GSEB Class 7 Maths Solutions Chapter 2 અપૂર્ણાંક અને દશાંશ સંખ્યાઓ InText Questions

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 34)

Question 1. શોધો :
(a) \( \frac { 2 }{ 7 } \times 3 \)
(b) \( \frac { 9 }{ 7 } \times 6 \)
(c) \( 3 \times \frac { 1 }{ 8 } \)
(d) \( \frac { 13 }{ 11 } \times 6 \)
જો તેનો જવાબ અશુદ્ધ અપૂર્ણાંકમાં છે, તો તેને મિશ્ર અપૂર્ણાંકમાં રજૂ કરો.
Answer:
(a) We need to multiply the fraction \( \frac { 2 }{ 7 } \) by 3.
\( \frac { 2 }{ 7 } \times 3 = \frac { 2 \times 3 }{ 7 } \)
\( \implies \frac { 6 }{ 7 } \)
(b) We need to multiply the fraction \( \frac { 9 }{ 7 } \) by 6.
\( \frac { 9 }{ 7 } \times 6 = \frac { 9 \times 6 }{ 7 } \)
\( \implies \frac { 54 }{ 7 } \)
To convert \( \frac { 54 }{ 7 } \) to a mixed fraction, we divide 54 by 7. \( 54 \div 7 = 7 \) with a remainder of 5.
\( \implies 7\frac { 5 }{ 7 } \)
In simple words: First, multiply the top number (numerator) of the fraction by the whole number. Keep the bottom number (denominator) the same. If your result is an improper fraction (top number is bigger), then change it into a mixed number by dividing.

Exam Tip: Remember to simplify fractions to their lowest terms and convert improper fractions to mixed numbers when required for full marks.

Answer:
(c) We need to multiply the whole number 3 by the fraction \( \frac { 1 }{ 8 } \).
\( 3 \times \frac { 1 }{ 8 } = \frac { 3 \times 1 }{ 8 } \)
\( \implies \frac { 3 }{ 8 } \)
(d) We need to multiply the fraction \( \frac { 13 }{ 11 } \) by 6.
\( \frac { 13 }{ 11 } \times 6 = \frac { 13 \times 6 }{ 11 } \)
\( \implies \frac { 78 }{ 11 } \)
To convert \( \frac { 78 }{ 11 } \) to a mixed fraction, we divide 78 by 11. \( 78 \div 11 = 7 \) with a remainder of 1.
\( \implies 7\frac { 1 }{ 11 } \)
In simple words: When you multiply a fraction by a whole number, only the top part of the fraction changes. If the top part ends up bigger than the bottom part, divide it to get a mixed number.

Exam Tip: Always check if the final answer needs to be converted into a mixed fraction, especially when the numerator is larger than the denominator.

Question 2. યિત્રાત્મક્ રજૂઆત કરો \( 2 \times \frac { 2 }{ 5 } = \frac { 4 }{ 5 } \)
Answer: The pictorial representation of multiplying 2 by \( \frac { 2 }{ 5 } \) to get \( \frac { 4 }{ 5 } \) is shown below. This shows that two groups of two-fifths combine to make four-fifths.

In simple words: When you multiply a fraction by a whole number, it means you have that many copies of the fraction. So, two times two-fifths means you have two separate two-fifths, and when you combine them, you get four-fifths in total.

Exam Tip: For pictorial questions, ensure your explanation directly relates to the visual elements provided and accurately represents the mathematical operation.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 34)

Question. શોધોઃ
(i) \( 5 \times 2\frac { 3 }{ 7 } \)
(ii) \( 1\frac { 4 }{ 9 } \times 6 \)
Answer:
(i) We need to calculate \( 5 \times 2\frac { 3 }{ 7 } \).
First, convert the mixed fraction \( 2\frac { 3 }{ 7 } \) to an improper fraction: \( 2\frac { 3 }{ 7 } = \frac { (7 \times 2) + 3 }{ 7 } = \frac { 14 + 3 }{ 7 } = \frac { 17 }{ 7 } \)
Now, multiply:
\( 5 \times \frac { 17 }{ 7 } = \frac { 5 \times 17 }{ 7 } \)
\( \implies \frac { 85 }{ 7 } \)
Convert the improper fraction back to a mixed fraction: \( 85 \div 7 = 12 \) with a remainder of 1.
\( \implies 12\frac { 1 }{ 7 } \)
(ii) We need to calculate \( 1\frac { 4 }{ 9 } \times 6 \).
First, convert the mixed fraction \( 1\frac { 4 }{ 9 } \) to an improper fraction: \( 1\frac { 4 }{ 9 } = \frac { (9 \times 1) + 4 }{ 9 } = \frac { 9 + 4 }{ 9 } = \frac { 13 }{ 9 } \)
Now, multiply:
\( \frac { 13 }{ 9 } \times 6 = \frac { 13 \times 6 }{ 9 } \)
We can simplify before multiplying: \( \frac { 13 \times 2 }{ 3 } \) (by dividing 6 and 9 by 3)
\( \implies \frac { 26 }{ 3 } \)
Convert the improper fraction back to a mixed fraction: \( 26 \div 3 = 8 \) with a remainder of 2.
\( \implies 8\frac { 2 }{ 3 } \)
In simple words: When multiplying a whole number by a mixed fraction, first change the mixed fraction into an improper fraction. Then, multiply the whole number by the top part of the improper fraction. Finally, change the answer back into a mixed fraction if it's an improper fraction. Remember to simplify if you can!

