Chapter 2 Fractions and Decimals

Given:

Total number of students in the class = 40.

Fraction of students who like English = $\frac{1}{5}$ of the total students.

Fraction of students who like Mathematics = $\frac{2}{5}$ of the total students.

Remaining students like Science.

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To Find:

(i) Number of students who like English.

(ii) Number of students who like Mathematics.

(iii) Fraction of students who like Science.

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Solution:

(i) Number of students who like to study English:

This is $\frac{1}{5}$ of the total number of students.

Number of students (English) = $\frac{1}{5} \times 40$

Number of students (English) = $\frac{40}{5}$

Number of students (English) = $\mathbf{8}$

So, 8 students like to study English.

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(ii) Number of students who like to study Mathematics:

This is $\frac{2}{5}$ of the total number of students.

Number of students (Mathematics) = $\frac{2}{5} \times 40$

Number of students (Mathematics) = $\frac{2 \times 40}{5}$

Number of students (Mathematics) = $\frac{80}{5}$

Number of students (Mathematics) = $\mathbf{16}$

So, 16 students like to study Mathematics.

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(iii) What fraction of the total number of students like to study Science?

The students who like Science are the remaining students.

First, find the fraction of students who like either English or Mathematics:

Fraction (English or Mathematics) = Fraction (English) + Fraction (Mathematics)

Fraction (English or Mathematics) = $\frac{1}{5} + \frac{2}{5} = \frac{1+2}{5} = \frac{3}{5}$

The total fraction representing all students is 1 (or $\frac{5}{5}$).

The fraction of students who like Science is the total fraction minus the fraction who like English or Mathematics.

Fraction (Science) = Total Fraction - Fraction (English or Mathematics)

Fraction (Science) = $1 - \frac{3}{5}$

Fraction (Science) = $\frac{5}{5} - \frac{3}{5} = \frac{5-3}{5} = \frac{2}{5}$

The fraction of the total number of students who like to study Science is $\mathbf{\frac{2}{5}}$.

Alternatively, we can find the number of students who like Science first.

Number of students (English) = 8

Number of students (Mathematics) = 16

Number of students (English or Mathematics) = $8 + 16 = 24$

Number of students (Science) = Total students - Number of students (English or Mathematics)

Number of students (Science) = $40 - 24 = 16$

Fraction (Science) = $\frac{\text{Number of students (Science)}}{\text{Total number of students}}$

Fraction (Science) = $\frac{16}{40}$

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 8.

Fraction (Science) = $\frac{\cancel{16}^{2}}{\cancel{40}_{5}} = \frac{2}{5}$

We need to match each mathematical expression with the drawing that represents it.

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Let's analyse each drawing:

Drawing (a) shows 3 groups of shapes. Each shape is divided into 3 equal parts, and 2 parts are shaded. This represents the fraction $\frac{2}{3}$. So, drawing (c) shows $3 \times \frac{2}{3}$.

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Drawing (b) shows 2 groups of shapes. Each shape is divided into 2 equal parts, and 1 part is shaded. This represents the fraction $\frac{1}{2}$. So, drawing (a) shows $2 \times \frac{1}{2}$.

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Drawing (c) shows 3 groups of shapes. Each shape is divided into 4 equal parts, and 1 part is shaded. This represents the fraction $\frac{1}{4}$. So, drawing (d) shows $3 \times \frac{1}{4}$.

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Drawing (d) shows 2 groups of shapes. Each shape is divided into 5 equal parts, and 1 part is shaded. This represents the fraction $\frac{1}{5}$. So, drawing (b) shows $2 \times \frac{1}{5}$.

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Now we match the given expressions with the drawings:

(i) $2 \times \frac{1}{5}$ matches drawing (d).

(ii) $2 \times \frac{1}{2}$ matches drawing (b).

(iii) $3 \times \frac{2}{3}$ matches drawing (a).

(iv) $3 \times \frac{1}{4}$ matches drawing (c).

We need to match each mathematical expression with the drawing that represents it along with the result.

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Let's analyse each drawing and its result:

Drawing (a) shows 2 identical groups. Each group represents $\frac{1}{3}$ (1 shaded part out of 3). The result shows $\frac{2}{3}$ (2 shaded parts out of 3). This represents $2 \times \frac{1}{3} = \frac{2}{3}$.

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Drawing (b) shows 3 identical groups. Each group represents $\frac{3}{4}$ (3 shaded parts out of 4). The result shows 2 whole shapes completely shaded and a third shape with 1 shaded part out of 4. This represents $2$ and $\frac{1}{4}$, or $2\frac{1}{4}$. This shows $3 \times \frac{3}{4} = \frac{9}{4} = 2\frac{1}{4}$.

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Drawing (c) shows 3 identical groups. Each group represents $\frac{1}{5}$ (1 shaded part out of 5). The result shows $\frac{3}{5}$ (3 shaded parts out of 5). This represents $3 \times \frac{1}{5} = \frac{3}{5}$.

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Now we match the given mathematical expressions with the drawings:

(i) $3 \times \frac{1}{5} = \frac{3}{5}$ is shown by drawing (c).

(ii) $2 \times \frac{1}{3} = \frac{2}{3}$ is shown by drawing (a).

(iii) $3 \times \frac{3}{4} = 2\frac{1}{4}$ is shown by drawing (b).

We multiply the integers by the fractions, then reduce the result to the lowest form and convert it into a mixed fraction if it is an improper fraction.

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(i) $7 \times \frac{3}{5}$

$7 \times \frac{3}{5} = \frac{7 \times 3}{5} = \frac{21}{5}$

This is an improper fraction. To convert it to a mixed fraction, divide 21 by 5.

$21 \div 5 = 4$ with a remainder of $1$.

$\frac{21}{5} = \mathbf{4\frac{1}{5}}$

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(ii) $4 \times \frac{1}{3}$

$4 \times \frac{1}{3} = \frac{4 \times 1}{3} = \frac{4}{3}$

This is an improper fraction. Divide 4 by 3.

$4 \div 3 = 1$ with a remainder of $1$.

$\frac{4}{3} = \mathbf{1\frac{1}{3}}$

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(iii) $2 \times \frac{6}{7}$

$2 \times \frac{6}{7} = \frac{2 \times 6}{7} = \frac{12}{7}$

This is an improper fraction. Divide 12 by 7.

$12 \div 7 = 1$ with a remainder of $5$.

$\frac{12}{7} = \mathbf{1\frac{5}{7}}$

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(iv) $5 \times \frac{2}{9}$

$5 \times \frac{2}{9} = \frac{5 \times 2}{9} = \frac{10}{9}$

This is an improper fraction. Divide 10 by 9.

$10 \div 9 = 1$ with a remainder of $1$.

$\frac{10}{9} = \mathbf{1\frac{1}{9}}$

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(v) $\frac{2}{3} \times 4$}

$\frac{2}{3} \times 4 = \frac{2 \times 4}{3} = \frac{8}{3}$

This is an improper fraction. Divide 8 by 3.

$8 \div 3 = 2$ with a remainder of $2$.

$\frac{8}{3} = \mathbf{2\frac{2}{3}}$

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(vi) $\frac{5}{2} \times 6$}

We can simplify before multiplying:

$\frac{5}{\cancel{2}_{1}} \times \cancel{6}^{3} = 5 \times 3 = 15$

Alternatively, multiply first: $\frac{5 \times 6}{2} = \frac{30}{2} = 15$.

The result is an integer. Converting to a mixed fraction is not applicable in the standard sense, but 15 can be written as $15\frac{0}{1}$ or similar.

The answer is $\mathbf{15}$.

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(vii) $11 \times \frac{4}{7}$}

$11 \times \frac{4}{7} = \frac{11 \times 4}{7} = \frac{44}{7}$

This is an improper fraction. Divide 44 by 7.

$44 \div 7 = 6$ with a remainder of $2$.

$\frac{44}{7} = \mathbf{6\frac{2}{7}}$

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(viii) $20 \times \frac{4}{5}$}

We can simplify before multiplying:

$\cancel{20}^{4} \times \frac{4}{\cancel{5}_{1}} = 4 \times 4 = 16$

Alternatively, multiply first: $\frac{20 \times 4}{5} = \frac{80}{5} = 16$.

The result is an integer.

The answer is $\mathbf{16}$.

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(ix) $13 \times \frac{1}{3}$}

$13 \times \frac{1}{3} = \frac{13 \times 1}{3} = \frac{13}{3}$

This is an improper fraction. Divide 13 by 3.

$13 \div 3 = 4$ with a remainder of $1$.

$\frac{13}{3} = \mathbf{4\frac{1}{3}}$

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(x) $15 \times \frac{3}{5}$}

We can simplify before multiplying:

$\cancel{15}^{3} \times \frac{3}{\cancel{5}_{1}} = 3 \times 3 = 9$

Alternatively, multiply first: $\frac{15 \times 3}{5} = \frac{45}{5} = 9$.

The result is an integer.

The answer is $\mathbf{9}$.

To shade the required fraction of shapes, we first count the total number of shapes in each box and then calculate how many shapes correspond to the given fraction. We will then shade that many shapes.

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(i) Shade $\frac{1}{2}$ of the circles in box (a).

Let's count the total number of circles in box (a). There are 3 rows and 4 columns of circles, so total circles are $3 \times 4 = 12$.

We need to shade $\frac{1}{2}$ of 12 circles.

Number of circles to shade = $\frac{1}{2} \times 12 = \frac{1 \times 12}{2} = \frac{12}{2} = 6$.

So, we need to shade 6 circles in box (a).

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(ii) Shade $\frac{2}{3}$ of the triangles in box (b).

Let's count the total number of triangles in box (b). There are 3 rows and 3 columns of triangles, so total triangles are $3 \times 3 = 9$.

