NCERT Exemplar for Class 7 Maths Solutions Chapter 2 Fractions & Decimals

Solved Examples

1. Savita is dividing $\mathbf{1}{\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}\mathbf{kg}$ of sweets equally among her seven friends. How much does each friend receive?

(a) ${\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}\mathbf{kg}$

(b) ${\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}\mathbf{kg}$

(c) ${\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}\mathbf{kg}$

(d) ${\displaystyle \frac{\mathbf{3}}{\mathbf{28}}}\mathbf{kg}$

Ans: Correct answer is (b)

The given number of sweets is in mixed fraction form. We will first convert it. Hence $1{\displaystyle \frac{3}{4}}={\displaystyle \frac{7}{4}}$

Now according to the given condition, Savita has to divide ${\displaystyle \frac{7}{4}}Kg$ sweets equally among her 7 friends.

Hence, the portion of sweet one person will get $\text{ = }{\displaystyle \frac{7}{4}}÷7={\displaystyle \frac{1}{4}}kg$

So, the amount of sweet each person will get is ${\displaystyle \frac{1}{4}}\text{kg}$.

2. If ${\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}$ of a number is $\mathbf{12},$the number is

(a) $\mathbf{9}$

(b) $\mathbf{16}$

(c) $\mathbf{18}$

(d) $\mathbf{32}$

Ans: Correct answer is (b)

The number is $16$ as ${\displaystyle \frac{3}{4}}$ of $16={\displaystyle \frac{3}{4}}\times 16=12$

3. Product of fractions ${\displaystyle \frac{\mathbf{2}}{\mathbf{7}}}$ and ${\displaystyle \frac{\mathbf{5}}{\mathbf{9}}}$ is

(a) ${\displaystyle \frac{\mathbf{2}\times \mathbf{5}}{\mathbf{7}+\mathbf{9}}}$

(b) ${\displaystyle \frac{\mathbf{2}\mathbf{+}\mathbf{5}}{\mathbf{2}\mathbf{+}\mathbf{9}}}$

(c) ${\displaystyle \frac{\mathbf{2}\times \mathbf{9}}{\mathbf{5}\times \mathbf{7}}}$

(d) ${\displaystyle \frac{\mathbf{2}\times \mathbf{5}}{\mathbf{7}\times \mathbf{9}}}$

Ans: Correct answer is (d)

The product of fraction is ${\displaystyle \frac{2\times 5}{7\times 9}}$

4 Given that $\mathbf{0}\mathbf{<}\mathbf{p}\mathbf{<}\mathbf{q}\mathbf{<}\mathbf{r}\mathbf{<}\mathbf{s}$ and $p,q,r,s$ are integers, which of the following is the smallest?

(a) ${\displaystyle \frac{p+q}{r+s}}$

(b) ${\displaystyle \frac{p+s}{q+r}}$

(c) ${\displaystyle \frac{q+s}{p+r}}$

(d) ${\displaystyle \frac{r+s}{p+q}}$

Ans: Correct answer is (a)

$s$ and $r$ are the largest and the second largest

$p$ and $q$ are the smallest and the second smallest

Smallest fraction $={\displaystyle \frac{(p+q)}{(r+s)}}$

So, the smallest fraction from the given option is ${\displaystyle \frac{p+q}{r+s}}$

5. The next number of the pattern $60,30,15$, is

(a) $\mathbf{10}$

(b) $\mathbf{5}$

(c) ${\displaystyle \frac{\mathbf{15}}{\mathbf{4}}}$

(d) ${\displaystyle \frac{\mathbf{15}}{\mathbf{2}}}$

Ans: Correct answer is (d)

${\displaystyle \frac{60}{2}}=30$ and ${\displaystyle \frac{30}{2}}=15$

The next number of the pattern is ${\displaystyle \frac{15}{2}}$.

6. The decimal expression for 8 rupees 8 paise (in Rupees) is

(a) $\mathbf{8}.\mathbf{8}$

(b) $\mathbf{8}.\mathbf{08}$

(c) $\mathbf{8}.\mathbf{008}$

(d) $\mathbf{88}.\mathbf{0}$

Ans: Correct answer is (b)

We know, 100paise=1 rupee

Also, 8 paise $={\displaystyle \frac{8}{100}}=0.08$

Thus, Decimal expansion in rupees is $(8+0.08)=8.08$

7. Each side of a regular hexagon is $\mathbf{3}.\mathbf{5}\mathbf{cm}$ long. The perimeter of the given polygon is

(a) $\mathbf{17}.\mathbf{5}\mathbf{cm}$

(b) $\mathbf{21}\mathbf{cm}$

(c) $\mathbf{18}.\mathbf{3}\mathbf{cm}$

(d) $\mathbf{20}\mathbf{cm}$

Ans: Correct answer is (b)

The side of a hexagon $=3.5\text{cm}$

Total number of sides in a Hexagon $=6$

The perimeter of hexagon $=$ sum of all six sides $=6\times 3.5=21\text{cm}$

Therefore, the perimeter of the given polygon is $21\text{cm}$.

8. $\mathbf{2}\mathbf{.5}÷\mathbf{1000}$ is equal to

(a) $\mathbf{0}\mathbf{.025}$

(b) $\mathbf{0}\mathbf{.0025}$

(c) $\mathbf{0}\mathbf{.2500}$

(d) $\mathbf{25000}$

Ans: Correct answer is (b)

The division of $2.5÷1000$is $0.0025$

9. Which of the following has the smallest value?

(a) $0.0002$

(b) ${\displaystyle \frac{2}{1000}}$

(c) ${\displaystyle \frac{{(0.2)}^2}{2}}$

(d) ${\displaystyle \frac{2}{100}}÷0.01$

Ans: Correct answer is (a)

The smallest value is $0.0002$ because ${\displaystyle \frac{2}{1000}}=0.002$, ${\displaystyle \frac{{(0.2)}^2}{2}}={\displaystyle \frac{0.04}{2}}=0.02$ and ${\displaystyle \frac{2}{100}}÷0.01=2$.

