Each shape in (a) has 3 equal parts, and two parts are shaded. There are three such shapes, showing three two-thirds.
Thus, (iii) matches drawing (a).

(iv) For \( 3 \times \frac { 1 }{ 4 } \), this means we add \( \frac { 1 }{ 4 } \) three times, so \( \frac { 1 }{ 4 } + \frac { 1 }{ 4 } + \frac { 1 }{ 4 } \). This is represented by drawing (c). Each shape in (c) has 4 equal parts, and one part is shaded. There are three such shapes, showing three one-fourths.
Thus, (iv) matches drawing (c).
In simple words: We looked at each fraction multiplication and found the picture that showed that sum. For instance, two times one-fifth means two shapes, each with one-fifth shaded.

Exam Tip: When matching visual representations to fractions, focus on two key things: the number of identical figures (which represents the multiplier) and the fraction shaded within each figure (the multiplicand).

Question 2. Some pictures (a) to (c) are given below. Tell which of them show:
(i) \( 3 \times \frac { 1 }{ 5 } \)
(ii) \( 2 \times \frac { 1 }{ 3 } = \frac { 2 }{ 3 } \)
(iii) \( 3 \times \frac { 3 }{ 4 } \) (The original document had a typo for (iii), corrected to match the solution.)
Answer:
(i) For \( 3 \times \frac { 1 }{ 5 } \): This means adding \( \frac { 1 }{ 5 } \) three times, which equals \( \frac { 1 }{ 5 } + \frac { 1 }{ 5 } + \frac { 1 }{ 5 } = \frac { 3 }{ 5 } \). This is represented by drawing (b). Drawing (b) shows three identical figures, each divided into five parts with one part shaded. So, (i) matches drawing (b).

(ii) For \( 2 \times \frac { 1 }{ 3 } = \frac { 2 }{ 3 } \): This implies adding \( \frac { 1 }{ 3 } \) two times, which equals \( \frac { 1 }{ 3 } + \frac { 1 }{ 3 } = \frac { 2 }{ 3 } \). This is represented by drawing (c). Drawing (c) shows two identical figures, each divided into three parts with one part shaded. So, (ii) matches drawing (c).

(iii) For \( 3 \times \frac { 3 }{ 4 } \): This means adding \( \frac { 3 }{ 4 } \) three times, which equals \( \frac { 3 }{ 4 } + \frac { 3 }{ 4 } + \frac { 3 }{ 4 } \). This is represented by drawing (a). Drawing (a) shows three identical figures, each divided into four parts with three parts shaded. So, (iii) matches drawing (a).
In simple words: We need to match the picture to the math problem. Each picture shows a certain number of shapes, and parts of those shapes are filled in. We count how many shapes there are and what fraction is filled in each to find the correct match.

Exam Tip: Always observe the number of figures and the shaded portion in each figure carefully to determine the correct fractional representation.