We need to shade $\frac{2}{3}$ of 9 triangles.

Number of triangles to shade = $\frac{2}{3} \times 9 = \frac{2 \times 9}{3} = \frac{18}{3} = 6$.

So, we need to shade 6 triangles in box (b).

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(iii) Shade $\frac{3}{5}$ of the squares in box (c).

Let's count the total number of squares in box (c). There are 3 rows and 5 columns of squares, so total squares are $3 \times 5 = 15$.

We need to shade $\frac{3}{5}$ of 15 squares.

Number of squares to shade = $\frac{3}{5} \times 15 = \frac{3 \times 15}{5} = \frac{45}{5} = 9$.

So, we need to shade 9 squares in box (c).

We calculate the fraction of the given numbers. "of" indicates multiplication.

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(a) $\frac{1}{2}$ of:

(i) 24: $\frac{1}{2} \times 24 = \frac{24}{2} = \mathbf{12}$

(ii) 46: $\frac{1}{2} \times 46 = \frac{46}{2} = \mathbf{23}$

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(b) $\frac{2}{3}$ of:

(i) 18: $\frac{2}{3} \times 18 = \frac{2 \times \cancel{18}^{6}}{\cancel{3}_{1}} = 2 \times 6 = \mathbf{12}$

(ii) 27: $\frac{2}{3} \times 27 = \frac{2 \times \cancel{27}^{9}}{\cancel{3}_{1}} = 2 \times 9 = \mathbf{18}$

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(c) $\frac{3}{4}$ of:

(i) 16: $\frac{3}{4} \times 16 = \frac{3 \times \cancel{16}^{4}}{\cancel{4}_{1}} = 3 \times 4 = \mathbf{12}$

(ii) 36: $\frac{3}{4} \times 36 = \frac{3 \times \cancel{36}^{9}}{\cancel{4}_{1}} = 3 \times 9 = \mathbf{27}$

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(d) $\frac{4}{5}$ of:

(i) 20: $\frac{4}{5} \times 20 = \frac{4 \times \cancel{20}^{4}}{\cancel{5}_{1}} = 4 \times 4 = \mathbf{16}$

(ii) 35: $\frac{4}{5} \times 35 = \frac{4 \times \cancel{35}^{7}}{\cancel{5}_{1}} = 4 \times 7 = \mathbf{28}$

We multiply the integers by the mixed fractions. First, convert the mixed fraction to an improper fraction, then multiply, and finally convert the result back to a mixed fraction.

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(a) $3 \times 5\frac{1}{5}$

Convert the mixed fraction to an improper fraction: $5\frac{1}{5} = \frac{5 \times 5 + 1}{5} = \frac{25 + 1}{5} = \frac{26}{5}$.

Now multiply:

$3 \times \frac{26}{5} = \frac{3 \times 26}{5} = \frac{78}{5}$.

Convert the improper fraction to a mixed fraction. Divide 78 by 5.

$78 \div 5 = 15$ with a remainder of $3$.

$\frac{78}{5} = \mathbf{15\frac{3}{5}}$

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(b) $5 \times 6\frac{3}{4}$

Convert the mixed fraction to an improper fraction: $6\frac{3}{4} = \frac{6 \times 4 + 3}{4} = \frac{24 + 3}{4} = \frac{27}{4}$.

Now multiply:

$5 \times \frac{27}{4} = \frac{5 \times 27}{4} = \frac{135}{4}$.

Convert the improper fraction to a mixed fraction. Divide 135 by 4.

$135 \div 4 = 33$ with a remainder of $3$.

$\frac{135}{4} = \mathbf{33\frac{3}{4}}$

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(c) $7 \times 2\frac{1}{4}$

Convert the mixed fraction to an improper fraction: $2\frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$.

Now multiply:

$7 \times \frac{9}{4} = \frac{7 \times 9}{4} = \frac{63}{4}$.

Convert the improper fraction to a mixed fraction. Divide 63 by 4.

$63 \div 4 = 15$ with a remainder of $3$.

$\frac{63}{4} = \mathbf{15\frac{3}{4}}$

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(d) $4 \times 6\frac{1}{3}$}

Convert the mixed fraction to an improper fraction: $6\frac{1}{3} = \frac{6 \times 3 + 1}{3} = \frac{18 + 1}{3} = \frac{19}{3}$.

Now multiply:

$4 \times \frac{19}{3} = \frac{4 \times 19}{3} = \frac{76}{3}$.

Convert the improper fraction to a mixed fraction. Divide 76 by 3.

$76 \div 3 = 25$ with a remainder of $1$.

$\frac{76}{3} = \mathbf{25\frac{1}{3}}$

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(e) $3\frac{1}{4} \times 6$}

Convert the mixed fraction to an improper fraction: $3\frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}$.

Now multiply:

$\frac{13}{4} \times 6 = \frac{13}{\cancel{4}_{2}} \times \cancel{6}^{3} = \frac{13 \times 3}{2} = \frac{39}{2}$.

Convert the improper fraction to a mixed fraction. Divide 39 by 2.

$39 \div 2 = 19$ with a remainder of $1$.

$\frac{39}{2} = \mathbf{19\frac{1}{2}}$

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(f) $3\frac{2}{5} \times 8$}

Convert the mixed fraction to an improper fraction: $3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$.

Now multiply:

$\frac{17}{5} \times 8 = \frac{17 \times 8}{5} = \frac{136}{5}$.

Convert the improper fraction to a mixed fraction. Divide 136 by 5.

$136 \div 5 = 27$ with a remainder of $1$.

$\frac{136}{5} = \mathbf{27\frac{1}{5}}$

We find the fraction of the given mixed fractions. "of" indicates multiplication. First, convert the mixed fractions to improper fractions, then multiply, and finally simplify the result.

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(a) $\frac{1}{2}$ of:

(i) $2\frac{3}{4}$: Convert the mixed fraction: $2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$.

Now multiply: $\frac{1}{2} \times \frac{11}{4} = \frac{1 \times 11}{2 \times 4} = \frac{11}{8}$.

The result is $\mathbf{\frac{11}{8}}$. (This is an improper fraction, sometimes conversion to mixed fraction might be expected: $1\frac{3}{8}$)

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(ii) $4\frac{2}{9}$: Convert the mixed fraction: $4\frac{2}{9} = \frac{4 \times 9 + 2}{9} = \frac{36 + 2}{9} = \frac{38}{9}$.

Now multiply: $\frac{1}{2} \times \frac{38}{9} = \frac{1}{\cancel{2}_{1}} \times \frac{\cancel{38}^{19}}{9} = \frac{19}{9}$.

The result is $\mathbf{\frac{19}{9}}$. (This is an improper fraction, sometimes conversion to mixed fraction might be expected: $2\frac{1}{9}$)

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(b) $\frac{5}{8}$ of:

(i) $3\frac{5}{6}$: Convert the mixed fraction: $3\frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}$.

Now multiply: $\frac{5}{8} \times \frac{23}{6} = \frac{5 \times 23}{8 \times 6} = \frac{115}{48}$.

The result is $\mathbf{\frac{115}{48}}$. (This is an improper fraction, sometimes conversion to mixed fraction might be expected: $2\frac{19}{48}$)

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(ii) $9\frac{2}{3}$: Convert the mixed fraction: $9\frac{2}{3} = \frac{9 \times 3 + 2}{3} = \frac{27 + 2}{3} = \frac{29}{3}$.

Now multiply: $\frac{5}{8} \times \frac{29}{3} = \frac{5 \times 29}{8 \times 3} = \frac{145}{24}$.

The result is $\mathbf{\frac{145}{24}}$. (This is an improper fraction, sometimes conversion to mixed fraction might be expected: $6\frac{1}{24}$)

Given:

Total quantity of water = 5 litres.

Fraction of water consumed by Vidya = $\frac{2}{5}$ of the total water.

Pratap consumed the remaining water.

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To Find:

(i) Quantity of water Vidya drank.

(ii) Fraction of water Pratap drank.

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Solution:

(i) How much water did Vidya drink?

Vidya consumed $\frac{2}{5}$ of the total water (5 litres).

Quantity of water Vidya drank = $\frac{2}{5} \times 5 \text{ litres}$

Quantity of water Vidya drank = $\frac{2 \times \cancel{5}^{1}}{\cancel{5}_{1}} = 2 \times 1 = 2$ litres.

Vidya drank 2 litres of water.

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(ii) What fraction of the total quantity of water did Pratap drink?

Pratap drank the remaining water. The total fraction of water is 1 (or $\frac{5}{5}$).

Fraction of water Pratap drank = Total fraction of water - Fraction Vidya drank

Fraction of water Pratap drank = $1 - \frac{2}{5}$

Fraction of water Pratap drank = $\frac{5}{5} - \frac{2}{5} = \frac{5 - 2}{5} = \frac{3}{5}$

Pratap drank $\mathbf{\frac{3}{5}}$ of the total quantity of water.

Alternatively, we can find the quantity of water Pratap drank first.

Quantity of water Pratap drank = Total water - Quantity Vidya drank

Quantity of water Pratap drank = $5 \text{ litres} - 2 \text{ litres} = 3$ litres.

Now find the fraction Pratap drank out of the total quantity:

Fraction of water Pratap drank = $\frac{\text{Quantity Pratap drank}}{\text{Total quantity of water}}$

Fraction of water Pratap drank = $\frac{3 \text{ litres}}{5 \text{ litres}} = \frac{3}{5}$

Given:

Part of the book read in 1 hour = $\frac{1}{3}$.

Time for which Sushant reads = $2\frac{1}{5}$ hours.

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To Find:

Part of the book read in $2\frac{1}{5}$ hours.

