MATHS M.C.Q. [1 Marks Each]

$\begin{array}{c|c}3&3,6,9\\\hline2&1,2,3\\\hline3&1,1,3\\\hline&1,1,1\end{array}$
$\frac{5}{6}+\frac{2}{3}-\frac{4}{9}$
$​​L.C.M.$ of $3, 6$ and $9 = (2 \times 3 \times 3) = 18$
$=\frac{(15+12-8)}{18}$
$\Big(\frac{18}{6}=3,3\times5=15\Big)$
$\Big(\frac{18}{3}=6,6\times2=12\Big)$
and $\Big(\frac{18}{9}=2,2\times4=8\Big)$
$=\frac{(27-8)}{18}$
$=\frac{19}{18}$
$=1\frac{1}{18}$

$9275$ Meters in km, as decimal fraction$=\frac{9275}{1000} \ \text{KM} =9.275 \ \text{KM}$

In $\dfrac{6}{5}$​, the numerator $6$ is greater than the denominator $5.$
Therefore, $\dfrac{6}{5}$ is not a proper fraction.

A fraction whose numerator is less than the denominator is called a proper fraction.
Here, $\frac{3}{5}$ is a proper fraction.
Hence, the correct option is $(a).$

since it is a mixed fraction.. it can be written as $\frac{9}{4}..$. then the reciprocal of $\frac{9}{4}$ is $\frac{4}{9}..$

Since, $0.7499$ is greater than $0.749$ and less than $0.75$. Therefore, $0.7499$ lies between $0.749$ and $0.75.$
$0.749 < 0.7499 < 0.75$

$\frac{26}{4}+\frac{14}{3}$
$=\frac{78+56}{12}=\frac{134}{12}=11\frac{2}{12}=11\frac{1}{6}$

Fractions that are greater than $00$ but less than $1$ are called proper fractions.
In proper fractions, the numerator is less than the denominator.
When a fraction has a numerator that is greater than or equal to the denominator, then the fraction is an improper fraction.
An improper fraction is always $1$ or greater than $1.$
Now looking at options
$\cfrac {2}{3} = .666 < 1$
$\cfrac {3}{4} = .75 < 1$
$\cfrac {5}{7} = .71 < 1$
$\cfrac {6}{5}= 1.2 < 1$
So $\cfrac {6}{5}$ is Not a Proper fraction.

The first decimal digit from the decimal point is the tenth.
$4.4$ has $4$ on the ones, after decimal point on the tenths is $4$ tenths.
$4.4$ is the sum of $4$ and$\displaystyle{\frac{4}{10}}$ or $\displaystyle{\frac{4}{10}}$.
So option $B$ is the correct answer.

Given, $3\frac{1}{2}\text{m}$ of cloth cost $Rs. 168.$
Then $4\frac{1}{3}\text{m}$ of cloth cost Rs. $\frac{168\times\frac{13}{3}}{\frac{7}{2}}=208.$

Since, numerator > denominator, $\frac{5}{4}$ is not a proper fraction.

Since, numerator > denominator, $\frac {5}{4}$ is not a proper fraction.

In an improper fraction, the numerator is greater than the denominator.
Of the given fractions, $\frac{15}{14}$ has numerator greater than the denominator.
Hence, $\frac{15}{14}$​ is a proper fraction.

The smallest possible decimal fraction upto three decimal places $=\dfrac{1}{1000}=.001$

Improper fraction of
$2\cfrac{1}{2}​= \cfrac{2 \times 2 + 1}{2}$
$= \cfrac{4 + 1}{2}$
$=\cfrac{5}{2}$

Number of dresses she had to stiches $= 35$
Number of dresses she has finished $= 21$
$\therefore$ Fraction of dresses she has finished = $\dfrac{21}{35} =\dfrac{3}{5}$

$\frac{1.5}{0.2 \ + \text{ x} } = 5$
cross multiplying$\Rightarrow 5 \times (0.2 + x) = 1.5$
$\Rightarrow 5 \times 0.2 + 5x = 1.5$
$\Rightarrow 1 \times 5x = 1.5$
$\Rightarrow 5x = 1.5 - 1 = 0.5$
$\Rightarrow 5x = 0.5$
$\Rightarrow x = 0.1$

Proper fractions are the one whose numerator is less than the denominator
In $\dfrac{3}{2}$ and $ \dfrac{8}{3},$ the numerator is greater than the denominatorSo, they are not proper fractionsWhereas in $\dfrac{2}{5}$​ and $\dfrac{1}{7}$ the numerator is less than the denominatorThus, they are proper fractions.Hence, $\dfrac{2}{5}$ and $\dfrac{1}{7}$are proper fractional numbers.