Exam Tip: Always convert mixed fractions to improper fractions before performing multiplication to avoid errors. Simplify results to their simplest mixed number form.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 35)

Question. શું તમે કહી શકો છો
(i) 10 of \( \frac {1}{2} \)
(ii) 16 of \( \frac {1}{4} \)
(iii) 25 of \( \frac {2}{5} \) કેટલા થાય?
Answer: We need to find the values for each expression:
(i) 10 of \( \frac {1}{2} \)
\( 10 \times \frac {1}{2} = \frac {10 \times 1}{2} = \frac {10}{2} = 5 \)
(ii) 16 of \( \frac {1}{4} \)
\( 16 \times \frac {1}{4} = \frac {16 \times 1}{4} = \frac {16}{4} = 4 \)
(iii) 25 of \( \frac {2}{5} \)
\( 25 \times \frac {2}{5} = \frac {25 \times 2}{5} = \frac {50}{5} = 10 \)
In simple words: When you see "of" between a whole number and a fraction, it means to multiply them. Multiply the whole number by the top part of the fraction, and then divide by the bottom part.

Exam Tip: "Of" in mathematics usually implies multiplication. Always perform multiplication before any addition or subtraction as per the order of operations.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 39)

Question. ખાલી જગ્યા પૂરોઃ
(i) \( \frac { 1 }{ 2 } \times \frac { 1 }{ 7 } = \frac { 1 \times 1 }{ 2 \times 7 } = \frac { \phantom{X} }{ \phantom{Y} } \)
(ii) \( \frac { 1 }{ 5 } \times \frac { 1 }{ 7 } = \frac { 1 \times 1 }{ 5 \times 7 } = \frac { \phantom{X} }{ \phantom{Y} } \)
(iii) \( \frac { 1 }{ 7 } \times \frac { 1 }{ 2 } = \frac { 1 \times 1 }{ 7 \times 2 } = \frac { \phantom{X} }{ \phantom{Y} } \)
(iv) \( \frac { 1 }{ 7 } \times \frac { 1 }{ 5 } = \frac { 1 \times 1 }{ 7 \times 5 } = \frac { \phantom{X} }{ \phantom{Y} } \)
Answer: We need to fill in the blanks by multiplying the fractions:
(i) \( \frac { 1 }{ 2 } \times \frac { 1 }{ 7 } = \frac { 1 \times 1 }{ 2 \times 7 } = \frac { 1 }{ 14 } \)
(ii) \( \frac { 1 }{ 5 } \times \frac { 1 }{ 7 } = \frac { 1 \times 1 }{ 5 \times 7 } = \frac { 1 }{ 35 } \)
(iii) \( \frac { 1 }{ 7 } \times \frac { 1 }{ 2 } = \frac { 1 \times 1 }{ 7 \times 2 } = \frac { 1 }{ 14 } \)
(iv) \( \frac { 1 }{ 7 } \times \frac { 1 }{ 5 } = \frac { 1 \times 1 }{ 7 \times 5 } = \frac { 1 }{ 35 } \)
In simple words: To multiply fractions, you just multiply the top numbers together and then multiply the bottom numbers together. It's a straightforward process.

Exam Tip: Remember that when multiplying fractions, the order of multiplication does not change the product (commutative property), as seen in parts (i) and (iii), and (ii) and (iv).

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 40)

Question. શોધો :
(i) \( \frac { 1 }{ 3 } \times \frac { 4 }{ 5 } \)
(ii) \( \frac { 2 }{ 3 } \times \frac { 1 }{ 5 } \)
Answer: We need to find the product of the given fractions:
(i) \( \frac { 1 }{ 3 } \times \frac { 4 }{ 5 } \)
\( = \frac { 1 \times 4 }{ 3 \times 5 } \)
\( \implies \frac { 4 }{ 15 } \)
(ii) We need to find the product of \( \frac { 2 }{ 3 } \times \frac { 1 }{ 5 } \).
\( = \frac { 2 \times 1 }{ 3 \times 5 } \)
\( \implies \frac { 2 }{ 15 } \)
In simple words: To multiply two fractions, multiply their top numbers (numerators) and their bottom numbers (denominators) separately.