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Solution:

If Sushant reads $\frac{1}{3}$ part of the book in 1 hour, then in $2\frac{1}{5}$ hours, he will read $2\frac{1}{5}$ times the amount he reads in 1 hour.

Part of the book read in $2\frac{1}{5}$ hours = (Part read in 1 hour) $\times$ (Time in hours)

Part of the book read = $\frac{1}{3} \times 2\frac{1}{5}$.

First, convert the mixed fraction to an improper fraction:

$2\frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}$.

Now multiply the fractions:

Part of the book read = $\frac{1}{3} \times \frac{11}{5}$

Multiply the numerators and multiply the denominators:

Part of the book read = $\frac{1 \times 11}{3 \times 5} = \frac{11}{15}$.

Sushant will read $\mathbf{\frac{11}{15}}$ part of the book in $2\frac{1}{5}$ hours.

We calculate the fraction of a fraction. "of" indicates multiplication.

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(i) $\frac{1}{4}$ of:

(a) $\frac{1}{4}$: $\frac{1}{4} \times \frac{1}{4} = \frac{1 \times 1}{4 \times 4} = \mathbf{\frac{1}{16}}$

(b) $\frac{3}{5}$: $\frac{1}{4} \times \frac{3}{5} = \frac{1 \times 3}{4 \times 5} = \mathbf{\frac{3}{20}}$

(c) $\frac{4}{3}$: $\frac{1}{4} \times \frac{4}{3} = \frac{1}{\cancel{4}_{1}} \times \frac{\cancel{4}^{1}}{3} = \frac{1 \times 1}{1 \times 3} = \mathbf{\frac{1}{3}}$

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(ii) $\frac{1}{7}$ of:

(a) $\frac{2}{9}$: $\frac{1}{7} \times \frac{2}{9} = \frac{1 \times 2}{7 \times 9} = \mathbf{\frac{2}{63}}$

(b) $\frac{6}{5}$: $\frac{1}{7} \times \frac{6}{5} = \frac{1 \times 6}{7 \times 5} = \mathbf{\frac{6}{35}}$

(c) $\frac{3}{10}$: $\frac{1}{7} \times \frac{3}{10} = \frac{1 \times 3}{7 \times 10} = \mathbf{\frac{3}{70}}$

We multiply the given fractions and mixed fractions, and then reduce the result to the lowest form where possible. If the result is an improper fraction, we will also convert it to a mixed fraction.

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(i) $\frac{2}{3} \times 2\frac{2}{3}$

First, convert the mixed fraction to an improper fraction:

$2\frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}$.

Now, multiply the fractions:

$\frac{2}{3} \times \frac{8}{3} = \frac{2 \times 8}{3 \times 3} = \frac{16}{9}$.

The fraction $\frac{16}{9}$ is an improper fraction. Let's convert it to a mixed fraction:

Divide 16 by 9. $16 \div 9 = 1$ with a remainder of $16 - (9 \times 1) = 7$.

So, $\frac{16}{9} = 1\frac{7}{9}$.

Result: $\mathbf{\frac{16}{9}}$ or $\mathbf{1\frac{7}{9}}$

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(ii) $\frac{2}{7} \times \frac{7}{9}$

Multiply the fractions:

$\frac{2}{7} \times \frac{7}{9} = \frac{2 \times 7}{7 \times 9}$.

We can cancel out the common factor 7:

$\frac{2 \times \cancel{7}^{1}}{\cancel{7}_{1} \times 9} = \frac{2 \times 1}{1 \times 9} = \frac{2}{9}$.

The fraction $\frac{2}{9}$ is a proper fraction and is in its lowest form.

Result: $\mathbf{\frac{2}{9}}$

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(iii) $\frac{3}{8} \times \frac{6}{4}$

Multiply the fractions:

$\frac{3}{8} \times \frac{6}{4} = \frac{3 \times 6}{8 \times 4} = \frac{18}{32}$.

Now, reduce the fraction to its lowest form. The greatest common factor of 18 and 32 is 2. Divide both numerator and denominator by 2:

$\frac{18 \div 2}{32 \div 2} = \frac{9}{16}$.

The fraction $\frac{9}{16}$ is a proper fraction and is in its lowest form.

Result: $\mathbf{\frac{9}{16}}$

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(iv) $\frac{9}{5} \times \frac{3}{5}$

Multiply the fractions:

$\frac{9}{5} \times \frac{3}{5} = \frac{9 \times 3}{5 \times 5} = \frac{27}{25}$.

The fraction $\frac{27}{25}$ is an improper fraction. Let's convert it to a mixed fraction:

Divide 27 by 25. $27 \div 25 = 1$ with a remainder of $27 - (25 \times 1) = 2$.

So, $\frac{27}{25} = 1\frac{2}{25}$.

Result: $\mathbf{\frac{27}{25}}$ or $\mathbf{1\frac{2}{25}}$

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(v) $\frac{1}{3} \times \frac{15}{8}$

Multiply the fractions:

$\frac{1}{3} \times \frac{15}{8} = \frac{1 \times 15}{3 \times 8}$.

We can cancel out the common factor 3:

$\frac{1 \times \cancel{15}^{5}}{\cancel{3}_{1} \times 8} = \frac{1 \times 5}{1 \times 8} = \frac{5}{8}$.

The fraction $\frac{5}{8}$ is a proper fraction and is in its lowest form.

Result: $\mathbf{\frac{5}{8}}$

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(vi) $\frac{11}{2} \times \frac{3}{10}$

Multiply the fractions:

$\frac{11}{2} \times \frac{3}{10} = \frac{11 \times 3}{2 \times 10} = \frac{33}{20}$.

The fraction $\frac{33}{20}$ is an improper fraction. Let's convert it to a mixed fraction:

Divide 33 by 20. $33 \div 20 = 1$ with a remainder of $33 - (20 \times 1) = 13$.

So, $\frac{33}{20} = 1\frac{13}{20}$.

Result: $\mathbf{\frac{33}{20}}$ or $\mathbf{1\frac{13}{20}}$

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(vii) $\frac{4}{5} \times \frac{12}{7}$

Multiply the fractions:

$\frac{4}{5} \times \frac{12}{7} = \frac{4 \times 12}{5 \times 7} = \frac{48}{35}$.

The fraction $\frac{48}{35}$ is an improper fraction. Let's convert it to a mixed fraction:

Divide 48 by 35. $48 \div 35 = 1$ with a remainder of $48 - (35 \times 1) = 13$.

So, $\frac{48}{35} = 1\frac{13}{35}$.

Result: $\mathbf{\frac{48}{35}}$ or $\mathbf{1\frac{13}{35}}$

We multiply the given fractions and mixed fractions. First, convert any mixed fraction to an improper fraction, then multiply the fractions, and simplify the result if possible.

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(i) $\frac{2}{5} \times 5\frac{1}{4}$

Convert $5\frac{1}{4}$ to an improper fraction: $5\frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{21}{4}$.

Multiply: $\frac{2}{5} \times \frac{21}{4} = \frac{\cancel{2}^{1}}{5} \times \frac{21}{\cancel{4}_{2}} = \frac{1 \times 21}{5 \times 2} = \frac{21}{10}$.

Result: $\mathbf{\frac{21}{10}}$ (or $2\frac{1}{10}$ as a mixed fraction).

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(ii) $6\frac{2}{5} \times \frac{7}{9}$}

Convert $6\frac{2}{5}$ to an improper fraction: $6\frac{2}{5} = \frac{6 \times 5 + 2}{5} = \frac{32}{5}$.

Multiply: $\frac{32}{5} \times \frac{7}{9} = \frac{32 \times 7}{5 \times 9} = \frac{224}{45}$.

Result: $\mathbf{\frac{224}{45}}$ (or $4\frac{44}{45}$ as a mixed fraction).

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(iii) $\frac{3}{2} \times 5\frac{1}{3}$}

Convert $5\frac{1}{3}$ to an improper fraction: $5\frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{16}{3}$.

Multiply: $\frac{3}{2} \times \frac{16}{3} = \frac{\cancel{3}^{1}}{\cancel{2}_{1}} \times \frac{\cancel{16}^{8}}{\cancel{3}_{1}} = \frac{1 \times 8}{1 \times 1} = 8$.

Result: $\mathbf{8}$.

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(iv) $\frac{5}{6} \times 2\frac{3}{7}$}

Convert $2\frac{3}{7}$ to an improper fraction: $2\frac{3}{7} = \frac{2 \times 7 + 3}{7} = \frac{14 + 3}{7} = \frac{17}{7}$.

Multiply: $\frac{5}{6} \times \frac{17}{7} = \frac{5 \times 17}{6 \times 7} = \frac{85}{42}$.

Result: $\mathbf{\frac{85}{42}}$ (or $2\frac{1}{42}$ as a mixed fraction).

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(v) $3\frac{2}{5} \times \frac{4}{7}$}

Convert $3\frac{2}{5}$ to an improper fraction: $3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$.

Multiply: $\frac{17}{5} \times \frac{4}{7} = \frac{17 \times 4}{5 \times 7} = \frac{68}{35}$.

Result: $\mathbf{\frac{68}{35}}$ (or $1\frac{33}{35}$ as a mixed fraction).

---

(vi) $2\frac{3}{5} \times 3$}

Convert $2\frac{3}{5}$ to an improper fraction: $2\frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}$.

Multiply: $\frac{13}{5} \times 3 = \frac{13 \times 3}{5} = \frac{39}{5}$.

Result: $\mathbf{\frac{39}{5}}$ (or $7\frac{4}{5}$ as a mixed fraction).

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(vii) $3\frac{4}{7} \times \frac{3}{5}$}

Convert $3\frac{4}{7}$ to an improper fraction: $3\frac{4}{7} = \frac{3 \times 7 + 4}{7} = \frac{21 + 4}{7} = \frac{25}{7}$.