10. Which of the following has the largest value?

(a) ${\displaystyle \frac{\mathbf{32}}{\mathbf{0}\mathbf{.05}}}$

(b) ${\displaystyle \frac{\mathbf{0}\mathbf{.320}}{\mathbf{50}}}$

(c) ${\displaystyle \frac{\mathbf{3}\mathbf{.2}}{\mathbf{0}\mathbf{.05}}}$

(d) ${\displaystyle \frac{\mathbf{3}\mathbf{.2}}{\mathbf{50}}}$

Ans: Correct answer is (a)

The division of ${\displaystyle \frac{32}{0.05}}=640$

So, the largest number is ${\displaystyle \frac{32}{0.05}}$

${\displaystyle \frac{\mathbf{0}\mathbf{.320}}{\mathbf{50}}}=0.0064$

${\displaystyle \frac{\mathbf{3}\mathbf{.2}}{\mathbf{0}\mathbf{.05}}}=64$

${\displaystyle \frac{\mathbf{3}\mathbf{.2}}{\mathbf{50}}}=0.064$

11: The largest of the following is

(a) $\mathbf{0}\mathbf{.0001}$

(b) ${\displaystyle \frac{\mathbf{1}}{\mathbf{1000}}}$

(c) $\mathbf{(}\mathbf{0}{\mathbf{.100}\mathbf{)}}^{\mathbf{2}}$

(d) ${\displaystyle \frac{\mathbf{1}}{\mathbf{10}}}÷\mathbf{0}\mathbf{.1}$

Ans: Correct answer is (d).

By division ${\displaystyle \frac{1}{10}}÷0.1=1$

The largest value is ${\displaystyle \frac{1}{10}}÷0.1$

 ${\displaystyle \frac{\mathbf{1}}{\mathbf{1000}}}=0.001$

$\mathbf{(}\mathbf{0}{\mathbf{.100}\mathbf{)}}^{\mathbf{2}}=0.01$

In Examples 12 to 19, fill in the blanks to make the statement true.

12: A fraction acts as an operator_____

Ans: The given true statement is:

A fraction acts as an operator of represents multiplication

13. Fraction which is reciprocal of ${\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}$ is_____

Ans: The given true statement is:

Fraction which is reciprocal of ${\displaystyle \frac{2}{3}}$ is ${\displaystyle \frac{3}{2}}$

14. Product of a proper and improper fraction is____ the improper fraction.

Ans: The given true statement is: 

Product of a proper and improper fraction is less than the improper fraction 

15. The two non-zero fractions whose product is 1, are called the____ of each other.

Ans: The given true statement is: 

The two non-zero fractions whose product is 1, are called the reciprocal of each other 

16: $$\mathbf{5}$$ rupees $$\mathbf{5}$$ paise $=Rs?$

Ans: $1$paisa is $0.01$rupees

So $5\text{ rupees }5\text{ paise }=5.05$rupees

17. $45\text{mm}=\mathrm{\_}\mathrm{\_}\mathrm{\_}\text{m}.$ 

Ans: $1\text{ mm}=0.001\text{m}$

$45\text{mm}=0.045\text{m}$

18. $\mathbf{2}\mathbf{.4}\times \mathbf{1000}\mathbf{=}$

Ans: By multiplying $2.4\times 1000=2400$

19. To divide a decimal number by $\mathbf{100}$, we shift the decimal point in the number to the ____by____ places. 

Ans: The given true statement is :

To divide a decimal number by $100,$ we shift the decimal point in the number to the left  by two  places.

In Examples 20 to 23 state whether the statements are True or False.

20: Reciprocal of an improper fraction is an improper fraction.

Ans: The given statement is False

The true statement is Reciprocal of an improper fraction is a proper fraction.

21: $\mathbf{2}{\displaystyle \frac{\mathbf{2}}{\mathbf{5}}}÷\mathbf{2}{\displaystyle \frac{\mathbf{1}}{\mathbf{5}}}=\mathbf{2}$

Ans: The given fraction is False

 The fraction of $2{\displaystyle \frac{2}{5}}÷2{\displaystyle \frac{1}{5}}={\displaystyle \frac{12}{5}}\times {\displaystyle \frac{5}{11}}={\displaystyle \frac{12}{11}}$

22. $\mathbf{0}\mathbf{.04}÷\mathbf{0}\mathbf{.2}\mathbf{=}\mathbf{0}\mathbf{.2}$

Ans: The given division is True.

$\mathbf{0}\mathbf{.04}÷\mathbf{0}\mathbf{.2}\mathbf{=}\mathbf{0}\mathbf{.2}$

23. $\mathbf{0}\mathbf{.2}\times \mathbf{0}\mathbf{.3}\mathbf{=}\mathbf{0}\mathbf{.6}$

Ans: The given statement is False. 

By multiplying the values we get $0.2\times 0.3=0.06$

24. Find ${\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}$ of $$\mathbf{6}$$ using circles with shaded parts.

Ans: From the following figure, try to find out ${\displaystyle \frac{2}{3}}$ of $$6$$ .

There are $12$ shaded parts out of $18$ parts which means $4$ wholes. 

Thus ${\displaystyle \frac{2}{3}}$ of $6$ is $4$

25. Find the value of ${\displaystyle \frac{\mathbf{1}}{\mathbf{4}{\displaystyle \frac{\mathbf{2}}{\mathbf{7}}}}}+{\displaystyle \frac{\mathbf{1}}{\mathbf{3}{\displaystyle \frac{\mathbf{11}}{\mathbf{13}}}}}+{\displaystyle \frac{\mathbf{1}}{\left({\displaystyle \frac{\mathbf{5}}{\mathbf{9}}}\right)}}$

Ans: $\text{Given expression = }$${\displaystyle \frac{1}{\left({\displaystyle \frac{30}{7}}\right)}}+{\displaystyle \frac{1}{\left({\displaystyle \frac{50}{13}}\right)}}+{\displaystyle \frac{1}{\left({\displaystyle \frac{5}{9}}\right)}}$

$={\displaystyle \frac{7}{30}}+{\displaystyle \frac{13}{50}}+{\displaystyle \frac{9}{5}}$

$={\displaystyle \frac{35}{150}}+{\displaystyle \frac{39}{150}}+{\displaystyle \frac{270}{150}}$

$={\displaystyle \frac{35+39+270}{150}}$

$={\displaystyle \frac{172}{75}}$

26. There is a $\mathbf{3}\times \mathbf{3}\times \mathbf{3}$ cube which consists of twenty seven $\mathbf{1}\times \mathbf{1}\times \mathbf{1}$ cubes (see Fig. 2.3). It is 'tunneled' by removing cubes from the coloured squares.