Question 3. Multiply and reduce to lowest form and convert into a mixed fraction:
(i) \( 7 \times \frac { 3 }{ 5 } \)
(ii) \( 4 \times \frac { 1 }{ 3 } \)
(iii) \( 2 \times \frac { 6 }{ 7 } \)
(iv) \( 5 \times \frac { 2 }{ 9 } \)
(v) \( \frac { 2 }{ 3 } \times 4 \)
(vi) \( \frac { 5 }{ 2 } \times 6 \)
(vii) \( 11 \times \frac { 4 }{ 7 } \)
(viii) \( 20 \times \frac { 4 }{ 5 } \)
(ix) \( 13 \times \frac { 1 }{ 3 } \)
(x) \( 15 \times \frac { 3 }{ 5 } \)
Answer:
(i) \( 7 \times \frac { 3 }{ 5 } = \frac { 7 \times 3 }{ 5 } = \frac { 21 }{ 5 } \)
\( \implies \) When converted to a mixed fraction, this becomes \( 4\frac { 1 }{ 5 } \).
(ii) \( 4 \times \frac { 1 }{ 3 } = \frac { 4 \times 1 }{ 3 } = \frac { 4 }{ 3 } \)
\( \implies \) When converted to a mixed fraction, this becomes \( 1\frac { 1 }{ 3 } \).
(iii) \( 2 \times \frac { 6 }{ 7 } = \frac { 2 \times 6 }{ 7 } = \frac { 12 }{ 7 } \)
\( \implies \) When converted to a mixed fraction, this becomes \( 1\frac { 5 }{ 7 } \).
(iv) \( 5 \times \frac { 2 }{ 9 } = \frac { 5 \times 2 }{ 9 } = \frac { 10 }{ 9 } \)
\( \implies \) When converted to a mixed fraction, this becomes \( 1\frac { 1 }{ 9 } \).
(v) \( \frac { 2 }{ 3 } \times 4 = \frac { 2 \times 4 }{ 3 } = \frac { 8 }{ 3 } \)
\( \implies \) When converted to a mixed fraction, this becomes \( 2\frac { 2 }{ 3 } \).
(vi) \( \frac { 5 }{ 2 } \times 6 = \frac { 5 \times 6 }{ 2 } = \frac { 30 }{ 2 } \)
\( \implies \) When simplified, this equals \( 15 \).
(vii) \( 11 \times \frac { 4 }{ 7 } = \frac { 11 \times 4 }{ 7 } = \frac { 44 }{ 7 } \)
\( \implies \) When converted to a mixed fraction, this becomes \( 6\frac { 2 }{ 7 } \).
(viii) \( 20 \times \frac { 4 }{ 5 } = \frac { 20 \times 4 }{ 5 } = \frac { 80 }{ 5 } \)
\( \implies \) When simplified, this equals \( 16 \).
(ix) \( 13 \times \frac { 1 }{ 3 } = \frac { 13 \times 1 }{ 3 } = \frac { 13 }{ 3 } \)
\( \implies \) When converted to a mixed fraction, this becomes \( 4\frac { 1 }{ 3 } \).
(x) \( 15 \times \frac { 3 }{ 5 } = \frac { 15 \times 3 }{ 5 } = \frac { 45 }{ 5 } \)
\( \implies \) When simplified, this equals \( 9 \).
In simple words: To multiply a whole number by a fraction, we multiply the whole number by the top part (numerator) of the fraction and keep the bottom part (denominator) the same. Then, if needed, we simplify the answer or change it into a mixed fraction.

Exam Tip: Remember to always simplify the fraction to its lowest terms before converting it into a mixed fraction. This makes the conversion process much easier and reduces the chance of errors.

Question 4. Shade:
(i) \( \frac { 1 }{ 2 } \) of the circles in box (a)
(ii) \( \frac { 2 }{ 3 } \) of the triangles in box (b)
(iii) \( \frac { 3 }{ 5 } \) of the squares in box (c)
Answer:
(i) To shade \( \frac { 1 }{ 2 } \) of the circles in box (a):
There are 12 circles in box (a).
\( \implies \frac { 1 }{ 2 } \) of \( 12 = \frac { 1 }{ 2 } \times 12 = 6 \).
So, we need to shade 6 circles. The student should shade any 6 of the 12 circles provided in the diagram.

(ii) To shade \( \frac { 2 }{ 3 } \) of the triangles in box (b):
There are 9 triangles in box (b).
\( \implies \frac { 2 }{ 3 } \) of \( 9 = \frac { 2 \times 9 }{ 3 } = 6 \).
So, we need to shade 6 triangles. The student should shade any 6 of the 9 triangles provided in the diagram.

(iii) To shade \( \frac { 3 }{ 5 } \) of the squares in box (c):
There are 15 squares in box (c).
\( \implies \frac { 3 }{ 5 } \) of \( 15 = \frac { 3 \times 15 }{ 5 } = 9 \).
So, we need to shade 9 squares. The student should shade any 9 of the 15 squares provided in the diagram.
In simple words: To shade a fraction of items, first figure out the total number of items. Then, multiply that total by the fraction given to find out how many items you need to color in.

Exam Tip: Make sure you perform the multiplication correctly to determine the exact number of items to shade. Double-check your calculation before marking the figures.

