GSEB Class 7 Maths Solutions Chapter 2 અપૂર્ણાંક અને દશાંશ સંખ્યાઓ Exercise 2.3

Question 1. Find the value of:
(i) Of \( \frac{1}{4} \): (a) \( \frac{1}{4} \) (b) \( \frac{3}{5} \) (c) \( \frac{4}{3} \)
(ii) Of \( \frac{1}{7} \): (a) \( \frac{2}{9} \) (b) \( \frac{6}{5} \) (c) \( \frac{3}{10} \)
Answer:
(i) (a) \( \frac{1}{4} \) of \( \frac{1}{4} = \frac{1}{4} \times \frac{1}{4} = \frac{1 \times 1}{4 \times 4} = \frac{1}{16} \)
(b) \( \frac{3}{5} \) of \( \frac{1}{4} = \frac{3}{5} \times \frac{1}{4} = \frac{3 \times 1}{5 \times 4} = \frac{3}{20} \)
(c) \( \frac{4}{3} \) of \( \frac{1}{4} = \frac{4}{3} \times \frac{1}{4} = \frac{4 \times 1}{3 \times 4} = \frac{1}{3} \)
(ii) (a) \( \frac{2}{9} \) of \( \frac{1}{7} = \frac{2}{9} \times \frac{1}{7} = \frac{2 \times 1}{9 \times 7} = \frac{2}{63} \)
(b) \( \frac{6}{5} \) of \( \frac{1}{7} = \frac{6}{5} \times \frac{1}{7} = \frac{6 \times 1}{5 \times 7} = \frac{6}{35} \)
(c) \( \frac{3}{10} \) of \( \frac{1}{7} = \frac{3}{10} \times \frac{1}{7} = \frac{3 \times 1}{10 \times 7} = \frac{3}{70} \)
In simple words: To find a fraction "of" another fraction, you simply multiply them together. Multiply the numerators (top numbers) and then multiply the denominators (bottom numbers) to get your result. Always try to simplify the fraction if possible.

Exam Tip: Remember that "of" in mathematics generally means multiplication. For mixed fractions, convert them to improper fractions before multiplying.

Question 2. Multiply and express in simplest form: (if possible)
(i) \( \frac{2}{3} \times 2\frac{2}{3} \)
(ii) \( \frac{2}{7} \times \frac{7}{9} \)
(iii) \( \frac{3}{8} \times \frac{6}{4} \)
(iv) \( \frac{9}{5} \times \frac{3}{5} \)
(v) \( \frac{1}{3} \times \frac{15}{8} \)
(vi) \( \frac{11}{2} \times \frac{3}{10} \)
(vii) \( \frac{4}{5} \times \frac{12}{7} \)
Answer:
(i) \( \frac{2}{3} \times 2\frac{2}{3} = \frac{2}{3} \times \frac{8}{3} = \frac{2 \times 8}{3 \times 3} = \frac{16}{9} = 1\frac{7}{9} \)
(ii) \( \frac{2}{7} \times \frac{7}{9} = \frac{2 \times 7}{7 \times 9} = \frac{2 \times 1}{1 \times 9} = \frac{2}{9} \)
(iii) \( \frac{3}{8} \times \frac{6}{4} = \frac{3 \times 6}{8 \times 4} = \frac{3 \times 3}{8 \times 2} = \frac{9}{16} \)
(iv) \( \frac{9}{5} \times \frac{3}{5} = \frac{9 \times 3}{5 \times 5} = \frac{27}{25} = 1\frac{2}{25} \)
(v) \( \frac{1}{3} \times \frac{15}{8} = \frac{1 \times 15}{3 \times 8} = \frac{1 \times 5}{1 \times 8} = \frac{5}{8} \)
(vi) \( \frac{11}{2} \times \frac{3}{10} = \frac{11 \times 3}{2 \times 10} = \frac{33}{20} = 1\frac{13}{20} \)
(vii) \( \frac{4}{5} \times \frac{12}{7} = \frac{4 \times 12}{5 \times 7} = \frac{48}{35} = 1\frac{13}{35} \)
In simple words: When multiplying fractions, multiply the top numbers together and the bottom numbers together. If you have a mixed number, change it into an improper fraction first. After multiplying, reduce the fraction to its smallest form, and if it's an improper fraction, convert it back to a mixed number.

Exam Tip: Always look for common factors in the numerator and denominator *before* multiplying to simplify the calculation process. For instance, in (ii), you can cancel the 7s directly.