$0.000000375=375\times { 10 }^{ -9 }=3.75\times { 10 }^{ -7 }$
$=\cfrac { 375 }{ 100 } \times { 10 }^{ -7 }$
$=\cfrac { 15 }{ 4 } \times { 10 }^{ -7 }=3\cfrac { 3 }{ 4 } \times 10^7.$

Fraction equivalent to a given fraction can be obtained by multiplying or dividing its numerator and denominator by a non-zero number.
Therefore, the fraction equivalent to $\frac{30}{45}$ is $\frac{30\div15}{45\div15}=\frac{2}{3}.$
Hence, the correct option is $(c).$

If the numerator is less than the denominator then the fraction is called as proper fraction.
Hence,$\frac{7}{8}$ is a proper fraction.

$\text{Addition of like fractions} =\frac{\text{Sum of the numerators}}{\text{ Common denominator}}$
$=\frac{5}{8}+\frac{1}{8}$
$=\frac{(5+1)}{8}$
$=\frac{6}{8}$
$=\frac{3}{4}$

A fraction whose numerator is less than the denominator is called a proper fraction, otherwise it is called an improper fraction.
Here, $\frac{34}{13}$ is an example of an improper fraction.
Hence, the correct option is $(b).$

$\Rightarrow2\frac{5}{7}\text{%} $ of $280$
$=\frac{19}{7}\text{%}$ of $280$
$=\frac{19}{7}\times\frac{280}{100}$
$=19\times\frac{4}{10}$
$\frac{76}{10}=7.6$
So $2\frac{5}{7}\text{%}$ of $280\text{cm}$ is $7.6\text{cm}.$

Here, Numerator > denominator only in option $A.$

Given, $\frac{11}{12}\times\frac{16}{4}\times\frac{9}{16}=\frac{11}{12}\times4\times\frac{9}{16}$
$\rightarrow\frac{11}{3}\times\frac{9}{16}$
$\rightarrow11\times\frac{3}{16}=\frac{33}{16}=2\frac{1} {16}$

In a proper fraction, the numerator is smaller than the denominator. Of the given fractions, $\displaystyle \frac{5}{7}$ has numerator smaller than the denominator.
Hence, $\displaystyle \frac{5}{7}$​ is a proper fraction.

Let us compare $3\frac{1}{3}$ and $\frac{33}{10}$
or $\frac{10}{3}$ and $\frac{33}{10}$
$10 \times 10 = 100$ and $3 ​\times 33 = 99$
Clearly,$ 100 > 99$
Therefore, $\frac{10}{3}<\frac{33}{10}$
or $3\frac{1}{3}<\frac{33}{10}$

Games won $= 6$
Total Games $= 6 + 4 = 10$
$\therefore$ required fraction $\frac{6}{10}$

According to number system , every fraction in $\frac{p}{q} $ form can be converted into decimal number. and vice versa.

In an improper fraction, the numerator is always greater than the denominator.
Eg $ \displaystyle \frac{9}{5}​,\displaystyle \frac{5}{3}.$

$\frac{7}{3},$ because in an improper fraction, the numerator is more than the denominator.

$0.231=\dfrac{0.231}{1}$​Multiplying numerator and denominator by $100.\dfrac{231}{1000}$​ Hence.

Converting fraction to decimal = $\frac{3}{10} = 0.3$

A Mixed Fraction is a whole number and a proper fraction combined.
Divide the numerator by the denominator.
Write down the whole number answerThen write down any remainder above the denominator.
$\dfrac{39}{12}=\dfrac{36}{12}+\dfrac{3}{12}$
$=3+\dfrac{3}{12}=3\dfrac{3}{12}$

$\frac{-1}{2}=-0.5$

We know that, if the numerator is less than the denominator, then the value of fraction is less than $1.$
Hence, the fraction $\frac57$ lies between $0$ and $1.$

$1\frac{3}{4}$ is a mixed fraction as:
$1+\frac{3}{4}=1\frac{3}{4}$

$\frac{15}{20}$ can be written in the simplest form $\frac{4}{3}.$
Now, look at the options, then option $D$ can also be written in the simplest form $\frac{4}{3}.$
That means option $D$ is equivalent to $\frac{15}{20}.$

$0.231=\frac{0.231}{1}$ Multiplying numerator and denominator by $100.\frac{231}{1000}$Hence.