Exam Tip: Always multiply numerators with numerators and denominators with denominators. Do not cross-multiply unless simplifying.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 40)

Question. શોધો :
(i) \( \frac { 8 }{ 3 } \times \frac { 4 }{ 7 } \)
(ii) \( \frac { 3 }{ 4 } \times \frac { 2 }{ 3 } \)
Answer: We need to find the product of the given fractions:
(i) \( \frac { 8 }{ 3 } \times \frac { 4 }{ 7 } \)
\( = \frac { 8 \times 4 }{ 3 \times 7 } \)
\( \implies \frac { 32 }{ 21 } \)
Convert the improper fraction to a mixed fraction: \( 32 \div 21 = 1 \) with a remainder of 11.
\( \implies 1\frac { 11 }{ 21 } \)
(ii) \( \frac { 3 }{ 4 } \times \frac { 2 }{ 3 } \)
\( = \frac { 3 \times 2 }{ 4 \times 3 } \)
We can simplify before multiplying: \( \frac { 1 \times 1 }{ 2 \times 1 } \) (by cancelling 3 from numerator and denominator, and dividing 2 and 4 by 2)
\( \implies \frac { 1 }{ 2 } \)
In simple words: When multiplying fractions, multiply the top numbers and the bottom numbers. Always simplify your answer to its simplest form, and if it's an improper fraction, change it to a mixed number.

Exam Tip: Simplify fractions before multiplying to work with smaller numbers, which helps in avoiding calculation errors and makes the process easier.

પાઠ્યપુસ્તકમાંથી : (પાઠ્યપુસ્તક પાન નંબર 40-41)

Question. બે શુદ્ધ, અપૂર્ણાકોના ગુણાકાર અંગે વિચારીએ. કોષ્ટકમાં ખાલી જગ્યાઓ પૂરો.
Answer: Let's consider the multiplication of two proper fractions and observe the results in the completed table below. We will notice that the product of two proper fractions is always smaller than both of the individual fractions.

Exam Tip: A key property of proper fractions (numerator < denominator) is that their product is always smaller than either of the original fractions. This can be used to quickly check your answers.

Question. ચાલો, હવે આપણે બે અશુદ્ધ અપૂર્ણાકોના ગુણાકાર વિશે જાણીએ. કોષ્ટકમાં ખાલી જગ્યાઓ પૂરો.
Answer: Let's now learn about multiplying two improper fractions and complete the table below. The product of two improper fractions is always greater than both of the individual fractions.

Exam Tip: For improper fractions (numerator > denominator), their product is always greater than either of the original fractions. This is a good way to verify if your calculations are reasonable.

વિચારો, ચર્ચા કરો અને લખો : (પાઠ્યપુસ્તક પાન નંબર 44)

Question. (i) શું શુદ્ધ અપૂર્ણાકનો વ્યસ્ત શુદ્ધ અપૂર્ણાંક છે?
(ii) શું અશુદ્ધ અપૂર્ણાકનો વ્યસ્ત અશુદ્ધ અપૂર્ણાંક છે?
Answer: Let's think about the properties of reciprocals for proper and improper fractions:
(i) No, the reciprocal of a proper fraction is always an improper fraction. For example, the reciprocal of \( \frac { 1 }{ 2 } \) is \( \frac { 2 }{ 1 } \), which is an improper fraction. If the original fraction is proper (less than 1), its reciprocal will be greater than 1.
(ii) No, the reciprocal of an improper fraction is always a proper fraction. For example, the reciprocal of \( \frac { 3 }{ 2 } \) is \( \frac { 2 }{ 3 } \), which is a proper fraction. If the original fraction is improper (greater than 1), its reciprocal will be less than 1.
Thus, we can say that:
(a) \( 1 \div \frac { 1 }{ 2 } = 1 \times \frac { 2 }{ 1 } \) (which is the reciprocal of \( \frac { 1 }{ 2 } \))
(b) \( 3 \div \frac { 1 }{ 4 } = 3 \times \frac { 4 }{ 1 } \) (which is the reciprocal of \( \frac { 1 }{ 4 } \))
(c) \( 3 \div \frac { 1 }{ 2 } = 3 \times \frac { 2 }{ 1 } \) (which is the reciprocal of \( \frac { 1 }{ 2 } \))
Therefore, \( 2 \div \frac { 3 }{ 4 } = 2 \times \frac { 4 }{ 3 } \) (which is the reciprocal of \( \frac { 3 }{ 4 } \))
(d) \( 5 \div \frac { 2 }{ 9 } = 5 \times \frac { 9 }{ 2 } \)
In simple words: A proper fraction's flip (reciprocal) will always be an improper fraction, meaning it's bigger than one. An improper fraction's flip (reciprocal) will always be a proper fraction, meaning it's smaller than one. When you divide by a fraction, you actually multiply by its reciprocal.