Question 5. Find:
(a) \( \frac { 1 }{ 2 } \) of (i) 24 (ii) 46
(b) \( \frac { 2 }{ 3 } \) of (i) 18 (ii) 27
(c) \( \frac { 3 }{ 4 } \) of (i) 16 (ii) 36
(d) \( \frac { 4 }{ 5 } \) of (i) 20 (ii) 35
Answer:
(a) (i) \( \frac { 1 }{ 2 } \) of \( 24 = \frac { 1 }{ 2 } \times 24 = \frac { 1 \times 24 }{ 2 } = \frac { 24 }{ 2 } = 12 \).
(ii) \( \frac { 1 }{ 2 } \) of \( 46 = \frac { 1 }{ 2 } \times 46 = \frac { 1 \times 46 }{ 2 } = \frac { 46 }{ 2 } = 23 \).
(b) (i) \( \frac { 2 }{ 3 } \) of \( 18 = \frac { 2 }{ 3 } \times 18 = \frac { 2 \times 18 }{ 3 } = \frac { 36 }{ 3 } = 12 \).
(ii) \( \frac { 2 }{ 3 } \) of \( 27 = \frac { 2 }{ 3 } \times 27 = \frac { 2 \times 27 }{ 3 } = \frac { 54 }{ 3 } = 18 \).
(c) (i) \( \frac { 3 }{ 4 } \) of \( 16 = \frac { 3 }{ 4 } \times 16 = \frac { 3 \times 16 }{ 4 } = \frac { 48 }{ 4 } = 12 \).
(ii) \( \frac { 3 }{ 4 } \) of \( 36 = \frac { 3 }{ 4 } \times 36 = \frac { 3 \times 36 }{ 4 } = \frac { 108 }{ 4 } = 27 \).
(d) (i) \( \frac { 4 }{ 5 } \) of \( 20 = \frac { 4 }{ 5 } \times 20 = \frac { 4 \times 20 }{ 5 } = \frac { 80 }{ 5 } = 16 \).
(ii) \( \frac { 4 }{ 5 } \) of \( 35 = \frac { 4 }{ 5 } \times 35 = \frac { 4 \times 35 }{ 5 } = \frac { 140 }{ 5 } = 28 \).
In simple words: To find a fraction "of" a number, you multiply the fraction by that number. Simplify the fraction if you can before multiplying, or simplify the result afterwards.

Exam Tip: Remember that "of" in mathematics often means multiplication. When multiplying fractions by whole numbers, you can often simplify by canceling common factors before performing the multiplication to make the numbers smaller and easier to handle.