Multiply: $\frac{25}{7} \times \frac{3}{5} = \frac{\cancel{25}^{5}}{7} \times \frac{3}{\cancel{5}_{1}} = \frac{5 \times 3}{7 \times 1} = \frac{15}{7}$.

Result: $\mathbf{\frac{15}{7}}$ (or $2\frac{1}{7}$ as a mixed fraction).

To determine which fraction is greater, we first calculate the value of each expression and then compare the resulting fractions.

---

(i) Comparing $\frac{2}{7}$ of $\frac{3}{4}$ and $\frac{3}{5}$ of $\frac{5}{8}$

Calculate the value of the first expression:

$\frac{2}{7}$ of $\frac{3}{4} = \frac{2}{7} \times \frac{3}{4} = \frac{\cancel{2}^{1}}{7} \times \frac{3}{\cancel{4}_{2}} = \frac{1 \times 3}{7 \times 2} = \frac{3}{14}$.

---

Calculate the value of the second expression:

$\frac{3}{5}$ of $\frac{5}{8} = \frac{3}{5} \times \frac{5}{8} = \frac{3}{\cancel{5}_{1}} \times \frac{\cancel{5}^{1}}{8} = \frac{3 \times 1}{1 \times 8} = \frac{3}{8}$.

---

Now compare $\frac{3}{14}$ and $\frac{3}{8}$.

We can compare fractions by finding a common denominator or by cross-multiplication. Using cross-multiplication:

Compare $\frac{3}{14}$ and $\frac{3}{8}$.

Multiply the numerator of the first fraction by the denominator of the second: $3 \times 8 = 24$.

Multiply the numerator of the second fraction by the denominator of the first: $3 \times 14 = 42$.

Since $24 < 42$, it means $\frac{3}{14} < \frac{3}{8}$.

Alternatively, when the numerators are the same, the fraction with the smaller denominator is greater. Since $8 < 14$, $\frac{3}{8} > \frac{3}{14}$.

Therefore, $\mathbf{\frac{3}{5}}$ of $\mathbf{\frac{5}{8}}$ is greater.

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(ii) Comparing $\frac{1}{2}$ of $\frac{6}{7}$ and $\frac{2}{3}$ of $\frac{3}{7}$

Calculate the value of the first expression:

$\frac{1}{2}$ of $\frac{6}{7} = \frac{1}{2} \times \frac{6}{7} = \frac{1}{\cancel{2}_{1}} \times \frac{\cancel{6}^{3}}{7} = \frac{1 \times 3}{1 \times 7} = \frac{3}{7}$.

---

Calculate the value of the second expression:

$\frac{2}{3}$ of $\frac{3}{7} = \frac{2}{3} \times \frac{3}{7} = \frac{2}{\cancel{3}_{1}} \times \frac{\cancel{3}^{1}}{7} = \frac{2 \times 1}{1 \times 7} = \frac{2}{7}$.

---

Now compare $\frac{3}{7}$ and $\frac{2}{7}$.

When the denominators are the same, the fraction with the larger numerator is greater. Since $3 > 2$, it means $\frac{3}{7} > \frac{2}{7}$.

Therefore, $\mathbf{\frac{1}{2}}$ of $\mathbf{\frac{6}{7}}$ is greater.

Given:

Number of saplings planted in a row = 4.

Distance between two adjacent saplings = $\frac{3}{4}$ m.

---

To Find:

The distance between the first and the last sapling.

---

Solution:

Let the saplings be denoted as S1, S2, S3, and S4 in the row.

The adjacent saplings are (S1, S2), (S2, S3), and (S3, S4).

The distance between S1 and S2 is $\frac{3}{4}$ m.

The distance between S2 and S3 is $\frac{3}{4}$ m.

The distance between S3 and S4 is $\frac{3}{4}$ m.

The distance between the first sapling (S1) and the last sapling (S4) is the sum of the distances between the adjacent saplings in between.

Number of gaps between 4 saplings in a row = Number of saplings - 1 = $4 - 1 = 3$ gaps.

Each gap has a distance of $\frac{3}{4}$ m.

Total distance between the first and the last sapling = Number of gaps $\times$ Distance per gap

Total distance = $3 \times \frac{3}{4}$ m.

Multiply the integer by the fraction:

$3 \times \frac{3}{4} = \frac{3 \times 3}{4} = \frac{9}{4}$ m.

The distance is $\frac{9}{4}$ m. We can express this as a mixed fraction.

Divide 9 by 4: $9 \div 4 = 2$ with a remainder of 1.

$\frac{9}{4} = 2\frac{1}{4}$ m.

The distance between the first and the last sapling is $\mathbf{2\frac{1}{4}}$ m.

Given:

Time Lipika spends reading everyday = $1\frac{3}{4}$ hours.

Number of days she takes to read the entire book = 6 days.

---

To Find:

Total number of hours required to read the book.

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Solution:

To find the total hours spent reading the book, we multiply the time spent reading per day by the number of days.

Total hours = (Time spent reading per day) $\times$ (Number of days)

Total hours = $1\frac{3}{4} \text{ hours/day} \times 6 \text{ days}$.

First, convert the mixed fraction to an improper fraction:

$1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}$.

Now multiply the fraction by the integer:

Total hours = $\frac{7}{4} \times 6$.

Multiply the numerator by the integer and keep the denominator:

Total hours = $\frac{7 \times 6}{4} = \frac{42}{4}$.

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2.

Total hours = $\frac{\cancel{42}^{21}}{\cancel{4}_{2}} = \frac{21}{2}$ hours.

The total time required is $\frac{21}{2}$ hours. We can convert this to a mixed number.

Divide 21 by 2: $21 \div 2 = 10$ with a remainder of 1.

Total hours = $10\frac{1}{2}$ hours.

It required $10\frac{1}{2}$ hours for her to read the book.

Given:

Distance covered by car using 1 litre of petrol = 16 km.

Quantity of petrol available = $2\frac{3}{4}$ litres.

---

To Find:

Distance covered using $2\frac{3}{4}$ litres of petrol.

---

Solution:

The distance covered is directly proportional to the amount of petrol used. To find the distance covered using $2\frac{3}{4}$ litres, we multiply the distance covered per litre by the total quantity of petrol.

Total distance covered = (Distance per litre) $\times$ (Total quantity of petrol)

Total distance covered = $16 \text{ km/litre} \times 2\frac{3}{4} \text{ litres}$.

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 the integer by the fraction:

Total distance covered = $16 \times \frac{11}{4}$.

Multiply the numerator by the integer and keep the denominator. We can simplify before multiplying:

Total distance covered = $\cancel{16}^{4} \times \frac{11}{\cancel{4}_{1}} = 4 \times 11 = 44$ km.

The car will cover 44 km using $2\frac{3}{4}$ litres of petrol.

We need to find the missing fraction in the multiplication equations.

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(a) (i) Provide the number in the box $\Box$, such that $\frac{2}{3} \times \Box = \frac{10}{30}$.

Let the number in the box be $x$. The equation is:

$\frac{2}{3} \times x = \frac{10}{30}$

To find the value of $x$, we need to isolate $x$. We can do this by dividing both sides of the equation by $\frac{2}{3}$.

$x = \frac{10}{30} \div \frac{2}{3}$

Remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

$x = \frac{10}{30} \times \frac{3}{2}$

Now, we multiply the two fractions:

$x = \frac{10 \times 3}{30 \times 2} = \frac{30}{60}$.

So, the number in the box is $\frac{30}{60}$.

We can simplify the fraction $\frac{30}{60}$. Both 30 and 60 are divisible by 6.

$\frac{30}{60} = \frac{\cancel{30}^{5}}{\cancel{60}_{10}}$

So, $\frac{30}{60}$ simplifies to $\frac{5}{10}$.

Thus, the number in the box $\Box$ is $\mathbf{\frac{5}{10}}$.

We can check this: $\frac{2}{3} \times \frac{5}{10} = \frac{2 \times 5}{3 \times 10} = \frac{10}{30}$, which is correct.

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(a) (ii) The simplest form of the number obtained in $\Box$ is _____.

The number obtained is $\frac{30}{60}$. To find the simplest form, divide the numerator and the denominator by their greatest common divisor, which is 30.

$\frac{30 \div 30}{60 \div 30} = \frac{1}{2}$

The simplest form of $\frac{30}{60}$ is $\mathbf{\frac{1}{2}}$.

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(b) (i) Provide the number in the box $\Box$, such that $\frac{3}{5} \times \Box = \frac{24}{75}$.

Let the number in the box be $y$. The equation is:

$\frac{3}{5} \times y = \frac{24}{75}$

To find $y$, we divide $\frac{24}{75}$ by $\frac{3}{5}$:

$y = \frac{24}{75} \div \frac{3}{5}$

Multiply by the reciprocal of $\frac{3}{5}$, which is $\frac{5}{3}$:

$y = \frac{24}{75} \times \frac{5}{3}$

Multiply the numerators and denominators, simplifying before or after:

$y = \frac{\cancel{24}^{8}}{\cancel{75}_{15}} \times \frac{\cancel{5}^{1}}{\cancel{3}_{1}} = \frac{8 \times 1}{15 \times 1} = \frac{8}{15}$

The number in the box $\Box$ is $\mathbf{\frac{8}{15}}$.

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(b) (ii) The simplest form of the number obtained in $\Box$ is _____.

The number obtained is $\frac{8}{15}$. The factors of 8 are 1, 2, 4, 8. The factors of 15 are 1, 3, 5, 15. The only common factor is 1. Therefore, the fraction is already in its simplest form.

The simplest form of $\frac{8}{15}$ is $\mathbf{\frac{8}{15}}$.

We find the value of each division expression. Dividing by a fraction is the same as multiplying by its reciprocal.