Given, $\displaystyle \frac{11}{12} \times \displaystyle \frac{16}{4} \times \displaystyle \frac{9}{16}$
$=\frac{11}{12} \times 4 \times \frac{9}{16}$
$\rightarrow \displaystyle \frac{11}{3} \times \frac{9}{16}$
$\rightarrow \displaystyle 11 \times \frac{3}{16} = \frac{33}{16} = 2 \frac{1}{16}$

Proper fraction of $6\cfrac{1}{5}​= \cfrac{5 \times 6 + 1}{5} ​= \cfrac{30 + 1}{5} ​= \cfrac{31}{5}$

$\Big(\frac{3}{4}=\frac{\text{x}}{20}\Big)$
We have,
$20 = 4 \times 5$
So, we have to multiply the numerator by $5.$
Therefore, $x = 3 \times 5 = 15$

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number).
In given options $\frac{11}{23}$ is proper fraction.

Given, $\frac{20}{50}$ To obtain the lowest form of given fraction, divide it by $10.$
Then the lowest form of $\frac {20}{50}$ is $ \frac{2}{5}​$

Fractions can be compared by converting them into like fractions and then arranging them in ascending or descending order.
$\frac{3}{5}=\frac{3}{5}\times\frac{6}{6}=\frac{18}{30}$
$\frac{2}{3}=\frac{2}{3}\times\frac{10}{10}=\frac{20}{30}$
$\frac{5}{6}=\frac{5}{6}\times\frac{5}{5}=\frac{25}{30}$
$\frac{7}{10}=\frac{7}{10}\times\frac{3}{3}=\frac{21}{30}$
We know,
$18 < 20 < 21 < 25$
$\Rightarrow\frac{18}{30}<\frac{20}{30}<\frac{21}{30}<\frac{25}{30}$
$\Rightarrow\frac{3}{5}<\frac{2}{3}<\frac{7}{10}<\frac{5}{6}$
$\therefore$ the smallest fraction is $\frac{3}{5}.$
Hence, the correct option is $(b).$

For rounding off to tenths place, we look at the hundredths place.
Here, the digit at hundredths place is $7$ which is greater than $5.$
So, the digit at the tenths place $(5)$ will be increased by $1$ and digits at the hundredths and thousandths place will be written as equal to zero.
Hence, rounding off $13.572$ to nearest tenths, we get $13.6.$

Since, for comparing fractions with same denominators, fraction with smaller numerator is
$\therefore\frac38<\frac58<\frac78<\frac98$
Hence, $\frac38$ is the smallest fraction.

Let us check with all options:
$(A) 14$ tenths $4$ thousandths $=\frac{14}{10}+\frac{4}{1000}=1.4+0.004=1.404.$
$(B) 2$ tenths $13$ hundredths $=\frac{2}{10}+\frac{13}{10}=0.33.$
$(C) 4$ hundredths $2$ tenths $=\frac{4}{100}+\frac{2}{10}=0.24.$
$(D) 7$ tenths $17$ hundredths $=\frac{7}{10}+\frac{17}{100}=0.87.$ Hence, it is correct.

$\frac{5}{12}=\frac{\text{x}}{3}$
On cross-multiplying, we get:
$5\times3=\text{x}\times12$
$\Rightarrow\text{x}=\frac{5\times3}{12}$
$\Rightarrow\text{x}=\frac{15}{12}$
$\text{x}=\frac{5}{4}$

Fraction equivalent to a given fraction can be obtained by multiplying or dividing its numerator and denominator by a non-zero number.
Therefore, the fraction equivalent to $\frac{2}{3}$ is $\frac{2\times5}{3\times5}.$
Hence, the correct option is $(c).$

$=\frac{5}{8}-\frac{1}{8}$
$=\frac{(5-1)}{8}$
$=\frac{4}{8}$
$=\frac{1}{2}$

Improper fraction of
$2\cfrac{3}{7}= \cfrac{7 \times 2 + 3}{7}​$
$= \cfrac{14 + 3}{7}​= \cfrac{17}{7}$

Proper fraction of
$6\frac{1}{5}=\frac{5\times6+1}{5}=\frac{30+1}{5}=\frac{31}{5}$

On dividing the numerator & denominator by $2$, we get $\frac{3}{5}.$

$L.C.M.$ of $3, 9, 2$ and $12 = ( 2 \times 2 \times 3 ​\times 3) = 36$
Now, we have:
$\frac{2}{3}=\frac{2\times12}{3\times12}=\frac{24}{36}$
$\frac{5}{9}=\frac{5\times4}{9\times4}=\frac{20}{36}$
$\frac{1}{2}=\frac{1\times18}{2\times18}=\frac{18}{36}$
$\frac{7}{12}=\frac{7\times3}{12\times3}=\frac{21}{36}$
Hence, $\frac{24}{36}=\frac{2}{3}$ is the largest fraction.