Question 6. Multiply and express as a mixed fraction:
(a) \( 3 \times 5\frac { 1 }{ 5 } \)
(b) \( 5 \times 6\frac { 3 }{ 4 } \)
(c) \( 7 \times 2\frac { 1 }{ 4 } \)
(d) \( 4 \times 6\frac { 1 }{ 3 } \)
(e) \( 3\frac { 1 }{ 4 } \times 6 \)
(f) \( 3\frac { 2 }{ 5 } \times 8 \)
Answer:
(a) \( 3 \times 5\frac { 1 }{ 5 } \)
First, change the mixed fraction to an improper fraction: \( 5\frac { 1 }{ 5 } = \frac { (5 \times 5) + 1 }{ 5 } = \frac { 25 + 1 }{ 5 } = \frac { 26 }{ 5 } \).
Then, multiply: \( 3 \times \frac { 26 }{ 5 } = \frac { 3 \times 26 }{ 5 } = \frac { 78 }{ 5 } \).
Convert the improper fraction back to a mixed fraction: \( \frac { 78 }{ 5 } = 15\frac { 3 }{ 5 } \).
(b) \( 5 \times 6\frac { 3 }{ 4 } \)
First, change the mixed fraction to an improper fraction: \( 6\frac { 3 }{ 4 } = \frac { (6 \times 4) + 3 }{ 4 } = \frac { 24 + 3 }{ 4 } = \frac { 27 }{ 4 } \).
Then, multiply: \( 5 \times \frac { 27 }{ 4 } = \frac { 5 \times 27 }{ 4 } = \frac { 135 }{ 4 } \).
Convert the improper fraction back to a mixed fraction: \( \frac { 135 }{ 4 } = 33\frac { 3 }{ 4 } \).
(c) \( 7 \times 2\frac { 1 }{ 4 } \)
First, change the mixed fraction to an improper fraction: \( 2\frac { 1 }{ 4 } = \frac { (2 \times 4) + 1 }{ 4 } = \frac { 8 + 1 }{ 4 } = \frac { 9 }{ 4 } \).
Then, multiply: \( 7 \times \frac { 9 }{ 4 } = \frac { 7 \times 9 }{ 4 } = \frac { 63 }{ 4 } \).
Convert the improper fraction back to a mixed fraction: \( \frac { 63 }{ 4 } = 15\frac { 3 }{ 4 } \).
(d) \( 4 \times 6\frac { 1 }{ 3 } \)
First, change the mixed fraction to an improper fraction: \( 6\frac { 1 }{ 3 } = \frac { (6 \times 3) + 1 }{ 3 } = \frac { 18 + 1 }{ 3 } = \frac { 19 }{ 3 } \).
Then, multiply: \( 4 \times \frac { 19 }{ 3 } = \frac { 4 \times 19 }{ 3 } = \frac { 76 }{ 3 } \).
Convert the improper fraction back to a mixed fraction: \( \frac { 76 }{ 3 } = 25\frac { 1 }{ 3 } \).
(e) \( 3\frac { 1 }{ 4 } \times 6 \)
First, change the mixed fraction to an improper fraction: \( 3\frac { 1 }{ 4 } = \frac { (3 \times 4) + 1 }{ 4 } = \frac { 12 + 1 }{ 4 } = \frac { 13 }{ 4 } \).
Then, multiply: \( \frac { 13 }{ 4 } \times 6 = \frac { 13 \times 6 }{ 4 } = \frac { 78 }{ 4 } \).
Simplify the fraction: \( \frac { 78 }{ 4 } = \frac { 39 }{ 2 } \).
Convert the improper fraction back to a mixed fraction: \( \frac { 39 }{ 2 } = 19\frac { 1 }{ 2 } \).
(f) \( 3\frac { 2 }{ 5 } \times 8 \)
First, change the mixed fraction to an improper fraction: \( 3\frac { 2 }{ 5 } = \frac { (3 \times 5) + 2 }{ 5 } = \frac { 15 + 2 }{ 5 } = \frac { 17 }{ 5 } \).
Then, multiply: \( \frac { 17 }{ 5 } \times 8 = \frac { 17 \times 8 }{ 5 } = \frac { 136 }{ 5 } \).
Convert the improper fraction back to a mixed fraction: \( \frac { 136 }{ 5 } = 27\frac { 1 }{ 5 } \).
In simple words: To multiply a mixed fraction by a whole number, first change the mixed fraction into an improper fraction. Then, multiply the numerator of this improper fraction by the whole number. Finally, change the resulting improper fraction back into a mixed fraction if needed.

Exam Tip: Always convert mixed fractions to improper fractions before performing multiplication. This simplifies the process and reduces potential calculation errors. Remember to convert the final answer back to a mixed fraction if the problem asks for it.