Find:

(i) Fraction of number of small cubes removed to the number of small cubes left in a given cube.

Ans: Number of small cubes removed $=1+1+1+1+1+1+$ $1=7$

So, required fraction $={\displaystyle \frac{7}{20}}$

(ii) Fraction of the number of small cubes removed to the total number of small cubes.

Ans: The Required fraction $={\displaystyle \frac{7}{27}}$

(iii) What part is (ii) of (i)?

Ans: Required part is ${\displaystyle \frac{7}{27}}÷{\displaystyle \frac{7}{20}}={\displaystyle \frac{7}{27}}\times {\displaystyle \frac{20}{7}}={\displaystyle \frac{20}{27}}$

27. Ramu finishes ${\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}$ part of a work in 1 hour. How much part of the work will be finished in $\mathbf{2}{\displaystyle \frac{\mathbf{1}}{\mathbf{5}}}$ hours?

Ans: The part of the work finished by Ramu in 1 hour $={\displaystyle \frac{1}{3}}$

So, the part of the work finished by Ramu in $2{\displaystyle \frac{1}{5}}$ hours $=2{\displaystyle \frac{1}{5}}\times {\displaystyle \frac{1}{3}}={\displaystyle \frac{11}{5}}\times {\displaystyle \frac{1}{3}}$

$={\displaystyle \frac{11\times 1}{5\times 3}}={\displaystyle \frac{11}{15}}$

Ramu will finish ${\displaystyle \frac{11}{15}}$ part of the work in $2{\displaystyle \frac{1}{5}}$ hours.

28. How many ${\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}\mathbf{kg}$ pieces can be cut from a cake of weight $\mathbf{4}\mathbf{kg}$ ?

Ans: The following figure represents 4 cakes each of $1\text{kg}$ and we need pieces of ${\displaystyle \frac{2}{3}}$ kg.

In the above figure 

That is, $4÷{\displaystyle \frac{2}{3}}=4\times {\displaystyle \frac{3}{2}}=6$

29. Harmeet purchased $\mathbf{3}\mathbf{.5}\mathbf{kg}$ of potatoes at the rate of ₹ $$\mathbf{13}.\mathbf{75}$$per kg. How much money should she pay in nearest rupees?

Ans: Cost of $1\text{kg}$ potatoes $=13.75$.

Cost of $3.5\text{kg}$ potatoes $=13.75\times 3.5$

By multiplying 

$13.75\times 3.5=48.125$

Hence, cost of $3.5\text{kg}$ potatoes $$=\text{Rs }48,$$ to the nearest rupees.

30. Kavita had a piece of rope of length $\mathbf{9}\mathbf{.5}\mathbf{m}$. She needed some small pieces of rope of length $\mathbf{1}\mathbf{.9}\mathbf{m}$ each. How many pieces of the required length will she get out of this rope?

Ans: Given 

The length of the rope $=9.5\text{m}$

The length of a small piece of rope $=1.9\text{m}$

Number of small pieces $=9.5\text{m}÷1.9\text{m}$

$={\displaystyle \frac{9.5}{1.9}}={\displaystyle \frac{9.5\times 10}{1.9\times 10}}$

$={\displaystyle \frac{95}{19}}=5$

So, she will get $5$ small pieces of rope.

31. Three boys earned a total of ₹$$\mathbf{235}.\mathbf{50}$$. What was the average amount earned per boy?

Ans: Three boys earned $=Rs235.50$

The average amount earned per boy $=Rs{\displaystyle \frac{235.50}{3}}$

By division

The average amount earned per boy is ₹$$78.50$$

32. Find the product of

(i) ${\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$ and ${\displaystyle \frac{\mathbf{5}}{\mathbf{8}}}$

Ans: The product of ${\displaystyle \frac{1}{2}}\times {\displaystyle \frac{5}{8}}={\displaystyle \frac{1\times 5}{2\times 8}}={\displaystyle \frac{5}{16}}$

(ii) ${\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}$ and ${\displaystyle \frac{\mathbf{7}}{\mathbf{5}}}$

Ans: The product of ${\displaystyle \frac{1}{3}}\times {\displaystyle \frac{7}{5}}={\displaystyle \frac{1\times 7}{3\times 5}}={\displaystyle \frac{7}{15}}$

(iii) ${\displaystyle \frac{\mathbf{4}}{\mathbf{3}}}$ and ${\displaystyle \frac{\mathbf{5}}{\mathbf{2}}}$

Ans: The product of  ${\displaystyle \frac{4}{3}}\times {\displaystyle \frac{5}{2}}={\displaystyle \frac{4\times 5}{3\times 2}}={\displaystyle \frac{20}{6}}={\displaystyle \frac{10}{3}}$

33. Observe the 3 products given in Example 32 and now give the answers of the following questions.

(i) Does interchanging the fractions in the example, ${\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}\times {\displaystyle \frac{\mathbf{5}}{\mathbf{8}}}$, affect the answer?

Ans: By interchanging ${\displaystyle \frac{1}{2}}\times {\displaystyle \frac{5}{8}}$ we get ${\displaystyle \frac{5}{8}}\times {\displaystyle \frac{1}{2}}$

${\displaystyle \frac{5}{8}}\times {\displaystyle \frac{1}{2}}={\displaystyle \frac{5\times 1}{8\times 2}}={\displaystyle \frac{5}{16}}$ which is same as the product we get

in Example 32 by multiplying ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{5}{8}}$. This means that interchanging the fractions does not affect the answer.