In an improper fraction, the numerator is greater than the denominator.
Of the given fractions, $\dfrac {15}{14}$ has numerator greater than the denominator.
Hence, $\dfrac {15}{14}$ is a proper fraction.

Improper fractions are the one who numerator is more than the denominator
In $ \frac{13}{2}​$ and $ \frac{7}{3}​, $ the numerator is greater than the denominator
So, they are not proper fractions Whereas in $\frac{2}{7}$​ and $\frac{7}{11},$ the numerator is less than the denominator
Thus, they are proper fractions.
Hence, $\frac{13}{2}​$ and $\frac{7}{3}​ $ are improper fractional numbers.

$1\frac{3}{4}$​ is a mixed fraction as:
$1\dfrac{3}{4}=1\dfrac{3}{4}$

Given that
$\frac{1.5}{0.2+\text{x}}=5$
cross multiplying
$\Rightarrow 5 \times (0.2 + x) = 1.5$
$\Rightarrow 5 \times 0.2 + 5x = 1.5$
$\Rightarrow 1 + 5x = 1.5$
$\Rightarrow 5x = 1.5 − 1 = 0.5$
$\Rightarrow 5x = 0.5$
$\Rightarrow x = 0.1$

Improper fractions are the one who numerator is more than the denominator
In $\displaystyle\frac{13}{2}$​ and $\dfrac{7}{3}$​, the numerator is greater than the denominator So, they are not proper fractions Whereas in $\dfrac{2}{7}$and $\dfrac{7}{11}$​, the numerator is less than the denominatorThus, they are proper fractions.Hence, $\dfrac{13}{2}$ and $\dfrac{7}{3}$ are improper fractional numbers.

Since, $0.023$ is greater than $0.02$ and less than $0.03.$
Therefore, $0.023$ lies between $0.02$ and $0.03.$
$0.02 < 0.023 < 0.03$

$\frac{3}{7}$ is a simple fraction because both numerator and denominator are integers.

$2 < 4 < 5 < 11$
$\Rightarrow\frac{2}{9}<\frac{4}{9}<\frac{5}{9}<\frac{11}{9}$
$\therefore$ the smallest fraction is $\frac{2}{9}.$
Hence, the correct option is $(d).$

This happens in only option
$\text{B}1+\frac{1}{10}=\frac{11}{10}=11\times1\times\frac{1}{10}$

To round to the nearest tenth, write down the number with a decimal point, and find the tenths place directly to the right of the decimal.
Then, to the right of the tenths place, look at the number in the hundredths place.
In $6050.287$ the number in the hundredths is $.2.$
So, $6050.287$ rounded to the nearest tenths will be $6050.3$

A proper or improper fraction which is completely divisible denotes the denominator is the divider and the numerator is the divident.
The quotient is the answer of that division. We see that option $D$ is the only option that shows this relationship through two different symbols.

Numerator $<$ Denominator in all $4$ cases.

Since, fractions with same denominators can be subtracted by simply subtracting the numerators and writing the common denominator as it is.
$\frac{19}{9}-\frac59=\frac{19-5}9{}=\frac{14}{9}$

Factors of $24$ are $1, 2, 3, 4, 6, 8, 12, 24.$
Factors of $36$ are $1, 2, 3, 4, 6, 9, 12, 18, 36.$
Common factors of $24$ and $36$ are $1, 2, 3, 4, 6, 12.$
$H.C.F. = 12$
Dividing both the numerator and the denominator by $12:$
$\frac{24}{36}$
$=\frac{24\div12}{36\div12}$
$=\frac{2}{3}$

$\Rightarrow2\dfrac { 5 }{ 7 } \%$ of $280$
$=\dfrac { 19 }{ 7 }\% \%$ of $280$
$=\dfrac{19}{7}\times \dfrac{280}{100}$
$=19\times \dfrac{4}{10}$
$=\dfrac{76}{10}=7.6$
So $2\dfrac { 5 }{ 7 }$ of $280\ cm$ is $7.67.6\ cm$

This happens in only option $\text{B1}+\frac{1}{10}= \frac{11}{10}​= 11\times 1\times \frac { 1 }{ 10 }$

$\frac{\text{W N } \times \text{ D + N}}{\text{D}}$
$\dfrac {12\times 6+1}{6}$
$=\dfrac {72+1}{6}$
$=\dfrac {73}{6}$