Exam Tip: Understanding reciprocals is crucial for fraction division. Remember that a number multiplied by its reciprocal always equals 1.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 45)

Question. શોધો :
(i) \( 7 \div \frac { 2 }{ 5 } \)
(ii) \( 6 \div \frac { 4 }{ 7 } \)
(iii) \( 2 \div \frac { 8 }{ 9 } \)
Answer: We need to find the result of each division problem:
(i) To calculate \( 7 \div \frac { 2 }{ 5 } \), we multiply 7 by the reciprocal of \( \frac { 2 }{ 5 } \).
The reciprocal of \( \frac { 2 }{ 5 } \) is \( \frac { 5 }{ 2 } \).
\( 7 \times \frac { 5 }{ 2 } = \frac { 7 \times 5 }{ 2 } = \frac { 35 }{ 2 } \)
Converting to a mixed fraction: \( 35 \div 2 = 17 \) with a remainder of 1.
\( \implies 17\frac { 1 }{ 2 } \)
In simple words: When you divide a whole number by a fraction, you flip the fraction and then multiply.

Exam Tip: Always remember the "Keep, Change, Flip" (KCF) method for dividing fractions: Keep the first number, Change division to multiplication, Flip the second fraction (use its reciprocal).

Answer:
(ii) To calculate \( 6 \div \frac { 4 }{ 7 } \), we multiply 6 by the reciprocal of \( \frac { 4 }{ 7 } \).
The reciprocal of \( \frac { 4 }{ 7 } \) is \( \frac { 7 }{ 4 } \).
\( 6 \times \frac { 7 }{ 4 } = \frac { 6 \times 7 }{ 4 } = \frac { 42 }{ 4 } \)
Simplify the fraction: \( \frac { 21 }{ 2 } \)
Converting to a mixed fraction: \( 21 \div 2 = 10 \) with a remainder of 1.
\( \implies 10\frac { 1 }{ 2 } \)
(iii) To calculate \( 2 \div \frac { 8 }{ 9 } \), we multiply 2 by the reciprocal of \( \frac { 8 }{ 9 } \).
The reciprocal of \( \frac { 8 }{ 9 } \) is \( \frac { 9 }{ 8 } \).
\( 2 \times \frac { 9 }{ 8 } = \frac { 2 \times 9 }{ 8 } = \frac { 18 }{ 8 } \)
Simplify the fraction: \( \frac { 9 }{ 4 } \)
Converting to a mixed fraction: \( 9 \div 4 = 2 \) with a remainder of 1.
\( \implies 2\frac { 1 }{ 4 } \)
In simple words: When you divide by a fraction, remember to flip the second fraction (find its reciprocal) and then multiply. Always simplify your final fraction and change improper fractions into mixed numbers.

Exam Tip: After performing the multiplication, always check if the resulting fraction can be simplified or converted to a mixed number for the final answer.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 45)

Question. શોધો :
(i) \( 6 \div 5\frac { 1 }{ 3 } \)
(ii) \( 7 \div 2\frac { 4 }{ 7 } \)
Answer: We need to calculate the value of each expression:
(i) To calculate \( 6 \div 5\frac { 1 }{ 3 } \), first convert the mixed fraction \( 5\frac { 1 }{ 3 } \) to an improper fraction.
\( 5\frac { 1 }{ 3 } = \frac { (3 \times 5) + 1 }{ 3 } = \frac { 15 + 1 }{ 3 } = \frac { 16 }{ 3 } \)
Now, divide 6 by \( \frac { 16 }{ 3 } \) by multiplying 6 by the reciprocal of \( \frac { 16 }{ 3 } \).
The reciprocal of \( \frac { 16 }{ 3 } \) is \( \frac { 3 }{ 16 } \).
\( 6 \times \frac { 3 }{ 16 } = \frac { 6 \times 3 }{ 16 } = \frac { 18 }{ 16 } \)
Simplify the fraction: \( \frac { 9 }{ 8 } \)
Converting to a mixed fraction: \( 9 \div 8 = 1 \) with a remainder of 1.
\( \implies 1\frac { 1 }{ 8 } \)
In simple words: When dividing by a mixed fraction, first change it into an improper fraction. Then, flip that fraction (find its reciprocal) and multiply it by the first number. Always simplify your answer.

Exam Tip: Remember to convert all mixed fractions into improper fractions before performing any division or multiplication operations to ensure accuracy.

Answer:
(ii) To calculate \( 7 \div 2\frac { 4 }{ 7 } \), first convert the mixed fraction \( 2\frac { 4 }{ 7 } \) to an improper fraction.
\( 2\frac { 4 }{ 7 } = \frac { (7 \times 2) + 4 }{ 7 } = \frac { 14 + 4 }{ 7 } = \frac { 18 }{ 7 } \)
Now, divide 7 by \( \frac { 18 }{ 7 } \) by multiplying 7 by the reciprocal of \( \frac { 18 }{ 7 } \).
The reciprocal of \( \frac { 18 }{ 7 } \) is \( \frac { 7 }{ 18 } \).
\( 7 \times \frac { 7 }{ 18 } = \frac { 7 \times 7 }{ 18 } = \frac { 49 }{ 18 } \)
Converting to a mixed fraction: \( 49 \div 18 = 2 \) with a remainder of 13.
\( \implies 2\frac { 13 }{ 18 } \)
In simple words: For division involving mixed numbers, always convert the mixed number to an improper fraction first. Then, flip the second fraction and multiply.