(ii) Is the value of the fraction in the product greater or less than the value of either fraction?

Ans: By observing the 3 products given in the solution of Example 32, we come to know that the value of the fractions in the products are as follows

(a) The product of two fractions whose value is less than 1 so, the product fractions is less than each of the fractions that are multiplied.

(b) The product of a proper and an improper fraction is less than the improper fractions and greater than the proper fraction.

(c) The product of two improper fractions is greater than each of the two fractions.

34. Reshma uses ${\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}\mathbf{m}$ of cloth to stitch a shirt. How many shirts can she make with $\mathbf{2}{\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}\mathbf{m}$ cloth?

Ans: Length of cloth needed to stitch 1 shirt $={\displaystyle \frac{3}{4}}\text{m}$

Length of cloth she has $=2{\displaystyle \frac{1}{4}}={\displaystyle \frac{9}{4}}\text{m}$

No. of shirts she can make by ${\displaystyle \frac{9}{4}}m$ cloth $={\displaystyle \frac{9}{4}}÷{\displaystyle \frac{3}{4}}$ shirts $={\displaystyle \frac{9}{4}}\times {\displaystyle \frac{4}{3}}$ shirts $=3$ shirts

Hence, she can make 3 shirts using $\mathbf{2}{\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}\mathbf{m}$ of cloth.

35: If the fraction of the frequencies of two notes have a common factor between the numerator and denominator, the two notes are harmonious. Use the graphic below to find the fraction of frequency of notes D and B.

Ans: Fraction of frequencies of notes $\text{D}$ and $\text{B}$ is 

$={\displaystyle \frac{\text{ Frequency of note D}}{\text{ Frequency of note B}}}={\displaystyle \frac{297}{495}}={\displaystyle \frac{3\times 3\times 3\times 11}{3\times 3\times 5\times 11}}$

So, the fraction of the frequencies of notes $D$ and $B$ is ${\displaystyle \frac{3}{5}}$. 

Clearly, the notes D and B are harmonies. 

36. Khilona said that we have gone about 120km or ${\displaystyle \frac{2}{3}}$ of the way to the campsite. So, how much farther do we have to go? 

Ans: Let total distance be $x$ km.

According to question,

${\displaystyle \frac{2}{3}}$ of $x=120$ 

$x=180$ km

Exercise

1. ${\displaystyle \frac{2}{5}}\times 5{\displaystyle \frac{1}{5}}$ is equal to:

(a) ${\displaystyle \frac{26}{25}}$

(b) ${\displaystyle \frac{52}{25}}$

(c) ${\displaystyle \frac{2}{5}}$

(d) $$\mathbf{6}$$

Ans: Option (b) is correct

Given that,

${\displaystyle \frac{2}{5}}\times 5{\displaystyle \frac{1}{5}}$

Converting mixed fraction into the whole fraction

${\displaystyle \frac{2}{5}}\times {\displaystyle \frac{26}{5}}={\displaystyle \frac{52}{25}}$

2. $3{\displaystyle \frac{3}{4}}÷{\displaystyle \frac{3}{4}}$is equal to:

(a)$\mathbf{3}$

(b) $\mathbf{4}$

(c) $\mathbf{5}$

(d) ${\displaystyle \frac{45}{16}}$

Ans: Option (c) is correct.

Converting mixed fraction into the whole fraction

${\displaystyle \frac{15}{4}}÷{\displaystyle \frac{3}{4}}$

Converting divide into multiplication so we reciprocal the fraction ${\displaystyle \frac{3}{4}}$

${\displaystyle \frac{15}{4}}\times {\displaystyle \frac{4}{3}}=5$

3. A ribbon of length $5{\displaystyle \frac{1}{4}}m$ is cut into small pieces each of length ${\displaystyle \frac{3}{4}}m$. Number of pieces will be:

(a) $\mathbf{5}$

(b) $\mathbf{6}$

(c) $\mathbf{7}$

(d) $\mathbf{8}$

Ans: Option (c) is correct.

Convert the mixed fraction

The length of ribbon $=5{\displaystyle \frac{1}{4}}={\displaystyle \frac{21}{4}}m$

Length of each small piece $={\displaystyle \frac{3}{4}}m$

No. of pieces $={\displaystyle \frac{21}{4}}÷{\displaystyle \frac{3}{4}}$

No. of pieces $={\displaystyle \frac{21}{4}}\times {\displaystyle \frac{4}{3}}=7$

4. The ascending arrangement of ${\displaystyle \frac{2}{3}},{\displaystyle \frac{6}{7}},{\displaystyle \frac{13}{21}}$ is:

(a) ${\displaystyle \frac{6}{7}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{13}{21}}$

(b) ${\displaystyle \frac{13}{21}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{6}{7}}$

(c) ${\displaystyle \frac{6}{7}},{\displaystyle \frac{13}{21}},{\displaystyle \frac{2}{3}}$

(d) ${\displaystyle \frac{2}{3}},{\displaystyle \frac{6}{7}},{\displaystyle \frac{13}{21}}$

Ans: Option (b) is correct.

The given numbers are ${\displaystyle \frac{2}{3}},{\displaystyle \frac{6}{7}},{\displaystyle \frac{13}{21}}$

21 is the LCM of denominators $(3,7,21)$

$\therefore {\displaystyle \frac{2}{3}}={\displaystyle \frac{2\times 7}{3\times 7}}={\displaystyle \frac{14}{21}};{\displaystyle \frac{6}{7}}={\displaystyle \frac{6\times 3}{7\times 3}}={\displaystyle \frac{18}{21}};{\displaystyle \frac{13}{21}}={\displaystyle \frac{13\times 1}{21\times 1}}={\displaystyle \frac{13}{21}}$

Now $13<14<18.$

${\displaystyle \frac{13}{21}}<{\displaystyle \frac{14}{21}}<{\displaystyle \frac{18}{21}}\Rightarrow {\displaystyle \frac{13}{21}}<{\displaystyle \frac{2}{3}}<{\displaystyle \frac{6}{7}}$

Hence, the ascending arrangement is

${\displaystyle \frac{13}{21}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{6}{7}}$

5. Reciprocal of the fraction ${\displaystyle \frac{2}{3}}$ is:

(a) $\mathbf{2}$

(b) $\mathbf{3}$

(c) ${\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}$

(d) ${\displaystyle \frac{\mathbf{3}}{\mathbf{2}}}$

Ans: Option (d) is correct.