$\frac{11}{7}=\frac{77}{\text{x}}$
On cross-multiplying, we get:
$11\times\text{x}=77\times4$
$\Rightarrow\text{x}=\frac{77\times4}{11}$
$\Rightarrow\text{x}=\frac{7\times11\times4}{11}$
$\text{x}=28$

Number of dresses she had to stiches $= 35$
Number of dresses she has finished $= 21$
$\therefore$ Fraction of dresses she has finished $=\frac{21}{35}=\frac{3}{5}$

$0.23=\frac{23}{100}$

$1008\times\frac{7}{8}-568\times\frac{3}{4}$
$=126\times7-142\times3$
$=882-426$
$=456$

A unit fraction is a rational number written as a fraction in which numerator is $11$ and denominator is a positive integer.
Example:
$\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{5}$ etc.

$\displaystyle 36.2 \Rightarrow 36 + 0.2\ = \frac {362}{10} = 36 \frac{2}{10}.$

Here, whole part of all numbers are same and tenths part of $0.0925$ and $0.038$ are same
i.e. $0$ and tenths part of $0.182 =\frac{1}{10}$
and tenths part of $0.29 =\frac{2}{10}$
Hence, $0.29$ is the greatest.

Here, numerator < denominator only in option $A.$

$\frac{232}{990}$ is equal to $0.234.$

$\displaystyle\frac{7}{11}$​ is not an improper fraction because the numerator is smaller than the denominator.
In Improper fraction, the Numerator is always larger than the denominator.

Let the blank part be $x$ Therefore,
$\frac{1}{7}+\frac{2}{7}+\frac{\text{x}}{7}=1\frac{3}{7}\Rightarrow\frac{3+\text{x}}{7}=\frac{10}{7}$
$\Rightarrow3+\text{x}=10$
$\Rightarrow\text{x}=10-3=7$

$\Box-\frac{5}{8}=\frac{1}{4}$
$\Rightarrow\Box=\frac{1}{4}+\frac{5}{8}$
$LCM$ of $4$ and $8$ is $8.$
$\Rightarrow\Box=\frac{1\times2}{4\times2}+\frac{5\times1}{8\times1}$
$\Rightarrow\Box=\frac{2}{8}+\frac{5}{8}$
$\Rightarrow\Box=\frac{2+5}{8}$
$\Box=\frac{7}{8}$

$\frac{1}{2\frac{1}{3}}+\frac{1}{1\frac{3}{4}}=\frac{1}{\frac{2\times3+1}{3}}+\frac{1}{\frac{1\times4+3}{4}}$
$=\frac{1}{\frac{7}{3}}+\frac{1}{\frac{7}{4}}$
$=\frac{3}{7}+\frac{4}{7}$
$=\frac{3+4}{7}$
$=\frac{7}{7}=1$

Vulgar fraction is a fraction expressed by numerator and denominator, and not in form of decimal.The given number is in decimal form: $0.23$ Here, the decimal point is before two digits. So, in order to obtain vulgar fraction, we need to multiply both the numerator and denominator by $100.$
$\displaystyle \frac{0.23}{1}\, =\, \frac{0.23 \times 100}{1 \times 100}\, =\, \frac{23}{100}$

$0.004 =\dfrac{0.004}{1}$ Here, we have three numbers after decimal point. So, we multiply by both numerator and denominator by 1000.
$=\dfrac{0.004\times 1000}{1\times 1000}$
$=\dfrac{4}{1000}$​
$\therefore\text{Fraction for 0.004 is } \dfrac{4}{1000}$

$\displaystyle\frac{26}{4}+\frac{14}{3} $
$= \dfrac{78+56}{12}=\dfrac{134}{12}$
$=11\dfrac{2}{12}=11\dfrac{1}{6}$

We have, improper fraction $=\frac{11}7$
Now,

$\text { 7)11(1 }$
     $ \frac{7}{4}$

$\therefore\frac{11}{7}=1\frac47$
Note: In order ot express an improper fraction as a mixed fraction, we first devide the numerator by denominator and obtain the quotient and remainder and then we write the mixed fraction as, $\text{Quotient}\ \frac{\text{Remainder}}{\text{Denominator}}$

The first decimal digit from the decimal point is the tenth, the second decimal digit from the decimal point is the hundredth and the third decimal digit from the decimal point is the thousandths digit.
Read the whole set of three decimal digits as a number, and say, "tenths ", "hundredths" and “thousandths.” $0.012$ has $0$ tenths, $1$ hundredth, and $2$ thousandths. While $0.012$ is the sum of $\frac{0}{10}$, $\frac{1}{100}$, and $\frac{2}{1000}$, it is also $\frac{12}{1000}$.