Exam Tip: Always convert mixed numbers to improper fractions before performing division. Simplify the resulting fraction to its lowest terms or convert it to a mixed number if necessary.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 45)

Question. શોધો :
(i) \( \frac { 3 }{ 5 } \div \frac { 1 }{ 2 } \)
(ii) \( \frac { 1 }{ 2 } \div \frac { 3 }{ 5 } \)
(iii) \( 2\frac { 1 }{ 2 } \div \frac { 3 }{ 5 } \)
Answer: We need to calculate the value of each division expression:
(i) To calculate \( \frac { 3 }{ 5 } \div \frac { 1 }{ 2 } \), we multiply \( \frac { 3 }{ 5 } \) by the reciprocal of \( \frac { 1 }{ 2 } \).
The reciprocal of \( \frac { 1 }{ 2 } \) is \( \frac { 2 }{ 1 } \).
\( \frac { 3 }{ 5 } \times \frac { 2 }{ 1 } = \frac { 3 \times 2 }{ 5 \times 1 } = \frac { 6 }{ 5 } \)
Converting to a mixed fraction: \( 6 \div 5 = 1 \) with a remainder of 1.
\( \implies 1\frac { 1 }{ 5 } \)
(ii) To calculate \( \frac { 1 }{ 2 } \div \frac { 3 }{ 5 } \), we multiply \( \frac { 1 }{ 2 } \) by the reciprocal of \( \frac { 3 }{ 5 } \).
The reciprocal of \( \frac { 3 }{ 5 } \) is \( \frac { 5 }{ 3 } \).
\( \frac { 1 }{ 2 } \times \frac { 5 }{ 3 } = \frac { 1 \times 5 }{ 2 \times 3 } = \frac { 5 }{ 6 } \)
(iii) To calculate \( 2\frac { 1 }{ 2 } \div \frac { 3 }{ 5 } \), first convert the mixed fraction \( 2\frac { 1 }{ 2 } \) to an improper fraction.
\( 2\frac { 1 }{ 2 } = \frac { (2 \times 2) + 1 }{ 2 } = \frac { 4 + 1 }{ 2 } = \frac { 5 }{ 2 } \)
Now, divide \( \frac { 5 }{ 2 } \) by \( \frac { 3 }{ 5 } \) by multiplying \( \frac { 5 }{ 2 } \) by the reciprocal of \( \frac { 3 }{ 5 } \).
The reciprocal of \( \frac { 3 }{ 5 } \) is \( \frac { 5 }{ 3 } \).
\( \frac { 5 }{ 2 } \times \frac { 5 }{ 3 } = \frac { 5 \times 5 }{ 2 \times 3 } = \frac { 25 }{ 6 } \)
Converting to a mixed fraction: \( 25 \div 6 = 4 \) with a remainder of 1.
\( \implies 4\frac { 1 }{ 6 } \)
In simple words: When dividing fractions, flip the second fraction (find its reciprocal) and then multiply the fractions. If there's a mixed number, change it to an improper fraction first.

Exam Tip: Pay close attention to the order of operations and the correct reciprocal for the divisor. Always simplify and convert to mixed numbers where appropriate.

Answer:
(iv) To calculate \( 5\frac { 1 }{ 6 } \div \frac { 9 }{ 2 } \), first convert the mixed fraction \( 5\frac { 1 }{ 6 } \) to an improper fraction.
\( 5\frac { 1 }{ 6 } = \frac { (6 \times 5) + 1 }{ 6 } = \frac { 30 + 1 }{ 6 } = \frac { 31 }{ 6 } \)
Now, divide \( \frac { 31 }{ 6 } \) by \( \frac { 9 }{ 2 } \) by multiplying \( \frac { 31 }{ 6 } \) by the reciprocal of \( \frac { 9 }{ 2 } \).
The reciprocal of \( \frac { 9 }{ 2 } \) is \( \frac { 2 }{ 9 } \).
\( \frac { 31 }{ 6 } \times \frac { 2 }{ 9 } = \frac { 31 \times 2 }{ 6 \times 9 } \)
Simplify before multiplying: \( \frac { 31 \times 1 }{ 3 \times 9 } \) (by dividing 2 and 6 by 2)
\( \implies \frac { 31 }{ 27 } \)
Converting to a mixed fraction: \( 31 \div 27 = 1 \) with a remainder of 4.
\( \implies 1\frac { 4 }{ 27 } \)
In simple words: When a problem involves a mixed number and division, first change the mixed number into an improper fraction. Then, flip the second fraction and multiply. Simplify the final answer if possible.

Exam Tip: Always look for opportunities to simplify fractions before multiplying to make calculations easier and reduce the chance of errors.