Reciprocal of the asked fraction is $1÷{\displaystyle \frac{2}{3}}={\displaystyle \frac{3}{2}}$

6. The product of ${\displaystyle \frac{\mathbf{11}}{\mathbf{13}}}$ and $$\mathbf{4}$$ is:

(a) $\mathbf{3}{\displaystyle \frac{\mathbf{5}}{\mathbf{13}}}$.

(b) $\mathbf{5}{\displaystyle \frac{\mathbf{3}}{\mathbf{13}}}$

(c) $\mathbf{13}{\displaystyle \frac{\mathbf{3}}{\mathbf{5}}}$

(d) $\mathbf{13}{\displaystyle \frac{\mathbf{5}}{\mathbf{3}}}$

Ans: Option (a) is correct.

The product of $\Rightarrow {\displaystyle \frac{11}{13}}\times 4={\displaystyle \frac{44}{13}}$

 $=3{\displaystyle \frac{5}{13}}$

7. The product of $$\mathbf{3}$$ and $\mathbf{4}{\displaystyle \frac{\mathbf{2}}{\mathbf{5}}}$ is.:

(a) $\mathbf{17}{\displaystyle \frac{\mathbf{2}}{\mathbf{5}}}$

(b) ${\displaystyle \frac{\mathbf{24}}{\mathbf{5}}}$

(c) $\mathbf{13}{\displaystyle \frac{\mathbf{1}}{\mathbf{5}}}$

(d) $\mathbf{5}{\displaystyle \frac{\mathbf{1}}{\mathbf{13}}}$

Ans: Option (c) is correct.

First, we have to convert the mixed fraction into improper fraction $4{\displaystyle \frac{2}{5}}={\displaystyle \frac{22}{5}}$

$\Rightarrow 3\times 4{\displaystyle \frac{2}{5}}=3\times {\displaystyle \frac{22}{5}}={\displaystyle \frac{66}{5}}$

$=13{\displaystyle \frac{1}{5}}$

8. Pictorial representation of $\mathbf{3}\times {\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}$ is:

Ans: Option (b) is correct.

$3\times {\displaystyle \frac{2}{3}}={\displaystyle \frac{2}{3}}+{\displaystyle \frac{2}{3}}+{\displaystyle \frac{2}{3}}$

Therefore, pictorial representation is

9. ${\displaystyle \frac{\mathbf{1}}{\mathbf{5}}}÷{\displaystyle \frac{\mathbf{4}}{\mathbf{5}}}$ equal to:

(a) ${\displaystyle \frac{\mathbf{4}}{\mathbf{5}}}$

(b) ${\displaystyle \frac{\mathbf{1}}{\mathbf{5}}}$

(c) ${\displaystyle \frac{\mathbf{5}}{\mathbf{4}}}$

(d) ${\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}$

Ans: Option (d) is correct.

By division $\Rightarrow {\displaystyle \frac{1}{5}}÷{\displaystyle \frac{4}{5}}={\displaystyle \frac{1}{5}}\times {\displaystyle \frac{5}{4}}$

$={\displaystyle \frac{1}{4}}$

10. The product of $\mathbf{0}\mathbf{.03}\times \mathbf{0}\mathbf{.9}$ is:

(a) $\mathbf{2}\mathbf{.7}$

(b) $\mathbf{0}\mathbf{.27}$

(c) $\mathbf{0}.\mathbf{027}$

(d) $\mathbf{0}.\mathbf{0027}$

Ans: Option (c) is correct.

The product of  $\mathbf{0}\mathbf{.03}\times \mathbf{0}\mathbf{.9}$

$={\displaystyle \frac{3}{100}}\times {\displaystyle \frac{9}{10}}$

$={\displaystyle \frac{27}{1000}}=0.027$

11. ${\displaystyle \frac{\mathbf{5}}{\mathbf{7}}}÷\mathbf{6}$ is equal to:

(a) ${\displaystyle \frac{\mathbf{30}}{\mathbf{7}}}$

(b) ${\displaystyle \frac{\mathbf{5}}{\mathbf{42}}}$

(c) ${\displaystyle \frac{\mathbf{30}}{\mathbf{42}}}$

(d) ${\displaystyle \frac{\mathbf{6}}{\mathbf{7}}}$

Ans: Option (b) is correct.

By division ${\displaystyle \frac{5}{7}}÷6$

$={\displaystyle \frac{5}{7}}\times {\displaystyle \frac{1}{6}}={\displaystyle \frac{5}{42}}$

12. $\mathbf{5}{\displaystyle \frac{\mathbf{1}}{\mathbf{6}}}÷{\displaystyle \frac{\mathbf{9}}{\mathbf{2}}}$ is equal to:

(a) ${\displaystyle \frac{\mathbf{31}}{\mathbf{6}}}$

(b) ${\displaystyle \frac{\mathbf{1}}{\mathbf{27}}}$

(c) $\mathbf{5}{\displaystyle \frac{\mathbf{1}}{\mathbf{27}}}$

(d) ${\displaystyle \frac{\mathbf{31}}{\mathbf{27}}}$

Ans: Option (d) is correct.

By division $5{\displaystyle \frac{1}{6}}÷{\displaystyle \frac{9}{2}}$

$={\displaystyle \frac{31}{6}}÷{\displaystyle \frac{9}{2}}={\displaystyle \frac{31}{6}}\times {\displaystyle \frac{2}{9}}$

$={\displaystyle \frac{31}{27}}$

13. Which of the following represents ${\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}$ of ${\displaystyle \frac{\mathbf{1}}{\mathbf{6}}}$ ?