Question 3. Multiply:
(i) \( \frac{2}{5} \times 5\frac{1}{4} \)
(ii) \( 6\frac{2}{5} \times \frac{7}{9} \)
(iii) \( \frac{3}{2} \times 5\frac{1}{3} \)
(iv) \( \frac{5}{6} \times 2\frac{3}{7} \)
(v) \( 3\frac{2}{5} \times \frac{4}{7} \)
(vi) \( 2\frac{3}{5} \times 3 \)
(vii) \( 3\frac{4}{7} \times \frac{3}{5} \)
Answer:
(i) \( \frac{2}{5} \times 5\frac{1}{4} = \frac{2}{5} \times \frac{21}{4} = \frac{1 \times 21}{5 \times 2} = \frac{21}{10} = 2\frac{1}{10} \)
(ii) \( 6\frac{2}{5} \times \frac{7}{9} = \frac{32}{5} \times \frac{7}{9} = \frac{32 \times 7}{5 \times 9} = \frac{224}{45} = 4\frac{44}{45} \)
(iii) \( \frac{3}{2} \times 5\frac{1}{3} = \frac{3}{2} \times \frac{16}{3} = \frac{1 \times 16}{2 \times 1} = \frac{16}{2} = 8 \)
(iv) \( \frac{5}{6} \times 2\frac{3}{7} = \frac{5}{6} \times \frac{17}{7} = \frac{5 \times 17}{6 \times 7} = \frac{85}{42} = 2\frac{1}{42} \)
(v) \( 3\frac{2}{5} \times \frac{4}{7} = \frac{17}{5} \times \frac{4}{7} = \frac{17 \times 4}{5 \times 7} = \frac{68}{35} = 1\frac{33}{35} \)
(vi) \( 2\frac{3}{5} \times 3 = \frac{13}{5} \times \frac{3}{1} = \frac{13 \times 3}{5 \times 1} = \frac{39}{5} = 7\frac{4}{5} \)
(vii) \( 3\frac{4}{7} \times \frac{3}{5} = \frac{25}{7} \times \frac{3}{5} = \frac{5 \times 3}{7 \times 1} = \frac{15}{7} = 2\frac{1}{7} \)
In simple words: When you need to multiply mixed numbers, first change them into improper fractions. Then, multiply the numerators and the denominators. Finally, simplify the resulting fraction and convert it back to a mixed number if it's an improper fraction.

Exam Tip: Always simplify fractions before performing the final multiplication to keep the numbers smaller and easier to handle, reducing chances of error.

Question 4. Which is greater?
(i) \( \frac{3}{4} \) of \( \frac{2}{7} \) or \( \frac{5}{8} \) of \( \frac{3}{5} \)
(ii) \( \frac{6}{7} \) of \( \frac{1}{2} \) or \( \frac{3}{7} \) of \( \frac{2}{3} \)
Answer:
(i) First, calculate each part:
\( \frac{3}{4} \) of \( \frac{2}{7} = \frac{3}{4} \times \frac{2}{7} = \frac{3 \times 2}{4 \times 7} = \frac{6}{28} = \frac{3}{14} \)
\( \frac{5}{8} \) of \( \frac{3}{5} = \frac{5}{8} \times \frac{3}{5} = \frac{5 \times 3}{8 \times 5} = \frac{15}{40} = \frac{3}{8} \)
Now, compare \( \frac{3}{14} \) and \( \frac{3}{8} \). To compare, find a common denominator, which is 56 (LCM of 14 and 8).
\( \frac{3}{14} = \frac{3 \times 4}{14 \times 4} = \frac{12}{56} \)
\( \frac{3}{8} = \frac{3 \times 7}{8 \times 7} = \frac{21}{56} \)
Since \( 21 > 12 \), then \( \frac{21}{56} > \frac{12}{56} \).
Therefore, \( \frac{5}{8} \) of \( \frac{3}{5} \) is greater than \( \frac{3}{4} \) of \( \frac{2}{7} \).
(ii) First, calculate each part:
\( \frac{6}{7} \) of \( \frac{1}{2} = \frac{6}{7} \times \frac{1}{2} = \frac{6 \times 1}{7 \times 2} = \frac{6}{14} = \frac{3}{7} \)
\( \frac{3}{7} \) of \( \frac{2}{3} = \frac{3}{7} \times \frac{2}{3} = \frac{3 \times 2}{7 \times 3} = \frac{6}{21} = \frac{2}{7} \)
Now, compare \( \frac{3}{7} \) and \( \frac{2}{7} \).
Since both fractions have the same denominator, compare their numerators.
Since \( 3 > 2 \), then \( \frac{3}{7} > \frac{2}{7} \).
Therefore, \( \frac{6}{7} \) of \( \frac{1}{2} \) is greater than \( \frac{3}{7} \) of \( \frac{2}{3} \).
In simple words: To find out which of two fractional expressions is larger, first calculate the value of each expression by multiplying the fractions. Once you have two single fractions, compare them by making their bottom numbers (denominators) the same. The fraction with the bigger top number (numerator) will be the larger one.

Exam Tip: When comparing fractions, finding a common denominator (preferably the Least Common Multiple, LCM) is the most reliable method. For fractions with the same numerator but different denominators, the fraction with the smaller denominator is larger.