પાઠ્યપુસ્તકમાંથી : (પાઠ્યપુસ્તક પાન નંબર 46)

Question. નીચેનું કોષ્ટક જુઓ અને ખાલી જગ્યા પૂરોઃ
Answer: Here is the completed table showing the place values for various numbers:

Exam Tip: Understanding place value is fundamental for working with decimals. Remember that moving a digit one place to the left increases its value tenfold, and moving it one place to the right decreases its value tenfold.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 50)

Question 1. શોધો :
(i) \( 2.7 \times 4 \)
(ii) \( 1.8 \times 1.2 \)
(iii) \( 2.3 \times 4.35 \)
Answer: Let's find the product for each of the given multiplications:
(i) To calculate \( 2.7 \times 4 \):
First, multiply the numbers as whole numbers: \( 27 \times 4 = 108 \).
Since there is one digit after the decimal point in 2.7 (and zero in 4), the result will have one digit after the decimal point.
So, \( 2.7 \times 4 = 10.8 \).
(ii) To calculate \( 1.8 \times 1.2 \):
First, multiply the numbers as whole numbers: \( 18 \times 12 = 216 \).
There is one digit after the decimal point in 1.8 and one digit after the decimal point in 1.2. So, the total number of digits after the decimal point in the product will be \( 1 + 1 = 2 \).
So, \( 1.8 \times 1.2 = 2.16 \).
In simple words: To multiply decimal numbers, first ignore the decimal points and multiply the numbers like whole numbers. Then, count the total number of digits after the decimal points in all the original numbers. Put the decimal point in your answer so that it has the same total number of digits after it.

Exam Tip: The number of decimal places in the product is equal to the sum of the decimal places in the numbers being multiplied. This is a crucial rule for decimal multiplication.

Answer:
(iii) To calculate \( 2.3 \times 4.35 \):
First, multiply the numbers as whole numbers: \( 23 \times 435 = 10005 \).
There is one digit after the decimal point in 2.3 and two digits after the decimal point in 4.35. So, the total number of digits after the decimal point in the product will be \( 1 + 2 = 3 \).
So, \( 2.3 \times 4.35 = 10.005 \).
In simple words: For multiplying decimals, multiply them without the points. Count how many decimal places are in all the numbers you started with, and that's how many places your answer will need.

Exam Tip: Carefully count the total number of digits after the decimal point in all numbers being multiplied. A small error in counting can lead to a significantly incorrect answer.

Question 2. ઉપરના પ્રશ્ન 1માં મળેલ જવાબને ઊતરતા ક્રમમાં ગોઠવો.
Answer: The answers obtained in Question 1 are 10.8, 2.16, and 10.005.
To arrange them in descending order, we compare them:
Comparing 10.8 and 10.005, the whole number parts (10) are the same. Now, compare the decimal parts. In 10.8, the tenths digit is 8. In 10.005, the tenths digit is 0.
Since \( 8 > 0 \), we know that \( 10.8 > 10.005 \).
Now compare with 2.16. Clearly, 10.8 and 10.005 are both greater than 2.16.
So, the numbers in descending order are: 10.8, 10.005, 2.16.
In simple words: To put decimal numbers in order, first look at the whole numbers. If they are the same, then look at the digit right after the decimal point, then the next digit, and so on. "Descending order" means going from the biggest number to the smallest number.

Exam Tip: When comparing decimals, always compare digits from left to right, starting with the largest place value. If the whole number parts are the same, move to the tenths, then hundredths, and so on.

પાઠ્યપુસ્તકમાંથી : (પાઠ્યપુસ્તક પાન નંબર 51)

Question. નીચે આપેલ કોષ્ટકને જુઓ અને ખાલી જગ્યા ભરોઃ
Answer: Here is the completed table showing the multiplication of decimal numbers by 10, 100, and 1000:

Exam Tip: Multiplying by powers of 10 involves shifting the decimal point to the right. The number of places to shift is equal to the number of zeros in the power of 10.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 51)

Question. શોધો :
(i) \( 0.3 \times 10 \)
(ii) \( 1.2 \times 100 \)
(iii) \( 56.3 \times 1000 \)
Answer: Let's find the product for each of these multiplications:
(i) To calculate \( 0.3 \times 10 \):
The multiplying number, 10, has one zero.
So, we will move the decimal point one place to the right.
\( 0.3 \times 10 = 3.0 = 3 \).
(ii) To calculate \( 1.2 \times 100 \):
The multiplying number, 100, has two zeros.
So, we will move the decimal point two places to the right.
Multiplying \( 1.2 \times 100 \): If we move the decimal two places right from 1.2, it becomes 120.
So, \( 1.2 \times 100 = 120 \).
In simple words: When multiplying a decimal by 10, 100, or 1000, count the number of zeros. That tells you how many places to move the decimal point to the right in the first number.

Exam Tip: Ensure you move the decimal point correctly. If there aren't enough digits, add zeros to the right of the number to complete the shift.

Answer:
(iii) To calculate \( 56.3 \times 1000 \):
The multiplying number, 1000, has three zeros.
So, we will move the decimal point three places to the right.
Multiplying \( 56.3 \times 1000 \): If we move the decimal three places right from 56.3, we get 56300.
So, \( 56.3 \times 1000 = 56300 \).
In simple words: To multiply a decimal by 10, 100, or 1000, move the decimal point to the right by the same number of places as there are zeros in the multiplier. Add trailing zeros if needed.