(a) ${\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}+{\displaystyle \frac{\mathbf{1}}{\mathbf{6}}}$

(b) ${\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}\mathbf{-}{\displaystyle \frac{\mathbf{1}}{\mathbf{6}}}$

(c) ${\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}\times {\displaystyle \frac{\mathbf{1}}{\mathbf{6}}}$

(d) ${\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}÷{\displaystyle \frac{\mathbf{1}}{\mathbf{6}}}$

Ans: Option (c) is correct.

$\Rightarrow {\displaystyle \frac{1}{3}}\text{ of }{\displaystyle \frac{1}{6}}={\displaystyle \frac{1}{3}}\times {\displaystyle \frac{1}{6}}$

14. ${\displaystyle \frac{\mathbf{3}}{\mathbf{7}}}$ of ${\displaystyle \frac{\mathbf{2}}{\mathbf{5}}}$ is equal to

(a) ${\displaystyle \frac{\mathbf{5}}{\mathbf{12}}}$

(b) ${\displaystyle \frac{\mathbf{5}}{\mathbf{35}}}$

(c) ${\displaystyle \frac{\mathbf{1}}{\mathbf{35}}}$

(d) ${\displaystyle \frac{\mathbf{6}}{\mathbf{35}}}$

Ans: Option (d) is correct.

${\displaystyle \frac{3}{7}}\text{ of }{\displaystyle \frac{2}{5}}={\displaystyle \frac{3}{7}}\times {\displaystyle \frac{2}{5}}={\displaystyle \frac{6}{35}}$

15. One packet of biscuits requires $\mathbf{2}{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$ cups of flour and $\mathbf{1}{\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}$ cups of sugar. Estimated total quantity of both ingredients used in 10 such packets of biscuits will be

(a) less than $\mathbf{30}$ cups

(b) between $\mathbf{30}$ cups and $\mathbf{40}$cups

(c) between $\mathbf{40}$ cups and $\mathbf{50}$ cups

(d) above $\mathbf{50}$ cups

Ans: Option (c) is correct.

The requirement of sugar and flour for one packet of biscuits $=\left(2{\displaystyle \frac{1}{2}}+1{\displaystyle \frac{2}{3}}\right)$

$=\left({\displaystyle \frac{5}{2}}+{\displaystyle \frac{5}{3}}\right)=\left({\displaystyle \frac{15+10}{6}}\right)={\displaystyle \frac{25}{6}}\text{ cups }$

Therefore, the requirement of both ingredients for $10$ packets

$=10\times {\displaystyle \frac{25}{6}}={\displaystyle \frac{125}{3}}=41{\displaystyle \frac{2}{3}}\text{ cups }$

Total estimated quantity will lie between $40$and $50$ cups.

16. The product of $\mathbf{7}$ and $\mathbf{6}{\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}$ is

(a) $\mathbf{42}{\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}$

(b) $\mathbf{47}{\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}$

(c) $\mathbf{42}{\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}$

(d) $\mathbf{47}{\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}$

Ans: Option (b) is correct.

The product of  $7\times 6{\displaystyle \frac{3}{4}}$

$={\displaystyle \frac{7}{1}}\times {\displaystyle \frac{27}{4}}={\displaystyle \frac{189}{4}}$

$=47{\displaystyle \frac{1}{4}}$

17. On dividing $\mathbf{7}$ by ${\displaystyle \frac{\mathbf{2}}{\mathbf{5}}}$, the result is

(a) ${\displaystyle \frac{\mathbf{14}}{\mathbf{2}}}$

(b) ${\displaystyle \frac{\mathbf{35}}{\mathbf{4}}}$

(c) ${\displaystyle \frac{\mathbf{14}}{\mathbf{5}}}$

(d) ${\displaystyle \frac{\mathbf{35}}{\mathbf{2}}}$

Ans: Option (d) is correct.

By dividing,

$7÷{\displaystyle \frac{2}{5}}=7\times {\displaystyle \frac{5}{2}}={\displaystyle \frac{35}{2}}$

18. $\mathbf{2}{\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}÷\mathbf{5}$ is equal to

(a) ${\displaystyle \frac{\mathbf{8}}{\mathbf{15}}}$

(b) ${\displaystyle \frac{\mathbf{40}}{\mathbf{3}}}$

(c) ${\displaystyle \frac{\mathbf{40}}{\mathbf{5}}}$

(d) ${\displaystyle \frac{\mathbf{8}}{\mathbf{3}}}$

Ans: Option (a) is correct.

First, we have to convert the mixed fraction into improper fraction $2{\displaystyle \frac{2}{3}}={\displaystyle \frac{8}{3}}$ 

Then

$2{\displaystyle \frac{2}{3}}÷{\displaystyle \frac{5}{1}}$

$={\displaystyle \frac{8}{3}}\times {\displaystyle \frac{1}{5}}={\displaystyle \frac{8}{15}}$

19. ${\displaystyle \frac{\mathbf{4}}{\mathbf{5}}}$ of $\mathbf{5}\mathbf{kg}$ apples were used on Monday. The next day ${\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}$ of what was left was used. Weight (in $\text{kg}$ ) of apples left now is

(a) ${\displaystyle \frac{\mathbf{2}}{\mathbf{7}}}$

(b) ${\displaystyle \frac{\mathbf{1}}{\mathbf{14}}}$

(c) ${\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}$

(d) ${\displaystyle \frac{\mathbf{4}}{\mathbf{21}}}$

Ans: Option (c) is correct.

The weight of apple used on Monday $={\displaystyle \frac{4}{5}}\times 5=4\text{Kg}$

Weight of apples left on next day $=(5-4)=1\text{kg}$

Weight of apples used on next day ${\displaystyle \frac{1}{3}}$ of $1\text{kg}={\displaystyle \frac{1}{3}}$

Therefore, weight of apples left $=\left(1-{\displaystyle \frac{1}{3}}\right)={\displaystyle \frac{2}{3}}\text{kg}$

20. The picture 

Interprets

(a) ${\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}÷\mathbf{3}$

(b) $\mathbf{3}\times {\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}$

(c) ${\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}\times \mathbf{3}$

(d) $\mathbf{3}÷{\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}$

Ans: Option (b) is correct.