Question 7. Find:
(a) \( \frac { 1 }{ 2 } \) of (i) \( 2\frac { 3 }{ 4 } \) (ii) \( 4\frac { 2 }{ 9 } \)
(b) \( \frac { 5 }{ 8 } \) of (i) \( 3\frac { 5 }{ 6 } \) (ii) \( 9\frac { 2 }{ 3 } \)
Answer:
(a) (i) Find \( \frac { 1 }{ 2 } \) of \( 2\frac { 3 }{ 4 } \):
First, convert the mixed fraction to an improper fraction: \( 2\frac { 3 }{ 4 } = \frac { (2 \times 4) + 3 }{ 4 } = \frac { 8 + 3 }{ 4 } = \frac { 11 }{ 4 } \).
Now, multiply: \( \frac { 1 }{ 2 } \times \frac { 11 }{ 4 } = \frac { 1 \times 11 }{ 2 \times 4 } = \frac { 11 }{ 8 } \).
Convert back to a mixed fraction: \( \frac { 11 }{ 8 } = 1\frac { 3 }{ 8 } \).
(ii) Find \( \frac { 1 }{ 2 } \) of \( 4\frac { 2 }{ 9 } \):
First, convert the mixed fraction to an improper fraction: \( 4\frac { 2 }{ 9 } = \frac { (4 \times 9) + 2 }{ 9 } = \frac { 36 + 2 }{ 9 } = \frac { 38 }{ 9 } \).
Now, multiply: \( \frac { 1 }{ 2 } \times \frac { 38 }{ 9 } = \frac { 1 \times 38 }{ 2 \times 9 } = \frac { 38 }{ 18 } \).
Simplify the fraction by dividing both numerator and denominator by 2: \( \frac { 38 \div 2 }{ 18 \div 2 } = \frac { 19 }{ 9 } \).
Convert back to a mixed fraction: \( \frac { 19 }{ 9 } = 2\frac { 1 }{ 9 } \).
(b) (i) Find \( \frac { 5 }{ 8 } \) of \( 3\frac { 5 }{ 6 } \):
First, convert the mixed fraction to an improper fraction: \( 3\frac { 5 }{ 6 } = \frac { (3 \times 6) + 5 }{ 6 } = \frac { 18 + 5 }{ 6 } = \frac { 23 }{ 6 } \).
Now, multiply: \( \frac { 5 }{ 8 } \times \frac { 23 }{ 6 } = \frac { 5 \times 23 }{ 8 \times 6 } = \frac { 115 }{ 48 } \).
Convert back to a mixed fraction: \( \frac { 115 }{ 48 } = 2\frac { 19 }{ 48 } \).
(ii) Find \( \frac { 5 }{ 8 } \) of \( 9\frac { 2 }{ 3 } \):
First, convert the mixed fraction to an improper fraction: \( 9\frac { 2 }{ 3 } = \frac { (9 \times 3) + 2 }{ 3 } = \frac { 27 + 2 }{ 3 } = \frac { 29 }{ 3 } \).
Now, multiply: \( \frac { 5 }{ 8 } \times \frac { 29 }{ 3 } = \frac { 5 \times 29 }{ 8 \times 3 } = \frac { 145 }{ 24 } \).
Convert back to a mixed fraction: \( \frac { 145 }{ 24 } = 6\frac { 1 }{ 24 } \).
In simple words: To find a fraction of a mixed fraction, first change the mixed fraction into a regular (improper) fraction. Then, simply multiply the two fractions together. If the answer is an improper fraction, change it back into a mixed fraction.

Exam Tip: When multiplying fractions, always look for opportunities to cross-cancel common factors between numerators and denominators. This can significantly simplify the multiplication process and make the final reduction easier.

Question 8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed \( \frac { 2 }{ 5 } \) of the water. Pratap consumed the remaining water.
(i) How much water did Vidya drink?
(ii) What fraction of the total quantity of water did Pratap drink?
Answer:
Total quantity of water in the bottle = 5 litres.
(i) Amount of water consumed by Vidya:
Vidya drank \( \frac { 2 }{ 5 } \) of the total water.
\( \implies \) Amount Vidya drank \( = \frac { 2 }{ 5 } \) of 5 litres \( = \frac { 2 }{ 5 } \times 5 = 2 \) litres.
So, Vidya drank 2 litres of water.

(ii) Fraction of total quantity of water Pratap drank:
Amount of water remaining after Vidya drank her share \( = \) Total water \( - \) Water Vidya drank
\( \implies \) Remaining water \( = 5 - 2 = 3 \) litres.
Pratap drank this remaining water, so Pratap drank 3 litres.
The fraction of water Pratap drank \( = \frac { \text{Amount Pratap drank} }{ \text{Total quantity of water} } = \frac { 3 }{ 5 } \).
So, Pratap drank \( \frac { 3 }{ 5 } \) of the total water.
In simple words: First, we found how much water Vidya drank by multiplying the total water by her fraction. Then, we subtracted that amount from the total to find out how much Pratap drank. Finally, we turned the amount Pratap drank into a fraction of the total.

Exam Tip: For word problems involving fractions, always start by identifying the total quantity. Then, use this total to calculate fractional parts. If a "remaining" amount is mentioned, subtract the consumed portion from the total to find it.