$\begin{array}{c|c}2&5,3,6,10\\\hline5&5,3,3,5\ \\\hline3&1,3,3,1\ \\\hline&1,1,1,1\ \end{array}$
$L.C.M$. of $5, 3, 6$ and $10 = (2 \times 3 \times 5) = 30$
Thus, we have:
$\frac{3}{5}=\frac{3\times6}{5\times6}=\frac{18}{30}$
$\frac{2}{3}=\frac{2\times10}{3\times10}=\frac{20}{30}$
$\frac{5}{6}=\frac{5\times5}{6\times5}=\frac{25}{30}$
$\frac{7}{10}=\frac{7\times3}{10\times3}=\frac{21}{30}$
Therefore, The smallest fraction $=\frac{18}{30}=\frac{3}{5}$

Dividing $23$ by $100$ will give the answer $0.23$

Dividing $23$ by $100$ will give the answer $0.23$

To write it as a decimal we divide the numerator from the denominator.
$9/1000 = 0.009$
So, $0.009$ is the decimal representation for $9/1000.$

$\frac{45}{60}=\frac{3}{\text{x}}$
$\Rightarrow\frac{45\div15}{60\div15}=\frac{3}{\text{x}}$
$\Rightarrow\frac{3}{4}=\frac{3}{\text{x}}$
$\text{x}=4$

$0.23 = \displaystyle\frac{23}{100}$

In an improper fraction, the numerator is greater than the denominator.
Of the given fractions, $\frac{23}{22}$ has numerator greater than the denominator.
Hence, $\frac{23}{22}$ is an improper fraction

Given, $\frac{5}{8}=\frac{20}{\text{p}}$
We know that, if two fractions $\frac{\text{a}}{\text{b}}$ and $\frac{\text{c}}{\text{d}}$ are equvalent.
Then, $\text{a}\times\text{d}=\text{b}\times\text{c}$
$\Rightarrow5\times\text{p}=8\times20$
$\Rightarrow\text{p}=\frac{8\times20}{5}$
$\Rightarrow\text{p}=\frac{160}{5}=32$
Hence, the value of $p$ is $32.$

$\cfrac { -1 }{ 2 } =-0.5$

$1008\times \dfrac{7}{8}-568\times \dfrac{3}{4}$
$126 \times 7 - 142 \times 3$
$= 882 - 426$
$= 456$

The first decimal digit from the decimal point is the tenth.
$4.4$ has $4$ on the ones, after decimal point on the tenths is $4$ tenths.
$4.4$ is the sum of $4$ and $\frac{4}{10}$or $\frac{44}{10}.$

Since it is a mixed fraction..it can be written as $\displaystyle \frac { 9 }{ 4 }$ then the reciprocal of $\displaystyle \frac { 9 }{ 4 }$ is $\displaystyle \frac { 4 }{ 9 }$

The $LCM$ of $8, 7, 6$ and $5$ is $840$, so we can convert each fraction into an equivalent fraction with denominator $840.$
$\frac{7}{8}=\frac{7}{8}\times\frac{105}{105}=\frac{735}{840}$
$\frac{6}{7}=\frac{6}{7}\times\frac{120}{120}=\frac{720}{840}$
$\frac{4}{5}=\frac{4}{5}\times\frac{168}{168}=\frac{672}{840}$
$\frac{5}{6}=\frac{5}{6}\times\frac{140}{140}=\frac{700}{840}$
When denominator is the same, the fraction with the largest numerator is the greatest.
Thus, $\frac{7}{8}$ is the greatest fraction among all.

Clearly from question denominator $= 3$ numerator $= 2$ Fraction = $\frac{2}{3}$

Converting the given decimals to like decimals, we have $15.80$ and $6.73.$
$\ \ 15.80\\ \underline{-\ 6.73\ \ }\\ \underline{\ \ \ \ \ 9.07\ \ }$
Note: Decimals having the same number of digits on the right of the decimal point are known as like decimals.

$36.2<\text{br}>\Rightarrow36+0.2=\frac{362}{10}=36\frac{2}{10}$

Proper fractions are the one whose numerator is less than the denominator
In $\frac{3}{2}​ $ and $\frac{8}{3}​,$ the numerator is greater than the denominator
So, they are not proper fractions Whereas in $\frac{2}{5}$​ and $ \frac{1}{7}​,$ the numerator is less than the denominator
Thus, they are proper fractions.Hence, $\frac{2}{5}$​ and $ \frac{1}{7} $ are proper fractional numbers.