${\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{4}}={\displaystyle \frac{3}{4}}$

In Questions 21 to 44, fill in the blanks to make the statements true.

21. Rani ${\displaystyle \frac{\mathbf{2}}{\mathbf{7}}}$ ate part of a cake while her brother Ravi ate ${\displaystyle \frac{\mathbf{4}}{\mathbf{5}}}$ of the remaining. Part of the cake left is _____

Ans: Let, there is only one cake,

Now the part of the cake ate by the Rani was ${\displaystyle \frac{2}{7}}$ 

Hence, the cake left

$\Rightarrow 1-{\displaystyle \frac{2}{7}}={\displaystyle \frac{5}{7}}$

Now the part of the cake ate by the Ravi was

$\Rightarrow {\displaystyle \frac{4}{5}}\times {\displaystyle \frac{5}{7}}={\displaystyle \frac{4}{7}}$

Therefore, the cake left

$\Rightarrow {\displaystyle \frac{5}{7}}-{\displaystyle \frac{4}{7}}={\displaystyle \frac{1}{7}}$

22. The reciprocal of ${\displaystyle \frac{\mathbf{3}}{\mathbf{7}}}$ is______

Ans: The reciprocal of ${\displaystyle \frac{3}{7}}$ is $${\displaystyle \frac{7}{3}}$$

23. ${\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}$ of $\mathbf{27}$ is_____

Ans: The multiply of ${\displaystyle \frac{2}{3}}\times 27$

$=2\times 9=18$

24. ${\displaystyle \frac{\mathbf{4}}{\mathbf{5}}}$ of $\mathbf{45}$ is____

Ans: The multiply of ${\displaystyle \frac{4}{5}}\times 45$

$=4\times 9=36$

25. $\mathbf{4}\times \mathbf{6}{\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}$ is equal to______

Ans: The product of $4\times 6{\displaystyle \frac{1}{3}}$

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

$=25{\displaystyle \frac{1}{3}}$

26. ${\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$ of $\mathbf{4}{\displaystyle \frac{\mathbf{2}}{\mathbf{7}}}$ is_____

Ans:  First we have to convert the mixed fraction into improper fraction $4{\displaystyle \frac{2}{7}}={\displaystyle \frac{30}{7}}$

${\displaystyle \frac{1}{2}}$ of $4{\displaystyle \frac{2}{7}}$

$={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{30}{7}}$

$={\displaystyle \frac{15}{7}}=2{\displaystyle \frac{1}{7}}$

27. ${\displaystyle \frac{\mathbf{1}}{\mathbf{9}}}$ of ${\displaystyle \frac{\mathbf{6}}{\mathbf{5}}}$ is________

Ans: The product the ${\displaystyle \frac{1}{9}}$ of ${\displaystyle \frac{6}{5}}$

$={\displaystyle \frac{1}{9}}\times {\displaystyle \frac{6}{5}}$

$={\displaystyle \frac{6}{45}}={\displaystyle \frac{2}{15}}$

28. The lowest form of the product $\mathbf{2}{\displaystyle \frac{\mathbf{3}}{\mathbf{7}}}\times {\displaystyle \frac{\mathbf{7}}{\mathbf{9}}}$ is______

Ans: The lowest form of $2{\displaystyle \frac{3}{7}}\times {\displaystyle \frac{7}{9}}$

$={\displaystyle \frac{17}{7}}\times {\displaystyle \frac{7}{9}}$

$={\displaystyle \frac{17}{9}}=1{\displaystyle \frac{8}{9}}$

29. ${\displaystyle \frac{\mathbf{4}}{\mathbf{5}}}÷\mathbf{4}$ is equal to_____

Ans: By division ${\displaystyle \frac{4}{5}}÷4$

$={\displaystyle \frac{4}{5}}\times {\displaystyle \frac{1}{4}}={\displaystyle \frac{1}{5}}$

30. ${\displaystyle \frac{\mathbf{2}}{\mathbf{5}}}$ of $25$ is____

Ans: The product is ${\displaystyle \frac{2}{5}}$ of $$25$$

$={\displaystyle \frac{2}{5}}\times 25$

$=2\times 5=10$

31. ${\displaystyle \frac{\mathbf{1}}{\mathbf{5}}}÷{\displaystyle \frac{\mathbf{5}}{\mathbf{6}}}={\displaystyle \frac{\mathbf{1}}{\mathbf{5}}}\mathrm{\_}\mathrm{\_}{\displaystyle \frac{\mathbf{6}}{\mathbf{5}}}$ 

Ans: While dividing one fraction by another fraction, we multiply the first fraction by the reciprocal of the other.

${\displaystyle \frac{1}{5}}÷{\displaystyle \frac{5}{6}}={\displaystyle \frac{1}{5}}\times {\displaystyle \frac{6}{5}}$

32. $\mathbf{3}\mathbf{.2}\times \mathbf{10}\mathbf{=}$

Ans: To multiply a decimal number by 10, we move the decimal point in the number to the right by as many places as many zeros $(0)$ are at the right of one

$3.2\times 10=32$

33. $\mathbf{25}\mathbf{.4}\times \mathbf{1000}\mathbf{=}$

Ans: To multiply a decimal number by $$1000$$ we move the decimal point in the number to the right by as many places as many zeros $(0)$ are at the right of one.

$25.4\times 1000=25400$

34. $\mathbf{93}\mathbf{.5}\times \mathbf{100}\mathbf{=}$

Ans: To multiply a decimal number by $$100$$ we move the decimal point in the number to the right by as many places as many zeros $(0)$ are at the right of one 

$93.5\times 100=9350$

35. $\mathbf{4}\mathbf{.7}÷\mathbf{10}\mathbf{=}$

Ans: To divide a decimal number by $$10$$, shift the decimal point in the decimal number to the left by as many places as there are zeros over 1, to get the quotient.

$4.7÷10=0.47$

36. $\mathbf{4}.\mathbf{7}÷\mathbf{100}=$

Ans: To divide a decimal number by $$100,$$ shift the decimal point in the decimal number to the left by as many places as there are zeros over 1, to get the quotient.