Exam Tip: Practise moving the decimal point for multiplication by powers of 10. This skill is vital for quick and accurate calculations with decimals.

પાઠ્યપુસ્તકમાંથી : (પાઠ્યપુસ્તક પાન નંબર 52)

Question. નીચે આપેલ કોષ્ટકને જુઓ અને ખાલી જગ્યા પૂરોઃ
Answer: Here is the completed table showing the division of decimal numbers by 10, 100, and 1000:

Exam Tip: Division by powers of 10 involves shifting the decimal point to the left. The number of places to shift is equal to the number of zeros in the power of 10.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 53)

Question. શોધો :
(i) \( 235.4 \div 10 \)
(ii) \( 235.4 \div 100 \)
(iii) \( 235.4 \div 1000 \)
Answer: Let's find the result for each of these division problems:
(i) To calculate \( 235.4 \div 10 \):
Since we are dividing by 10 (which has one zero), we move the decimal point one place to the left.
So, \( 235.4 \div 10 = 23.54 \).
(ii) To calculate \( 235.4 \div 100 \):
Since we are dividing by 100 (which has two zeros), we move the decimal point two places to the left.
So, \( 235.4 \div 100 = 2.354 \).
(iii) To calculate \( 235.4 \div 1000 \):
Since we are dividing by 1000 (which has three zeros), we move the decimal point three places to the left.
So, \( 235.4 \div 1000 = 0.2354 \).
In simple words: When dividing a decimal number by 10, 100, or 1000, simply shift the decimal point to the left by the number of zeros in the divisor.

Exam Tip: Always count the zeros in the divisor (10, 100, or 1000) to know how many places to move the decimal point to the left in the dividend.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 53)

Question. શોધો :
(i) \( 35.7 \div 3 \)
(ii) \( 25.5 \div 3 \)
Answer: Let's find the result for each of these division problems:
(i) To calculate \( 35.7 \div 3 \):
We can write this as \( \frac { 35.7 }{ 3 } \). Convert the decimal to a fraction: \( \frac { 357 }{ 10 } \).
So, \( \frac { 357 }{ 10 } \div 3 = \frac { 357 }{ 10 } \times \frac { 1 }{ 3 } \)
\( = \frac { 357 \times 1 }{ 10 \times 3 } = \frac { 357 }{ 30 } \)
\( = \frac { 119 }{ 10 } \) (by dividing numerator and denominator by 3)
\( \implies 11.9 \)
(ii) To calculate \( 25.5 \div 3 \):
We can write this as \( \frac { 25.5 }{ 3 } \). Convert the decimal to a fraction: \( \frac { 255 }{ 10 } \).
So, \( \frac { 255 }{ 10 } \div 3 = \frac { 255 }{ 10 } \times \frac { 1 }{ 3 } \)
\( = \frac { 255 \times 1 }{ 10 \times 3 } = \frac { 255 }{ 30 } \)
\( = \frac { 85 }{ 10 } \) (by dividing numerator and denominator by 3)
\( \implies 8.5 \)
In simple words: To divide a decimal by a whole number, you can divide them just like whole numbers and then place the decimal point in the answer directly above the decimal point in the number being divided. Alternatively, convert the decimal to a fraction, then divide.

Exam Tip: When dividing a decimal by a whole number, ensure the decimal point in the quotient is placed exactly above the decimal point in the dividend.

Question 1. (i) Find the value of \( 43.15 \div 5 \).
Answer: We need to calculate the division of \( 43.15 \) by \( 5 \). First, we convert the decimal to a fraction: \( 43.15 = \frac{4315}{100} \). Then, dividing by \( 5 \) is the same as multiplying by \( \frac{1}{5} \). \[ 43.15 \div 5 = \frac{4315}{100} \times \frac{1}{5} \] \[ = \frac{4315 \times 1}{100 \times 5} \] \[ = \frac{4315}{500} \] When we simplify this, we can divide the numerator and denominator by 5: \[ = \frac{863}{100} \] Converting back to a decimal, we get the final answer: \[ = 8.63 \]In simple words: To divide \( 43.15 \) by \( 5 \), change \( 43.15 \) into a fraction, multiply it by \( \frac{1}{5} \), then simplify the fraction and change it back to a decimal.

Exam Tip: When dividing decimals by whole numbers, convert the decimal to a fraction first, or simply perform long division and place the decimal point correctly in the quotient.