---

(i) $12 \div \frac{3}{4}$

$12 \div \frac{3}{4} = 12 \times \frac{4}{3}$

Multiply the integer by the reciprocal fraction. We can simplify before multiplying:

$12 \times \frac{4}{3} = \cancel{12}^{4} \times \frac{4}{\cancel{3}_{1}} = 4 \times 4 = \mathbf{16}$

---

(ii) $14 \div \frac{5}{6}$

$14 \div \frac{5}{6} = 14 \times \frac{6}{5}$

Multiply the integer by the reciprocal fraction:

$14 \times \frac{6}{5} = \frac{14 \times 6}{5} = \frac{84}{5}$.

Result: $\mathbf{\frac{84}{5}}$ (or $16\frac{4}{5}$ as a mixed fraction).

---

(iii) $8 \div \frac{7}{3}$

$8 \div \frac{7}{3} = 8 \times \frac{3}{7}$

Multiply the integer by the reciprocal fraction:

$8 \times \frac{3}{7} = \frac{8 \times 3}{7} = \frac{24}{7}$.

Result: $\mathbf{\frac{24}{7}}$ (or $3\frac{3}{7}$ as a mixed fraction).

---

(iv) $4 \div \frac{8}{3}$

$4 \div \frac{8}{3} = 4 \times \frac{3}{8}$

Multiply the integer by the reciprocal fraction. We can simplify before multiplying:

$4 \times \frac{3}{8} = \cancel{4}^{1} \times \frac{3}{\cancel{8}_{2}} = 1 \times \frac{3}{2} = \frac{3}{2}$.

Result: $\mathbf{\frac{3}{2}}$ (or $1\frac{1}{2}$ as a mixed fraction).

---

(v) $3 \div 2\frac{1}{3}$

Convert the mixed fraction to an improper fraction: $2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$.

Now divide the integer by the fraction:

$3 \div \frac{7}{3} = 3 \times \frac{3}{7}$.

Multiply the integer by the reciprocal fraction:

$3 \times \frac{3}{7} = \frac{3 \times 3}{7} = \frac{9}{7}$.

Result: $\mathbf{\frac{9}{7}}$ (or $1\frac{2}{7}$ as a mixed fraction).

---

(vi) $5 \div 3\frac{4}{7}$

Convert the mixed fraction to an improper fraction: $3\frac{4}{7} = \frac{3 \times 7 + 4}{7} = \frac{21 + 4}{7} = \frac{25}{7}$.

Now divide the integer by the fraction:

$5 \div \frac{25}{7} = 5 \times \frac{7}{25}$.

Multiply the integer by the reciprocal fraction. We can simplify before multiplying:

$5 \times \frac{7}{25} = \cancel{5}^{1} \times \frac{7}{\cancel{25}_{5}} = 1 \times \frac{7}{5} = \frac{7}{5}$.

Result: $\mathbf{\frac{7}{5}}$ (or $1\frac{2}{5}$ as a mixed fraction).

The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. We will find the reciprocal of each given fraction and then classify it.

Classification rules:

Proper fraction: Numerator < Denominator

Improper fraction: Numerator $\geq$ Denominator

Whole number: An integer (which can be written as a fraction with denominator 1)

---

(i) $\frac{3}{7}$

Reciprocal of $\frac{3}{7}$ is $\frac{7}{3}$.

In $\frac{7}{3}$, the numerator (7) is greater than the denominator (3). So it is an improper fraction.

---

(ii) $\frac{5}{8}$

Reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$.

In $\frac{8}{5}$, the numerator (8) is greater than the denominator (5). So it is an improper fraction.

---

(iii) $\frac{9}{7}$

Reciprocal of $\frac{9}{7}$ is $\frac{7}{9}$.

In $\frac{7}{9}$, the numerator (7) is less than the denominator (9). So it is a proper fraction.

---

(iv) $\frac{6}{5}$

Reciprocal of $\frac{6}{5}$ is $\frac{5}{6}$.

In $\frac{5}{6}$, the numerator (5) is less than the denominator (6). So it is a proper fraction.

---

(v) $\frac{12}{7}$

Reciprocal of $\frac{12}{7}$ is $\frac{7}{12}$.

In $\frac{7}{12}$, the numerator (7) is less than the denominator (12). So it is a proper fraction.

---

(vi) $\frac{1}{8}$

Reciprocal of $\frac{1}{8}$ is $\frac{8}{1}$.

$\frac{8}{1}$ is equal to 8, which is a whole number.

---

(vii) $\frac{1}{11}$

Reciprocal of $\frac{1}{11}$ is $\frac{11}{1}$.

$\frac{11}{1}$ is equal to 11, which is a whole number.

We evaluate each division expression. Dividing by an integer is the same as multiplying by the reciprocal of the integer (which is 1 divided by the integer).

---

(i) $\frac{7}{3} \div 2$

$\frac{7}{3} \div 2 = \frac{7}{3} \times \frac{1}{2} = \frac{7 \times 1}{3 \times 2} = \mathbf{\frac{7}{6}}$.

---

(ii) $\frac{4}{9} \div 5$

$\frac{4}{9} \div 5 = \frac{4}{9} \times \frac{1}{5} = \frac{4 \times 1}{9 \times 5} = \mathbf{\frac{4}{45}}$.

---

(iii) $\frac{6}{13} \div 7$

$\frac{6}{13} \div 7 = \frac{6}{13} \times \frac{1}{7} = \frac{6 \times 1}{13 \times 7} = \mathbf{\frac{6}{91}}$.

---

(iv) $4\frac{1}{3} \div 3$

Convert the mixed fraction to an improper fraction: $4\frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{13}{3}$.

Now divide the fraction by the integer:

$\frac{13}{3} \div 3 = \frac{13}{3} \times \frac{1}{3} = \frac{13 \times 1}{3 \times 3} = \mathbf{\frac{13}{9}}$.

---

(v) $3\frac{1}{2} \div 4$

Convert the mixed fraction to an improper fraction: $3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}$.

Now divide the fraction by the integer:

$\frac{7}{2} \div 4 = \frac{7}{2} \times \frac{1}{4} = \frac{7 \times 1}{2 \times 4} = \mathbf{\frac{7}{8}}$.

---

(vi) $4\frac{3}{7} \div 7$

Convert the mixed fraction to an improper fraction: $4\frac{3}{7} = \frac{4 \times 7 + 3}{7} = \frac{28 + 3}{7} = \frac{31}{7}$.

Now divide the fraction by the integer:

$\frac{31}{7} \div 7 = \frac{31}{7} \times \frac{1}{7} = \frac{31 \times 1}{7 \times 7} = \mathbf{\frac{31}{49}}$.

We evaluate each division expression involving fractions and mixed fractions. Dividing by a fraction is the same as multiplying by its reciprocal. Convert mixed fractions to improper fractions before performing the operation.

---

(i) $\frac{2}{5}\div\frac{1}{2}$

$\frac{2}{5}\div\frac{1}{2} = \frac{2}{5} \times \frac{2}{1} = \frac{2 \times 2}{5 \times 1} = \mathbf{\frac{4}{5}}$.

---

(ii) $\frac{4}{9}\div\frac{2}{3}$

$\frac{4}{9}\div\frac{2}{3} = \frac{4}{9} \times \frac{3}{2} = \frac{\cancel{4}^{2}}{\cancel{9}_{3}} \times \frac{\cancel{3}^{1}}{\cancel{2}_{1}} = \frac{2 \times 1}{3 \times 1} = \mathbf{\frac{2}{3}}$.

---

(iii) $\frac{3}{7}\div\frac{8}{7}$

$\frac{3}{7}\div\frac{8}{7} = \frac{3}{7} \times \frac{7}{8} = \frac{3}{\cancel{7}_{1}} \times \frac{\cancel{7}^{1}}{8} = \frac{3 \times 1}{1 \times 8} = \mathbf{\frac{3}{8}}$.

---

(iv) $2\frac{1}{3}\div\frac{3}{5}$

Convert mixed fraction: $2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$.

Divide: $\frac{7}{3}\div\frac{3}{5} = \frac{7}{3} \times \frac{5}{3} = \frac{7 \times 5}{3 \times 3} = \mathbf{\frac{35}{9}}$.

---

(v) $3\frac{1}{2}\div\frac{8}{3}$

Convert mixed fraction: $3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}$.

Divide: $\frac{7}{2}\div\frac{8}{3} = \frac{7}{2} \times \frac{3}{8} = \frac{7 \times 3}{2 \times 8} = \mathbf{\frac{21}{16}}$.

---

(vi) $\frac{2}{5} \div 1\frac{1}{2}$

Convert mixed fraction: $1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}$.

Divide: $\frac{2}{5} \div \frac{3}{2} = \frac{2}{5} \times \frac{2}{3} = \frac{2 \times 2}{5 \times 3} = \mathbf{\frac{4}{15}}$.

---

(vii) $3\frac{1}{5}\div 1\frac{2}{3}$

Convert mixed fractions:

$3\frac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{16}{5}$.

$1\frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3}$.

Divide: $\frac{16}{5}\div\frac{5}{3} = \frac{16}{5} \times \frac{3}{5} = \frac{16 \times 3}{5 \times 5} = \mathbf{\frac{48}{25}}$.

---

(viii) $2\frac{1}{5}\div 1\frac{1}{5}$

Convert mixed fractions:

$2\frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{11}{5}$.

$1\frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{6}{5}$.

Divide: $\frac{11}{5}\div\frac{6}{5} = \frac{11}{5} \times \frac{5}{6} = \frac{11}{\cancel{5}_{1}} \times \frac{\cancel{5}^{1}}{6} = \frac{11 \times 1}{1 \times 6} = \mathbf{\frac{11}{6}}$.