$4.7÷100=0.047$

37. $\mathbf{4}.\mathbf{7}÷\mathbf{1000}=$

Ans: To divide a decimal number by $$1000$$, shift the decimal point in the decimal number to

the left by as many places as there are zeros over 1, to get the quotient.

$4.7÷1000=0.0047$

38. The product of two proper fractions is______than each of the fractions that are multiplied.

Ans: The product of two proper fractions is less than each of the fractions that are multiplied.

39. While dividing a fraction by another fraction, we _________ the first fraction by the _______ of the other fraction.

Ans: While dividing a fraction by another fraction, we multiply the first fraction by the reciprocal of the other fraction.

40. $8.4÷\mathrm{\_}\mathrm{\_}\mathrm{\_}=2.1$ 

Ans: $8.4÷\underset{-}{4}=2.1$

Let us assume the missing fraction be $x$,

Then,

$8.4÷x=2.1$

$8.4/x=2.1$

By cross multiplication, we get,

$x=8.4/2.1$

$x=84/21$

$x=28/7$

$x=4$

41. $\mathbf{52}\mathbf{.7}÷\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}=\mathbf{0}\mathbf{.527}$

Ans: Let consider $x$ be the missing variable

 $52.7÷\text{x}=0.527$

$\text{x}={\displaystyle \frac{52.7}{0.527}}=100$

$x=100$

42. $\mathbf{0}\mathbf{.5}\mathbf{\_}\mathbf{\_}\mathbf{\_}\mathbf{0}\mathbf{.7}\mathbf{=}\mathbf{0}\mathbf{.35}$

Ans: Let determine the exponent 

$0.5\text{of}0.7=0.35$

So, $0.5\times 0.7=0.35$

43. $\mathbf{2}\mathrm{\_}\mathrm{\_}\mathrm{\_}{\displaystyle \frac{\mathbf{5}}{\mathbf{3}}}\mathbf{=}{\displaystyle \frac{\mathbf{10}}{\mathbf{3}}}$

Ans: The missing exponent is 

  $\Rightarrow 2\times {\displaystyle \frac{5}{3}}={\displaystyle \frac{10}{3}}$

  $=2\times {\displaystyle \frac{5}{3}}={\displaystyle \frac{10}{3}}$

44. $\mathbf{2}\mathbf{.001}÷\mathbf{0}\mathbf{.003}\mathbf{=}$

Ans: By division

   $\Rightarrow 2.001÷0.003=$

   $={\displaystyle \frac{2001}{1000}}\times {\displaystyle \frac{1000}{3}}$

   $=667$

In each of the Questions 45 to 54, state whether the statement is True or False.

45. The reciprocal of a proper fraction is a proper fraction.

Ans: The given statement is False, 

So, the correct statement is the reciprocal of a proper fraction is an improper fraction.

46. The reciprocal of an improper fraction is an improper fraction.

Ans: The given statement is False, 

So, the correct statement is the reciprocal of an improper fraction is a proper fraction.

47. $\text{Product of two fractions = }{\displaystyle \frac{\text{ Product of their denominators }}{\text{ Product of their numerators }}}$

Ans: The given statement is false. 

The correct statement is $\text{Product}\text{of}\text{two}\text{fractions}={\displaystyle \frac{\text{ Product of their numerators }}{\text{ Product of their denominators }}}$

48. The product of two improper fractions is less than both the fractions.

Ans: The given statement is False,

Correct statement is the product of two improper fractions is greater than both the fractions.

49. A reciprocal of a fraction is obtained by inverting it upside down.

Ans: The given statement is True

50. To multiply a decimal number by $1000$ , we move the decimal point in the number to the right by three places.

Ans: The given statement is True

51. To divide a decimal number by $100$ , we move the decimal point in the number to the left by two places. 

Ans: The given statement is True

52. 1 is the only number which is its own reciprocal. 

Ans: The given statement is True

53. ${\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}$ of $$\mathbf{8}$$ is same as ${\displaystyle \frac{\mathbf{2}}{\mathbf{3}}}÷\mathbf{8}$

Ans: The given statement is False

The product of ${\displaystyle \frac{2}{3}}\times 8={\displaystyle \frac{16}{3}}$

And

The division of  ${\displaystyle \frac{2}{3}}÷8={\displaystyle \frac{2}{3}}\times {\displaystyle \frac{1}{8}}={\displaystyle \frac{1}{12}}$

${\displaystyle \frac{2}{3}}\text{ of }8\ne {\displaystyle \frac{2}{3}}÷8$

54. The reciprocal of ${\displaystyle \frac{\mathbf{4}}{\mathbf{7}}}$ is ${\displaystyle \frac{\mathbf{4}}{\mathbf{7}}}$

Ans: The given statement is False, 

The reciprocal of ${\displaystyle \frac{4}{7}}$ is ${\displaystyle \frac{7}{4}}$

55. If $\mathbf{5}$ is added to both the numerator and the denominator of the fraction ${\displaystyle \frac{5}{9}}$, will the value of the fraction be changed? If so, will the value increase or decrease?

Ans: Given, ${\displaystyle \frac{5}{9}}$ and ${\displaystyle \frac{5+5}{9+5}}={\displaystyle \frac{10}{14}}={\displaystyle \frac{5}{7}}$

Here, ${\displaystyle \frac{5}{9}}\ne {\displaystyle \frac{5}{7}}$

Now, ${\displaystyle \frac{5}{9}}<{\displaystyle \frac{5}{7}}$

Therefore, the value is increased.

56. What happens to the value of a fraction if the denominator of the fraction is decreased while the numerator is kept unchanged?

Ans: If the denominator of the fraction is decreased, the value of a fraction is increased while the numerator remains unchanged.

57. Which letter comes ${\displaystyle \frac{\mathbf{2}}{\mathbf{5}}}$ of the way among $\text{A}$ and$$\mathbf{J}$$?

Ans: The total letter in number between $\text{A}$ and $\text{J}$ is $10.$