Question 5. Shaili planted 4 saplings in a row in her garden. She left a distance of \( \frac{3}{4} \) meter between two saplings. Find the distance between the first and last saplings.
Answer:
Total number of saplings planted by Shaili = 4
The distance between two adjacent saplings = \( \frac{3}{4} \) meter
When 4 saplings are planted in a row, there are 3 gaps between the first and the last sapling. Each gap is \( \frac{3}{4} \) meter long. Therefore, the distance between the first and the last (fourth) sapling will be the sum of these 3 distances.
Distance between the first and last (fourth) sapling \( = 3 \times \frac{3}{4} \) meters
\( = \frac{9}{4} \) meters
\( = 2\frac{1}{4} \) meters
Thus, the distance between the first and the last (fourth) sapling is \( 2\frac{1}{4} \) meters.
In simple words: If you plant 4 trees in a line, there are 3 spaces between them. Since each space is \( \frac{3}{4} \) of a meter, you just multiply 3 by \( \frac{3}{4} \) to get the total distance from the first tree to the last tree.

Exam Tip: For problems involving objects placed in a row, remember that the number of gaps between the objects is always one less than the number of objects themselves. For example, 4 objects create 3 gaps.

Question 6. Lipika reads a book for \( 1\frac{3}{4} \) hours daily. She finishes the book in 6 days. How many hours did she dedicate to reading this book in total?
Answer:
Number of days for reading = 6
Time spent reading per day = \( 1\frac{3}{4} \) hours \( = \frac{7}{4} \) hours
Total time spent reading the entire book = (Time per day \( \times \) Number of days)
\( = \frac{7}{4} \times 6 \) hours
\( = \frac{7 \times 6}{4} \) hours
\( = \frac{42}{4} \) hours
\( = \frac{21}{2} \) hours
\( = 10\frac{1}{2} \) hours
So, Lipika dedicated a total of \( 10\frac{1}{2} \) hours to finish reading the entire book.
In simple words: To find the total time Lipika spent reading, you need to multiply the number of hours she reads each day by the total number of days she spent reading. Convert the daily reading time to an improper fraction first, then multiply.

Exam Tip: Always convert mixed fractions into improper fractions before performing multiplication or division to avoid errors in calculation.

Question 7. A car covers a distance of 16 km using 1 liter of petrol. How much distance will it cover using \( 2\frac{3}{4} \) liters of petrol?
Answer:
Distance covered by the car using 1 liter of petrol = 16 km
Quantity of petrol to be used = \( 2\frac{3}{4} \) liters
Convert the mixed fraction to an improper fraction: \( 2\frac{3}{4} = \frac{(4 \times 2) + 3}{4} = \frac{8+3}{4} = \frac{11}{4} \) liters
Distance covered by the car using \( \frac{11}{4} \) liters of petrol = Distance per liter \( \times \) Quantity of petrol
\( = 16 \times \frac{11}{4} \) km
\( = \frac{16 \times 11}{4} \) km
\( = 4 \times 11 \) km
\( = 44 \) km
Therefore, the car will cover a distance of 44 km using \( 2\frac{3}{4} \) liters of petrol.
In simple words: If you know how far a car goes on one liter of petrol, you can find out how far it goes on more liters by multiplying the distance for one liter by the total liters used. Remember to turn any mixed numbers into simple fractions first.

Exam Tip: For unit rate problems, identify the "per unit" value first. Then, multiply this rate by the total number of units to find the overall quantity.

Question 8. (a) (i) Write the number in the box (khaana) such that \( \frac{2}{3} \times \boxed{\phantom{X}} = \frac{10}{30} \).
(ii) Find the simplest form of the number obtained in (i).
(b) (i) Write the number in the box (khaana) such that \( \frac{3}{5} \times \boxed{\phantom{X}} = \frac{24}{75} \).
(ii) Find the simplest form of the number obtained in (i).
Answer:
(a) (i) Let the missing number be \( x \).
\( \frac{2}{3} \times x = \frac{10}{30} \)
To find \( x \), we can think: \( 2 \times 5 = 10 \) and \( 3 \times 10 = 30 \).
So, \( x = \frac{5}{10} \)
(ii) The simplest form of \( \frac{5}{10} \) is \( \frac{5 \div 5}{10 \div 5} = \frac{1}{2} \).
(b) (i) Let the missing number be \( y \).
\( \frac{3}{5} \times y = \frac{24}{75} \)
To find \( y \), we can think: \( 3 \times 8 = 24 \) and \( 5 \times 15 = 75 \).
So, \( y = \frac{8}{15} \)
(ii) The simplest form of \( \frac{8}{15} \) is \( \frac{8}{15} \). It is already in its simplest form because 8 and 15 have no common factors other than 1.
In simple words: For part (a)(i), to find the missing fraction, you need to figure out what you multiply 2 by to get 10, and what you multiply 3 by to get 30. For (a)(ii), then simplify that new fraction as much as possible. Do the same for part (b).

Exam Tip: When finding a missing fraction in a multiplication problem, divide the product by the known fraction. To simplify a fraction, divide both the numerator and denominator by their greatest common factor (GCF).