In $\frac{6}{5}​,$ the numerator 6 is greater than the denominator $5.$
Therefore, $\frac{6}{5} $ is not a proper fraction.

$0.004\frac{0.004}{1}$ Here, we have three numbers after decimal point.
So, we multiply by both numerator and denominator by $1000.$
$=\frac{0.004\times1000}{1\times1000}$
$=\frac{4}{1000}$
$\therefore $ Fraction for $0.0040.004$ is $\frac{4}{1000}$

To get the Equivalent fraction of $\dfrac {9}{11}$ we will multiply numerator and denominator by $26.$
$\dfrac{9}{11} \times \dfrac{26}{26} =\dfrac{234}{286}$

Here, whole part of numbers $0.27, 0.082$ and $0.103$ are same and is less than $1.5.$
Now, we will compare the tenths part of $0.27, 0.082$ and $0.103.$
Tenths part of $0.27 = \frac{2}{10}$
Tenths part of $0.082 = \frac{0}{10}$ and tenths part of $0.103 =\frac{1}{10}$
Clearly, tenths part of $0.082$ is smallest.
Hence, $0.082$ is the smallest decimal.

We have, $\frac{3}{5}$
$\therefore\frac{3}{5}=\frac{3}{5}\times\frac{2}{2}=\frac{6}{10}$

To get the Equivalent fraction of $\frac{9}{11}$ we will multiply numerator and denominator by $26.$
$\frac{9}{11}\times\frac{26}{26}=\frac{234}{286}$

$\begin{array}{c|c}2&4,6,12,3\\\hline2&2,3,6,3\ \ \\\hline3&1,3,3,3\ \ \\\hline&1,1,1,1\ \ \end{array}$
$​​L.C.M.$ of $4, 6, 12$ and $3 = (2 \times 2 \times 3) = 12$
Thus, we have:
$\frac{3}{4}=\frac{3\times3}{4\times3}=\frac{9}{12}$
$\frac{5}{6}=\frac{5\times2}{6\times2}=\frac{10}{12}$
$\frac{7}{12}=\frac{7\times1}{12\times1}=\frac{7}{12}$
$\frac{2}{3}=\frac{2\times4}{3\times4}=\frac{8}{12}$
Clearly, $\frac{7}{12}$ is the smallest fraction.

$?-\frac{8}{21}=\frac{8}{21}$
$?=\frac{8}{21}+\frac{8}{21}$
$?=\frac{16}{21}$

$\Big(\frac{45}{60}=\frac{\text{3}}{\text{x}}\Big)$
Now,
$3=\frac{45}{15}$
So, we have to multiply the denominator by $15$
Therefore, $\text{x}=\frac{60}{15}$
$\text{x}=4$

$0.000000375=375\times10^{-9}=3.75\times10^{-7}$ $=\frac{375}{100}\times10^{-7}=\frac{15}{4}\times10^{-7}=3\frac{3}{4}\times10^{-7}$

We hve, mixed fraction $=5\frac47$
Converting mixed fraction into improper fraction by using that formula, mixed fraction = Improper fraction
$=\Big(\frac{\text{Whole number}\times\text{Denominator}+\text{Numerator}}{\text{Denominator}}\Big)$
$=\frac{5\times7+4}{7}=\frac{39}{7}$

To round to the nearest tenth, write down the number with a decimal point, and find the tenths place directly to the right of the decimal.
Then, to the right of the tenths place, look at the number in the hundredths place.
In $6050.287$ the number in the hundredths is $2.$
So, $6050.287$ rounded to the nearest tenths will be $6050.3$

$\frac{\text{a}}{\text{b}}=4$
$\Rightarrow\text{a}=4\text{b}$
On putting the value of a in $\frac{\text{a}+\text{b}}{\text{a}-\text{b}},$ we get:
$\frac{\text{a}+\text{b}}{\text{a}-\text{b}}=\frac{4\text{b}+\text{b}}{4\text{b}-\text{b}}=\frac{5\text{b}}{3\text{b}}$
On dividing the numerator & denominator by b, we get $\frac{5}{3}.$

Proper fractionof $5\cfrac{4}{9}​= \cfrac{9 \times 5 + 4}{9}​= \cfrac{45 + 4}{9} = \cfrac{49}{9}$​

Since, fractions with same denominators can be added by simply adding the numerators and writing the common denominator as it is.
$\frac{4}{17}+\frac{15}{17}=\frac{4+15}{17}=\frac{19}{17}$