Given:

Side length of an equilateral triangle ($s$) = 3.5 cm.

---

To Find:

The perimeter of the equilateral triangle.

---

Solution:

An equilateral triangle is a triangle in which all three sides are of equal length.

The perimeter of any polygon is the sum of the lengths of its sides.

For an equilateral triangle with side length $s$, the perimeter (P) is the sum of the three equal sides:

$P = s + s + s = 3 \times s$

Given $s = 3.5$ cm.

Perimeter = $3 \times 3.5$ cm.

Calculating the product:

$3 \times 3.5 = 10.5$

Perimeter = $10.5$ cm.

---

The perimeter of the equilateral triangle is 10.5 cm.

Given:

Length of the rectangle ($l$) = 7.1 cm.

Breadth (width) of the rectangle ($b$) = 2.5 cm.

---

To Find:

The area of the rectangle.

---

Solution:

The area of a rectangle is given by the formula:

Area = Length $\times$ Breadth

Area = $l \times b$

Area = $7.1 \text{ cm} \times 2.5 \text{ cm}$.

Area = $17.75 \text{ cm}^2$.

---

The area of the rectangle is $17.75 \text{ cm}^2$.

We multiply the decimal numbers by the given integers.

---

(i) $0.2 \times 6$

Multiply ignoring decimal points: $2 \times 6 = 12$.

There is 1 digit after the decimal point in 0.2.

Place the decimal point 1 place from the right in 12.

Result: $\mathbf{1.2}$

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(ii) $8 \times 4.6$

Multiply ignoring decimal points: $8 \times 46$.

$8 \times 46 = 368$.

There is 1 digit after the decimal point in 4.6.

Place the decimal point 1 place from the right in 368.

Result: $\mathbf{36.8}$

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(iii) $2.71 \times 5$

Multiply ignoring decimal points: $271 \times 5$.

$271 \times 5 = 1355$.

There are 2 digits after the decimal point in 2.71.

Place the decimal point 2 places from the right in 1355.

Result: $\mathbf{13.55}$

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(iv) $20.1 \times 4$

Multiply ignoring decimal points: $201 \times 4$.

$201 \times 4 = 804$.

There is 1 digit after the decimal point in 20.1.

Place the decimal point 1 place from the right in 804.

Result: $\mathbf{80.4}$

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(v) $0.05 \times 7$

Multiply ignoring decimal points: $5 \times 7 = 35$.

There are 2 digits after the decimal point in 0.05.

Place the decimal point 2 places from the right in 35. Add leading zeros if needed.

Result: $\mathbf{0.35}$

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(vi) $211.02 \times 4$

Multiply ignoring decimal points: $21102 \times 4$.

$21102 \times 4 = 84408$.

There are 2 digits after the decimal point in 211.02.

Place the decimal point 2 places from the right in 84408.

Result: $\mathbf{844.08}$

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(vii) $2 \times 0.86$

Multiply ignoring decimal points: $2 \times 86$.

$2 \times 86 = 172$.

There are 2 digits after the decimal point in 0.86.

Place the decimal point 2 places from the right in 172.

Result: $\mathbf{1.72}$

Given:

Length of the rectangle ($l$) = 5.7 cm.

Breadth of the rectangle ($b$) = 3 cm.

---

To Find:

The area of the rectangle.

---

Solution:

The area of a rectangle is calculated by the formula:

Area = Length $\times$ Breadth

Area = $l \times b$

Area = $5.7 \text{ cm} \times 3 \text{ cm}$.

We need to multiply the decimal number 5.7 by the integer 3.

Multiply the numbers ignoring the decimal point: $57 \times 3$.

$57 \times 3 = 171$.

Now, count the total number of digits after the decimal point in the original numbers. In 5.7, there is 1 digit after the decimal. In 3, there are 0 digits after the decimal. The total is $1 + 0 = 1$ digit.

Place the decimal point 1 place from the right in the product 171.

17.1

The units for the area will be $\text{cm} \times \text{cm} = \text{cm}^2$.

Area = $17.1 \text{ cm}^2$.

---

The area of the rectangle is $17.1 \text{ cm}^2$.

To multiply a decimal number by 10, 100, 1000, etc., we shift the decimal point to the right by the number of zeros in the multiplier.

---

(i) $1.3 \times 10$

10 has one zero. Shift the decimal point 1 place to the right.

$1.3 \times 10 = \mathbf{13.0}$ or $\mathbf{13}$

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(ii) $36.8 \times 10$

10 has one zero. Shift the decimal point 1 place to the right.

$36.8 \times 10 = \mathbf{368.0}$ or $\mathbf{368}$

---

(iii) $153.7 \times 10$

10 has one zero. Shift the decimal point 1 place to the right.

$153.7 \times 10 = \mathbf{1537.0}$ or $\mathbf{1537}$

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(iv) $168.07 \times 10$

10 has one zero. Shift the decimal point 1 place to the right.

$168.07 \times 10 = \mathbf{1680.7}$

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(v) $31.1 \times 100$

100 has two zeros. Shift the decimal point 2 places to the right.

$31.1 \times 100 = 31.10 \times 100 = \mathbf{3110.0}$ or $\mathbf{3110}$

---

(vi) $156.1 \times 100$

100 has two zeros. Shift the decimal point 2 places to the right.

$156.1 \times 100 = 156.10 \times 100 = \mathbf{15610.0}$ or $\mathbf{15610}$

---

(vii) $3.62 \times 100$

100 has two zeros. Shift the decimal point 2 places to the right.

$3.62 \times 100 = \mathbf{362.0}$ or $\mathbf{362}$

---

(viii) $43.07 \times 100$

100 has two zeros. Shift the decimal point 2 places to the right.

$43.07 \times 100 = \mathbf{4307.0}$ or $\mathbf{4307}$

---

(ix) $0.5 \times 10$

10 has one zero. Shift the decimal point 1 place to the right.

$0.5 \times 10 = \mathbf{5.0}$ or $\mathbf{5}$

---

(x) $0.08 \times 10$

10 has one zero. Shift the decimal point 1 place to the right.

$0.08 \times 10 = \mathbf{0.8}$

---

(xi) $0.9 \times 100$

100 has two zeros. Shift the decimal point 2 places to the right.

$0.9 \times 100 = 0.90 \times 100 = \mathbf{90.0}$ or $\mathbf{90}$

---

(xii) $0.03 \times 1000$

1000 has three zeros. Shift the decimal point 3 places to the right.

$0.03 \times 1000 = 0.030 \times 1000 = \mathbf{30.0}$ or $\mathbf{30}$

Given:

Distance covered by a two-wheeler in 1 litre of petrol = 55.3 km.

Quantity of petrol available = 10 litres.

---

To Find:

Distance covered in 10 litres of petrol.

---

Solution:

The distance covered is proportional to the amount of petrol used. To find the distance covered using 10 litres, we multiply the distance covered per litre by the total quantity of petrol.

Total distance covered = (Distance per litre) $\times$ (Quantity of petrol)

Total distance covered = $55.3 \text{ km/litre} \times 10 \text{ litres}$.

We need to multiply the decimal number 55.3 by 10.

To multiply a decimal number by 10, shift the decimal point 1 place to the right (because 10 has one zero).

$55.3 \times 10 = 553.0$

Total distance covered = 553 km.

---

The two-wheeler will cover 553 km in 10 litres of petrol.

We multiply the decimal numbers. Multiply the numbers as whole numbers, then place the decimal point in the product such that the number of decimal places in the product is the sum of the number of decimal places in the original numbers.

---

(i) $2.5 \times 0.3$

$25 \times 3 = 75$.

2.5 has 1 decimal place, 0.3 has 1 decimal place. Total = $1 + 1 = 2$ decimal places.

Place the decimal point 2 places from the right in 75.

Result: $\mathbf{0.75}$

---

(ii) $0.1 \times 51.7$

$1 \times 517 = 517$.

0.1 has 1 decimal place, 51.7 has 1 decimal place. Total = $1 + 1 = 2$ decimal places.

Place the decimal point 2 places from the right in 517.

Result: $\mathbf{5.17}$

---

(iii) $0.2 \times 316.8$

$2 \times 3168 = 6336$.

0.2 has 1 decimal place, 316.8 has 1 decimal place. Total = $1 + 1 = 2$ decimal places.

Place the decimal point 2 places from the right in 6336.

Result: $\mathbf{63.36}$

---

(iv) $1.3 \times 3.1$

$13 \times 31 = 403$.

1.3 has 1 decimal place, 3.1 has 1 decimal place. Total = $1 + 1 = 2$ decimal places.

Place the decimal point 2 places from the right in 403.

Result: $\mathbf{4.03}$

---

(v) $0.5 \times 0.05$

$5 \times 5 = 25$.

0.5 has 1 decimal place, 0.05 has 2 decimal places. Total = $1 + 2 = 3$ decimal places.

Place the decimal point 3 places from the right in 25. Add a leading zero.

Result: $\mathbf{0.025}$

---

(vi) $11.2 \times 0.15$

$112 \times 15 = 1680$.

11.2 has 1 decimal place, 0.15 has 2 decimal places. Total = $1 + 2 = 3$ decimal places.

Place the decimal point 3 places from the right in 1680.

Result: $\mathbf{1.680}$ or $\mathbf{1.68}$

---

(vii) $1.07 \times 0.02$

$107 \times 2 = 214$.

1.07 has 2 decimal places, 0.02 has 2 decimal places. Total = $2 + 2 = 4$ decimal places.

Place the decimal point 4 places from the right in 214. Add leading zeros.

Result: $\mathbf{0.0214}$

---

(viii) $10.05 \times 1.05$

$1005 \times 105 = 105525$.

10.05 has 2 decimal places, 1.05 has 2 decimal places. Total = $2 + 2 = 4$ decimal places.

