Decimal Fraction
Name: _____________________
Date: _____________________
Instructions: Answer all questions. Write your answers clearly in the space provided.
Solve 1599 ÷ 39.99 + $$frac{4}{5}$$ × 2449 - 120.05 = ?
If $$frac{144}{0.144}$$ = $$frac{14.4}{x},$$ xa0then the value of $$x$$ is :
999.99 + 99.99 + 9.99 = ?
Which of the following fractions is the largest ? $$frac{3}{2}$$, $$frac{7}{3}$$, $$frac{5}{4}$$, $$frac{7}{2}$$
$$left( {8.3overline 1 + 0.overline 6 + 0.00overline 2 }
ight)$$ xa0 xa0is equal to :
Solve $$frac{?}{529}$$ = $$frac{329}{?}$$
$$left( {0.34overline {67} + 0.13overline {33} }
ight)$$ xa0 is equal to :
Solve $$sqrt {197} $$ × 6.99 + 626.96 = ?
636.66 + 366.36 + 363.33 = ?
Solve : $$frac{{{{left( {36.54}
ight)}^2} - {{left( {3.46}
ight)}^2}}}{?} = 40$$
The rational number for the recurring decimal 0.125125..... is :
Which of the following fractions lies between $$frac{2}{3}$$ and $$frac{3}{5}$$ = ?
$$frac{5}{9}$$ of a number is equal to twenty five percent of second number. Second number is equal to $$frac{1}{4}$$ of third number. The value of third number is 2960. What is 30% of first number ?
Solve this, $$frac{3.5 × 1.4}{0.7}$$xa0 = ?
Express $$frac{1999}{2111}$$ in decimal :
The value of $$left( {frac{{0.943 imes 0.943 - 0.943 imes 0.057 + 0.057 imes 0.057}}{{0.943 imes 0.943 imes 0.943 + 0.057 imes 0.057 imes 0.057}}}
ight)$$ xa0 xa0 xa0 xa0 is :
[(?) 2 + (18) 2 ] ÷ 125 = 3.56
534.596 + 61.472 - 496.708 = ? + 27.271
Solve $${left( {frac{{18}}{4}}
ight)^2} imes $$ $$left( {frac{{455}}{{19}}}
ight) div $$xa0 $$left( {frac{{61}}{{799}}}
ight) = ?$$
Solve $$frac{294 ÷ 14 × 5 + 11}{?}$$ xa0xa0 = 8 2 ÷ 5 + 1.7
Solve : $$7frac{1}{2} - $$ xa0$$left[ {2frac{1}{4} ÷ left{ {1frac{1}{4} - frac{1}{2}left( {1frac{1}{2} - frac{1}{3} - frac{1}{6}}
ight)}
ight}}
ight]$$ xa0 xa0 xa0 = ?
Which part contains the fractions in ascending order ?
4.036 divided by 0.04 gives :
The place value of 9 in 0.06945 is :
The rational numbers lying between $$frac{1}{3}$$ and $$frac{3}{4}$$ are :
The value of $$frac{{3.157 imes 4126 imes 3.198}}{{63.972 imes 2835.121}}$$ xa0 xa0 is closest to :
47.7 × 12.4 × 8.6 = ?
Given, 168 × 32 = 5376 , then 5.376 ÷ 16.8 is equal to :
Which of the following has fractions in ascending order ?
The value of $$left( {frac{{0.1 imes 0.1 imes 0.1 + 0.02 imes 0.02 imes 0.02}}{{0.2 imes 0.2 imes 0.2 + 0.04 imes 0.04 imes 0.04}}}
ight)$$ xa0 xa0 xa0 is :
11.71 - 0.86 + 1.78 - 9.20 = ?
The product of 0.09 and 0.007 is :
29.92 × 2.4 + 21.28 × 4.5 = ?
$$1.overline {27} $$ xa0 in the from $$frac{p}{q}$$ is equal to :
$$frac{{5.3472 imes 324.23}}{{3.489 imes 5.42}}$$ xa0 xa0is the same as :
$$0.4overline {23} $$ xa0 is equivalent to the fraction :
4.4 × 5.8 × 11.5 - 141.27 = ?
Simplify : $$frac{{5.32 imes 56 + 5.32 imes 44}}{{{{left( {7.66}
ight)}^2} - {{left( {2.34}
ight)}^2}}}$$
Express $$0.29overline {56} $$ xa0 in the form $$frac{{ ext{p}}}{{ ext{q}}}$$ (vulgar fraction)
The value of $$left( {0.overline 2 + 0.overline 3 + 0.overline {32} }
ight)$$ xa0 is :
555.05 + 55.5 + 5.55 + 5 + 0.55 = ?
$${2.8overline {768} }$$ xa0 is equal to :
40.04 ÷ 0.4 = ? × 0.05
Solve : 48.2 × 2.5 × 2.2 + ? = 270
(78.95) 2 - (43.35) 2 = ?
Solve 2.5 × 4.8 + 7.2 × 1.5 - 1.2 × 14 = ?
10.0001 + 9.9999 - 8.9995 = ?
Solve : $$2frac{2}{9}$$ + $$4frac{1}{18}$$ - $$1frac{1}{2}$$ = ?
Which of the following is closest to zero ?