Question 1. (ii) Find the value of \( 82.44 \div 6 \).
Answer: To find the value of \( 82.44 \) divided by \( 6 \), we first represent \( 82.44 \) as a fraction: \( 82.44 = \frac{8244}{100} \). Dividing by \( 6 \) is similar to multiplying by \( \frac{1}{6} \). \[ 82.44 \div 6 = \frac{8244}{100} \times \frac{1}{6} \] \[ = \frac{8244 \times 1}{100 \times 6} \] \[ = \frac{8244}{600} \] Simplifying the fraction by dividing the numerator and denominator by 6 gives: \[ = \frac{1374}{100} \] Converting this fraction back to a decimal number, we get: \[ = 13.74 \]In simple words: To divide \( 82.44 \) by \( 6 \), write \( 82.44 \) as a fraction, multiply it by \( \frac{1}{6} \), and then simplify the fraction to get the final decimal result.

Exam Tip: Always double-check your multiplication and division steps to avoid calculation mistakes when dealing with decimals and fractions.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 53)

Question 2. Find the values for the following divisions:
(i) \( 15.5 \div 5 \)
(i) \( 126.35 \div 7 \)
Answer:
(i) To divide \( 15.5 \) by \( 5 \): \[ 15.5 \div 5 = \frac{155}{10} \times \frac{1}{5} \] \[ = \frac{155 \times 1}{10 \times 5} \] \[ = \frac{155}{50} \] Simplifying the fraction by dividing the numerator and denominator by 5: \[ = \frac{31}{10} \] Converting back to a decimal gives: \[ = 3.1 \] (i) To divide \( 126.35 \) by \( 7 \): \[ 126.35 \div 7 = \frac{12635}{100} \times \frac{1}{7} \] \[ = \frac{12635 \times 1}{100 \times 7} \] \[ = \frac{12635}{700} \] Simplifying the fraction by dividing the numerator and denominator by 7: \[ = \frac{1805}{100} \] Converting back to a decimal gives: \[ = 18.05 \]In simple words: For both problems, change the decimal to a fraction, then multiply by the reciprocal of the divisor. Simplify the resulting fraction and change it back to a decimal.

Exam Tip: Remember that dividing by a number is equivalent to multiplying by its reciprocal. This method works well for both whole numbers and fractions.

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર 54)

Question 1. (i) Find the value of \( \frac{7.75}{0.25} \).
Answer: To find the value of \( \frac{7.75}{0.25} \), we first convert both decimal numbers into fractions. We have \( 7.75 = \frac{775}{100} \) and \( 0.25 = \frac{25}{100} \). Now, we can rewrite the division as: \[ 7.75 \div 0.25 = \frac{775}{100} \div \frac{25}{100} \] To divide by a fraction, we multiply by its reciprocal: \[ = \frac{775}{100} \times \frac{100}{25} \] The \( 100 \) in the numerator and denominator cancel out, so we simplify: \[ = \frac{775}{25} \] Performing the division, we get: \[ = 31 \] Thus, \( \frac{7.75}{0.25} = 31 \).In simple words: Change both decimals into fractions. Then, flip the second fraction and multiply. The 100s cancel out, leaving a simple division to solve.

Exam Tip: When dividing by a decimal, it is often easier to first convert both numbers into fractions or to shift the decimal point in both numbers until the divisor is a whole number.

Question 1. (ii) Find the value of \( \frac{42.8}{0.02} \).
Answer: To evaluate \( \frac{42.8}{0.02} \), we convert the decimal numbers into fractions. We have \( 42.8 = \frac{428}{10} \) and \( 0.02 = \frac{2}{100} \). The division can be written as: \[ 42.8 \div 0.02 = \frac{428}{10} \div \frac{2}{100} \] Now, we multiply the first fraction by the reciprocal of the second fraction: \[ = \frac{428}{10} \times \frac{100}{2} \] We can simplify this expression. \( \frac{100}{10} = 10 \) and \( \frac{428}{2} = 214 \). \[ = 214 \times 10 \] Performing the multiplication gives: \[ = 2140 \] Therefore, \( \frac{42.8}{0.02} = 2140 \).In simple words: Turn both decimals into fractions. Then, multiply the first fraction by the flipped second fraction. Simplify by dividing numbers before multiplying to make it easier.

Exam Tip: Simplify fractions or cancel common factors before multiplying to make calculations faster and reduce potential errors.

Question 1. (iii) Find the value of \( \frac{5.6}{1.4} \).
Answer: To calculate \( \frac{5.6}{1.4} \), we convert the decimal numbers into fractions. We have \( 5.6 = \frac{56}{10} \) and \( 1.4 = \frac{14}{10} \). Now, the division expression becomes: \[ 5.6 \div 1.4 = \frac{56}{10} \div \frac{14}{10} \] To divide by a fraction, we multiply by its reciprocal: \[ = \frac{56}{10} \times \frac{10}{14} \] The \( 10 \) in the numerator and denominator cancel each other out, leaving: \[ = \frac{56}{14} \] Performing the division, we get: \[ = 4 \] Thus, \( \frac{5.6}{1.4} = 4 \).In simple words: Convert the decimals into fractions. Multiply the first fraction by the inverse of the second fraction. The tens will cancel, making the division much simpler.

Exam Tip: When the divisor and dividend have the same number of decimal places, you can remove the decimal points and divide the whole numbers directly.