Place the decimal point 4 places from the right in 105525.

Result: $\mathbf{10.5525}$

---

(ix) $101.01 \times 0.01$

$10101 \times 1 = 10101$.

101.01 has 2 decimal places, 0.01 has 2 decimal places. Total = $2 + 2 = 4$ decimal places.

Place the decimal point 4 places from the right in 10101.

Result: $\mathbf{1.0101}$

---

(x) $100.01 \times 1.1$

$10001 \times 11 = 110011$.

100.01 has 2 decimal places, 1.1 has 1 decimal place. Total = $2 + 1 = 3$ decimal places.

Place the decimal point 3 places from the right in 110011.

Result: $\mathbf{110.011}$

Given:

The numbers are 4.2, 3.8, and 7.6.

There are 3 numbers.

---

To Find:

The average of these numbers.

---

Solution:

The average of a set of numbers is the sum of the numbers divided by the count of the numbers.

Average = $\frac{\text{Sum of the numbers}}{\text{Count of the numbers}}$

First, calculate the sum of the numbers:

Sum = $4.2 + 3.8 + 7.6$

Sum = 15.6

Now, divide the sum by the count (which is 3):

Average = $\frac{15.6}{3}$.

Average = 5.2.

---

The average of 4.2, 3.8, and 7.6 is 5.2.

Given:

Length of each side of a regular polygon = 2.5 cm.

Perimeter of the polygon = 12.5 cm.

The polygon is regular, meaning all sides are equal in length.

---

To Find:

The number of sides the polygon has.

---

Solution:

The perimeter of a polygon is the total length of its boundary, which is the sum of the lengths of all its sides.

For a regular polygon with $n$ sides, each of length $s$, the perimeter (P) is given by:

$P = n \times s$

We are given $P = 12.5$ cm and $s = 2.5$ cm. We need to find $n$.

The equation is: $12.5 = n \times 2.5$.

To find $n$, divide the perimeter by the length of one side:

$n = \frac{12.5}{2.5}$.

We need to divide the decimal number 12.5 by the decimal number 2.5.

To divide a decimal by another decimal, shift the decimal point in the divisor to the right until it becomes a whole number. Shift the decimal point in the dividend by the same number of places to the right.

The divisor is 2.5. Shift the decimal 1 place to the right to get 25.

The dividend is 12.5. Shift the decimal 1 place to the right to get 125.

Now perform the division $125 \div 25$.

$125 \div 25 = 5$.

$n = 5$.

The polygon has 5 sides.

---

The polygon has 5 sides. (A regular polygon with 5 sides is called a regular pentagon).

Given:

Distance covered by the car = 89.1 km.

Time taken to cover the distance = 2.2 hours.

---

To Find:

The average distance covered by the car in 1 hour (this is the average speed).

---

Solution:

Average distance covered in 1 hour = $\frac{\text{Total Distance Covered}}{\text{Time Taken}}$

Average distance per hour = $\frac{89.1 \text{ km}}{2.2 \text{ hours}}$.

We need to divide the decimal number 89.1 by the decimal number 2.2.

To divide a decimal by another decimal, shift the decimal point in the divisor to the right until it becomes a whole number. Shift the decimal point in the dividend by the same number of places to the right.

The divisor is 2.2. Shift the decimal 1 place to the right to get 22.

The dividend is 89.1. Shift the decimal 1 place to the right to get 891.

Now perform the division $891 \div 22$.

$891 \div 22 = 40.5$.

The average distance covered in 1 hour is 40.5 km.

---

The average distance covered by the car in 1 hour is 40.5 km.

To divide a decimal number by a whole number, we can convert the decimal number into a fraction, divide the fraction by the whole number, and then convert the resulting fraction back into a decimal. We can use cancellation when multiplying fractions.

---

(i) $0.4 \div 2$

First, convert the decimal to a fraction: $0.4 = \frac{4}{10}$.

Now, the division becomes $\frac{4}{10} \div 2$. Dividing by a whole number is the same as multiplying by its reciprocal:

$\frac{4}{10} \times \frac{1}{2}$

Multiply the fractions and use cancellation:

$\frac{\cancel{4}^{2}}{10} \times \frac{1}{\cancel{2}_{1}} = \frac{2 \times 1}{10 \times 1} = \frac{2}{10}$.

Finally, convert the fraction back to a decimal: $\frac{2}{10} = 0.2$.

Result: $\mathbf{0.2}$

---

(ii) $0.35 \div 5$

Convert the decimal to a fraction: $0.35 = \frac{35}{100}$.

The division is $\frac{35}{100} \div 5$. Multiply by the reciprocal of 5:

$\frac{35}{100} \times \frac{1}{5}$

Multiply the fractions and use cancellation:

$\frac{\cancel{35}^{7}}{100} \times \frac{1}{\cancel{5}_{1}} = \frac{7 \times 1}{100 \times 1} = \frac{7}{100}$.

Convert the fraction back to a decimal: $\frac{7}{100} = 0.07$.

Result: $\mathbf{0.07}$

---

(iii) $2.48 \div 4$

Convert the decimal to a fraction: $2.48 = \frac{248}{100}$.

The division is $\frac{248}{100} \div 4$. Multiply by the reciprocal of 4:

$\frac{248}{100} \times \frac{1}{4}$

Multiply the fractions and use cancellation ($248 \div 4 = 62$):

$\frac{\cancel{248}^{62}}{100} \times \frac{1}{\cancel{4}_{1}} = \frac{62 \times 1}{100 \times 1} = \frac{62}{100}$.

Convert the fraction back to a decimal: $\frac{62}{100} = 0.62$.

Result: $\mathbf{0.62}$

---

(iv) $65.4 \div 6$

Convert the decimal to a fraction: $65.4 = \frac{654}{10}$.

The division is $\frac{654}{10} \div 6$. Multiply by the reciprocal of 6:

$\frac{654}{10} \times \frac{1}{6}$

Multiply the fractions and use cancellation ($654 \div 6 = 109$):

$\frac{\cancel{654}^{109}}{10} \times \frac{1}{\cancel{6}_{1}} = \frac{109 \times 1}{10 \times 1} = \frac{109}{10}$.

Convert the fraction back to a decimal: $\frac{109}{10} = 10.9$.

Result: $\mathbf{10.9}$

---

(v) $651.2 \div 4$

Convert the decimal to a fraction: $651.2 = \frac{6512}{10}$.

The division is $\frac{6512}{10} \div 4$. Multiply by the reciprocal of 4:

$\frac{6512}{10} \times \frac{1}{4}$

Multiply the fractions and use cancellation ($6512 \div 4 = 1628$):

$\frac{\cancel{6512}^{1628}}{10} \times \frac{1}{\cancel{4}_{1}} = \frac{1628 \times 1}{10 \times 1} = \frac{1628}{10}$.

Convert the fraction back to a decimal: $\frac{1628}{10} = 162.8$.

Result: $\mathbf{162.8}$

---

(vi) $14.49 \div 7$

Convert the decimal to a fraction: $14.49 = \frac{1449}{100}$.

The division is $\frac{1449}{100} \div 7$. Multiply by the reciprocal of 7:

$\frac{1449}{100} \times \frac{1}{7}$

Multiply the fractions and use cancellation ($1449 \div 7 = 207$):

$\frac{\cancel{1449}^{207}}{100} \times \frac{1}{\cancel{7}_{1}} = \frac{207 \times 1}{100 \times 1} = \frac{207}{100}$.

Convert the fraction back to a decimal: $\frac{207}{100} = 2.07$.

Result: $\mathbf{2.07}$

---

(vii) $3.96 \div 4$

Convert the decimal to a fraction: $3.96 = \frac{396}{100}$.

The division is $\frac{396}{100} \div 4$. Multiply by the reciprocal of 4:

$\frac{396}{100} \times \frac{1}{4}$

Multiply the fractions and use cancellation ($396 \div 4 = 99$):

$\frac{\cancel{396}^{99}}{100} \times \frac{1}{\cancel{4}_{1}} = \frac{99 \times 1}{100 \times 1} = \frac{99}{100}$.

Convert the fraction back to a decimal: $\frac{99}{100} = 0.99$.

Result: $\mathbf{0.99}$

---

(viii) $0.80 \div 5$

Convert the decimal to a fraction: $0.80 = \frac{80}{100}$.

The division is $\frac{80}{100} \div 5$. Multiply by the reciprocal of 5:

$\frac{80}{100} \times \frac{1}{5}$

Multiply the fractions and use cancellation ($80 \div 5 = 16$):

$\frac{\cancel{80}^{16}}{100} \times \frac{1}{\cancel{5}_{1}} = \frac{16 \times 1}{100 \times 1} = \frac{16}{100}$.

Convert the fraction back to a decimal: $\frac{16}{100} = 0.16$.

Result: $\mathbf{0.16}$

To divide a decimal number by 10, we shift the decimal point one place to the left.

---

(i) $4.8 \div 10$

Shift the decimal point in 4.8 one place to the left.

Result: $\mathbf{0.48}$

---

(ii) $52.5 \div 10$

Shift the decimal point in 52.5 one place to the left.

Result: $\mathbf{5.25}$

---

(iii) $0.7 \div 10$

Shift the decimal point in 0.7 one place to the left. Add a leading zero if necessary.

Result: $\mathbf{0.07}$

---

(iv) $33.1 \div 10$

Shift the decimal point in 33.1 one place to the left.

Result: $\mathbf{3.31}$

---

(v) $272.23 \div 10$

Shift the decimal point in 272.23 one place to the left.

Result: $\mathbf{27.223}$

---

(vi) $0.56 \div 10$

Shift the decimal point in 0.56 one place to the left. Add a leading zero if necessary.