Largest mixed number using these digits will be $11\frac{9}{7}$
Smallest mixed number will be $7\frac{9}{11}$
Their sum $=11\frac{9}{7}+7\frac{9}{11}=\frac{86}{7}+\frac{86}{11}$
$=\frac{11\times86+7\times86}{77}$
$=\frac{1548}{77}$
$=20\frac{8}{77}$

$\frac{1}{3}+\frac{1}{2}+\frac{1}{\text{x}}=4$
$\Rightarrow\frac{1}{\text{x}}=4-\frac{1}{3}-\frac{1}{2}$
$\Rightarrow\frac{1}{\text{x}}=\frac{4\times6}{1\times6}-\frac{1\times2}{3\times2}-\frac{1\times3}{2\times3}$
$\Rightarrow\frac{1}{\text{x}}=\frac{24}{6}-\frac{2}{6}-\frac{3}{6}$
$\Rightarrow\frac{1}{\text{x}}=\frac{24-2-3}{6}$
$\Rightarrow\frac{1}{\text{x}}=\frac{19}{6}$
$\text{x}=\frac{6}{19}$

Vulgar fraction is a fraction expressed by numerator and denominator, and not in form of decimal.
The given number is in decimal form: $0.23$ Here, the decimal point is before two digits.
So, in order to obtain vulgar fraction, we need to multiply both the numerator and denominator by $100.$
$\frac{0.23}{1}=\frac{0.23\times100}{1\times100}=\frac{23}{100}$

Fractions that are greater than $0$ but less than $1$ are called proper fractions.
In proper fractions, the numerator is less than the denominator.
When a fraction has a numerator that is greater than or equal to the denominator, then the fraction is an improper fraction.
An improper fraction is always $1$ or greater than $1.$
Now looking at options
$\frac{2}{3}=.666<1$
$\frac{3}{4}=.75<1$
$\frac{5}{7}=71<1$
$\frac{6}{5}=1.2>1$
So $\frac{6}{5}$ is Not a Proper fraction.

$12\dfrac16=\dfrac {12\times6+1}{6}$
$=\dfrac {73}{6}$

Two sevenths $\frac{2}{7}$

Converting the given decimals to like decimals, we have $0.070$ and $0.008.$
$\ \ \ \ 0.070\\\underline{+\ 0.008\ \ }\\\underline{\ \ \ \ 0.078\ \ }$
Note: Decimals having the same number of digits on the right of the decimal point are known as like decimals.

When the prime factorization of the denominator of a fraction has only factors of $2$ and factors of $5$, then the number is a terminating decimal.
If there are prime factors in the denominator other than $2$ or $5,$ then the decimals repeat.

Considering $\frac{3}{8}$and $\frac{5}{12}$
On cross multiplying, we get:
$3 \times 12 = 36$ and $8 \times 5 = 40$
Clearly, $36 < 40$
$\therefore\frac{3}{8}<\frac{5}{12}$

The smallest possible decimal fraction upto three decimal places $=\frac{1}{1000}=.001.$

$\frac{1}{\text{k}}=\frac{1 }{3}+\frac{1}{4}$ Multiply by $k$
$\therefore1=\frac{\text{k}}{3}+\frac{\text{k}}{4}\Rightarrow12=4\text{k}+3\text{k}$
$\Rightarrow7\text{k}=12\Rightarrow\text{k}=1\frac{5}{7}$

$\frac{1}{5}-\frac{1}{6}=\frac{4}{\text{x}}$
$\Rightarrow\frac{1\times6}{5\times6}-\frac{1\times5}{6\times5}=\frac{4}{\text{x}}$
$\Rightarrow\frac{6}{30}-\frac{5}{30}=\frac{4}{\text{x}}$
$\Rightarrow\frac{6-5}{30}=\frac{4}{\text{x}}$
$\Rightarrow\frac{1}{30}=\frac{4}{\text{x}}$
$\Rightarrow1\times\text{x}=4\times30$
$\text{x}=120$

Given, fraction $=\frac14$
In order to make the denominator as $12$, we will multiply the denominator by $3$ and we will also multiply the numerator by $3$, to make it an equivalent fraction.
$\frac14=\frac{1\times3}{4\times3}=\frac{3}{12}$
Hence, when denominator of $\frac14$ is $12,$ then its numerator will be $3.$

Improper fraction of $2\frac{3}{7}=\frac{7\times2+3}{7}=\frac{14+3}{7}=\frac{17}{7}$

$\frac { 7 }{ 11 }​$ is not an improper fraction because the numerator is smaller than the denominator.
In Improper fraction, the Numerator is always larger than the denominator.