Result: $\mathbf{0.056}$

---

(vii) $3.97 \div 10$

Shift the decimal point in 3.97 one place to the left.

Result: $\mathbf{0.397}$

To divide a decimal number by 100, we shift the decimal point two places to the left (because 100 has two zeros).

---

(i) $2.7 \div 100$

Shift the decimal point in 2.7 two places to the left. Add a leading zero.

Result: $\mathbf{0.027}$

---

(ii) $0.3 \div 100$

Shift the decimal point in 0.3 two places to the left. Add leading zeros.

Result: $\mathbf{0.003}$

---

(iii) $0.78 \div 100$

Shift the decimal point in 0.78 two places to the left. Add leading zeros.

Result: $\mathbf{0.0078}$

---

(iv) $432.6 \div 100$

Shift the decimal point in 432.6 two places to the left.

Result: $\mathbf{4.326}$

---

(v) $23.6 \div 100$

Shift the decimal point in 23.6 two places to the left.

Result: $\mathbf{0.236}$

---

(vi) $98.53 \div 100$

Shift the decimal point in 98.53 two places to the left.

Result: $\mathbf{0.9853}$

To divide a decimal number by 1000, we shift the decimal point three places to the left (because 1000 has three zeros).

---

(i) $7.9 \div 1000$

Shift the decimal point in 7.9 three places to the left. Add leading zeros.

Result: $\mathbf{0.0079}$

---

(ii) $26.3 \div 1000$

Shift the decimal point in 26.3 three places to the left. Add leading zeros.

Result: $\mathbf{0.0263}$

---

(iii) $38.53 \div 1000$

Shift the decimal point in 38.53 three places to the left. Add leading zeros.

Result: $\mathbf{0.03853}$

---

(iv) $128.9 \div 1000$

Shift the decimal point in 128.9 three places to the left.

Result: $\mathbf{0.1289}$

---

(v) $0.5 \div 1000$

Shift the decimal point in 0.5 three places to the left. Add leading zeros.

Result: $\mathbf{0.0005}$

To divide a decimal number by another decimal number, we can first convert both numbers into fractions. Then we divide the first fraction by the second fraction, which is the same as multiplying the first fraction by the reciprocal of the second fraction. Finally, we convert the resulting fraction back into a decimal. We can use cancellation to simplify calculations.

---

(i) $7 \div 3.5$

Convert to fractions: $7 = \frac{7}{1}$ and $3.5 = \frac{35}{10}$.

Now divide: $\frac{7}{1} \div \frac{35}{10}$. This is equal to $\frac{7}{1} \times \frac{10}{35}$.

Multiply the fractions and cancel common factors (7 and 35 share a factor of 7):

$\frac{\cancel{7}^{1}}{1} \times \frac{10}{\cancel{35}_{5}} = \frac{1 \times 10}{1 \times 5} = \frac{10}{5}$.

Simplify the fraction: $\frac{10}{5} = 2$.

Result: $\mathbf{2}$

---

(ii) $36 \div 0.2$

Convert to fractions: $36 = \frac{36}{1}$ and $0.2 = \frac{2}{10}$.

Now divide: $\frac{36}{1} \div \frac{2}{10}$. This is equal to $\frac{36}{1} \times \frac{10}{2}$.

Multiply the fractions and cancel common factors (36 and 2 share a factor of 2):

$\frac{\cancel{36}^{18}}{1} \times \frac{10}{\cancel{2}_{1}} = \frac{18 \times 10}{1 \times 1} = \frac{180}{1} = 180$.

Result: $\mathbf{180}$

---

(iii) $3.25 \div 0.5$

Convert to fractions: $3.25 = \frac{325}{100}$ and $0.5 = \frac{5}{10}$.

Now divide: $\frac{325}{100} \div \frac{5}{10}$. This is equal to $\frac{325}{100} \times \frac{10}{5}$.

Multiply the fractions and cancel common factors (325 and 5 share a factor of 5, 10 and 100 share a factor of 10):

$\frac{\cancel{325}^{65}}{\cancel{100}_{10}} \times \frac{\cancel{10}^{1}}{\cancel{5}_{1}} = \frac{65 \times 1}{10 \times 1} = \frac{65}{10}$.

Convert the fraction back to a decimal: $\frac{65}{10} = 6.5$.

Result: $\mathbf{6.5}$

---

(iv) $30.94 \div 0.7$

Convert to fractions: $30.94 = \frac{3094}{100}$ and $0.7 = \frac{7}{10}$.

Now divide: $\frac{3094}{100} \div \frac{7}{10}$. This is equal to $\frac{3094}{100} \times \frac{10}{7}$.

Multiply the fractions and cancel common factors (3094 and 7 share a factor of 7, $3094 \div 7 = 442$; 10 and 100 share a factor of 10):

$\frac{\cancel{3094}^{442}}{\cancel{100}_{10}} \times \frac{\cancel{10}^{1}}{\cancel{7}_{1}} = \frac{442 \times 1}{10 \times 1} = \frac{442}{10}$.

Convert the fraction back to a decimal: $\frac{442}{10} = 44.2$.

Result: $\mathbf{44.2}$

---

(v) $0.5 \div 0.25$

Convert to fractions: $0.5 = \frac{5}{10}$ and $0.25 = \frac{25}{100}$.

Now divide: $\frac{5}{10} \div \frac{25}{100}$. This is equal to $\frac{5}{10} \times \frac{100}{25}$.

Multiply the fractions and cancel common factors (5 and 25 share a factor of 5; 10 and 100 share a factor of 10):

$\frac{\cancel{5}^{1}}{\cancel{10}_{1}} \times \frac{\cancel{100}^{10}}{\cancel{25}_{5}} = \frac{1 \times 10}{1 \times 5} = \frac{10}{5}$.

Simplify the fraction: $\frac{10}{5} = 2$.

Result: $\mathbf{2}$

---

(vi) $7.75 \div 0.25$

Convert to fractions: $7.75 = \frac{775}{100}$ and $0.25 = \frac{25}{100}$.

Now divide: $\frac{775}{100} \div \frac{25}{100}$. This is equal to $\frac{775}{100} \times \frac{100}{25}$.

Multiply the fractions and cancel common factors (100 and 100 cancel out; 775 and 25 share a factor of 25, $775 \div 25 = 31$):

$\frac{\cancel{775}^{31}}{\cancel{100}_{1}} \times \frac{\cancel{100}^{1}}{\cancel{25}_{1}} = \frac{31 \times 1}{1 \times 1} = \frac{31}{1} = 31$.

Result: $\mathbf{31}$

---

(vii) $76.5 \div 0.15$

Convert to fractions: $76.5 = \frac{765}{10}$ and $0.15 = \frac{15}{100}$.

Now divide: $\frac{765}{10} \div \frac{15}{100}$. This is equal to $\frac{765}{10} \times \frac{100}{15}$.

Multiply the fractions and cancel common factors (10 and 100 share a factor of 10; 765 and 15 share a factor of 15, $765 \div 15 = 51$):

$\frac{\cancel{765}^{51}}{\cancel{10}_{1}} \times \frac{\cancel{100}^{10}}{\cancel{15}_{1}} = \frac{51 \times 10}{1 \times 1} = \frac{510}{1} = 510$.

Result: $\mathbf{510}$

---

(viii) $37.8 \div 1.4$

Convert to fractions: $37.8 = \frac{378}{10}$ and $1.4 = \frac{14}{10}$.

Now divide: $\frac{378}{10} \div \frac{14}{10}$. This is equal to $\frac{378}{10} \times \frac{10}{14}$.

Multiply the fractions and cancel common factors (10 and 10 cancel out; 378 and 14 share a factor of 14, $378 \div 14 = 27$):

$\frac{\cancel{378}^{27}}{\cancel{10}_{1}} \times \frac{\cancel{10}^{1}}{\cancel{14}_{1}} = \frac{27 \times 1}{1 \times 1} = \frac{27}{1} = 27$.

Result: $\mathbf{27}$

---

(ix) $2.73 \div 1.3$

Convert to fractions: $2.73 = \frac{273}{100}$ and $1.3 = \frac{13}{10}$.

Now divide: $\frac{273}{100} \div \frac{13}{10}$. This is equal to $\frac{273}{100} \times \frac{10}{13}$.

Multiply the fractions and cancel common factors (273 and 13 share a factor of 13, $273 \div 13 = 21$; 10 and 100 share a factor of 10):

$\frac{\cancel{273}^{21}}{\cancel{100}_{10}} \times \frac{\cancel{10}^{1}}{\cancel{13}_{1}} = \frac{21 \times 1}{10 \times 1} = \frac{21}{10}$.

Convert the fraction back to a decimal: $\frac{21}{10} = 2.1$.

Result: $\mathbf{2.1}$

Given:

Distance covered by the vehicle = 43.2 km.

Quantity of petrol used = 2.4 litres.

---

To Find:

Distance covered in one litre of petrol.

---

Solution:

To find the distance covered by the vehicle in one litre of petrol, we need to divide the total distance covered by the total quantity of petrol used.

Distance covered in one litre = $\frac{\text{Total Distance Covered}}{\text{Total Quantity of Petrol Used}}$

Distance per litre = $\frac{43.2 \text{ km}}{2.4 \text{ litres}}$.

We need to perform the division $43.2 \div 2.4$.

To divide a decimal number by another decimal number, we first shift the decimal point in the divisor to the right until it becomes a whole number. Then, we shift the decimal point in the dividend by the same number of places to the right.

The divisor is 2.4. To make it a whole number, shift the decimal point 1 place to the right to get 24.

The dividend is 43.2. Shift the decimal point 1 place to the right to get 432.

Now, we divide 432 by 24.

$432 \div 24 = 18$.

The distance covered in one litre of petrol is 18 km.

---

The vehicle will cover 18 km in one litre of petrol.