(55.25) 2 - 637.5625 = ?
Solve : $$4frac{2}{3}$$ + $$3frac{1}{2}$$ - $$1frac{2}{3}$$ = ?
One hundred th of centimetre when written in fractions of kilometres, is equal to :
(5420 + 3312 + ?) ÷ 600 = 25.93
The expression : $$frac{3}{4}$$ + $$frac{5}{36}$$ + $$frac{7}{144}$$ + ..... + $$frac{17}{5184}$$ + $$frac{19}{8100}$$ is equal to :
Solve : $$1frac{1}{8}$$ + $$1frac{6}{7}$$ + $$3frac{3}{5}$$ = ?
Given expression : (11.6 ÷ 0.8) (13.5 ÷ 2) = ?
$$frac{96.54 - 89.63}{96.54 + 89.63}$$ xa0 ÷ $$frac{965.4 - 896.3}{9.654 + 8.963}$$ xa0 = ?
$$5.5 - left[ {6.5 - left{ {3.5 div left( {6.5 - overline {5.5 - 2.5} }
ight)}
ight}}
ight]$$ xa0 xa0 xa0 is equal to :
$$left( {frac{{1.49 imes 14.9 - 0.51 imes 5.1}}{{14.9 - 5.1}}}
ight)$$ xa0 xa0 is equal to :
The value of $$frac{1}{4}$$ + $$frac{1}{4 × 5}$$ + $$frac{1}{4 × 5 × 6}$$ xa0 correct to 4 decimal places is :
(833.25 - 384.45) ÷ 24 = ?
Vishal donates blood thrice in 2 years-each time 350 ml. How many litres of blood will he donate in 6 years ?
6425 ÷ 125 × 8 = ?
The value of $$frac{5.71 × 5.71 × 5.71 - 2.79 × 2.79 × 2.79}{5.71 × 5.71 + 5.71 × 2.79 + 2.79 × 2.79}$$ xa0 xa0 xa0 is :
7777 ÷ 77 ÷ 5 = ?
$$frac{{{{left( {3.63}
ight)}^2} - {{left( {2.37}
ight)}^2}}}{{3.63 + 2.37}}$$ xa0 xa0is simplified to :
(0.75 × 4.4 × 2.4) ÷ 0.6 = ?
Solve : 1576 ÷ 45.02 + 23.99 × $$sqrt {255} $$xa0 = ?
Solve : $${left( {frac{{0.05}}{{0.25}} + frac{{0.25}}{{0.05}}}
ight)^3} =, ?$$
0.04 x 0.0162 is equal to:
$$frac{{4.2 imes 4.2 - 1.9 imes 1.9}}{{2.3 imes 6.1}}$$ xa0xa0 is equal to:
If $$frac{{144}}{{0.144}} = frac{{14.4}}{x},$$ xa0 then the value of x is:
The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. If in 2001, the price of commodity X was Rs. 4.20 and that of Y was Rs. 6.30, in which year commodity X will cost 40 paise more than the commodity Y ?
Which of the following are in descending order of their value ?
Which of the following fractions is greater than $$frac{{3}}{{4}}$$ and less than $$frac{{5}}{{6}}$$ ?
The rational number for recurring decimal 0.125125.... is:
617 + 6.017 + 0.617 + 6.0017 = ?
The value of $$frac{{489.1375 imes 0.0483 imes 1.956}}{{0.0873 imes 92.581 imes 99.749}}$$ xa0 xa0 is closest to:
0.002 x 0.5 = ?
34.95 + 240.016 + 23.98 = ?
Which of the following is equal to 3.14 x 10 6 ?
$$frac{{5 imes 1.6 - 2 imes 1.4}}{{1.3}} = ?$$
How many digits will be there to the right of the decimal point in the product of 95.75 and .02554 ?
The correct expression of $$6.overline {46},$$ in the fractional form is:
The fraction $$101frac{{27}}{{100000}}$$ xa0 in decimal for is
$$frac{{0.0203 imes 2.92}}{{0.0073 imes 14.5 imes 0.7}} = ?$$
$$3.overline {87} - 2.overline {59} = ?$$
When 52416 is divided by 312, the quotient is 168. What will be the quotient when 52.416 is divided by 0.0168 ?
The value of (1.25) 3 - 2.25 (1.25) 2 + 3.75 (0.75) 2 - (0.75) 3 is :
Solve $$frac{{21.5}}{5} + frac{{21}}{6}$$ $$ - frac{{13.5}}{{15}}$$ $$ = left{ {frac{{{{left( ?
ight)}^{frac{1}{3}}}}}{4}}
ight}$$ xa0$$ + frac{{17}}{{30}}$$
Out of the fractions $$frac{9}{31}$$, $$frac{3}{17}$$, $$frac{6}{23}$$, $$frac{4}{11}$$ and $$frac{7}{25}$$ which is the largest ?
$$0.overline {142857} div 0.overline {285714} $$ xa0 xa0 is equal to :
58.621 - 13.829 - 7.302 - 1.214 = ?
The numerator of a fraction is decreased by 25% and the denominator is increased by 250%. If the resultant fraction is $$frac{6}{5}$$, what is the original fraction ?
The arrangement of rational numbers, $$frac{- 7}{10}$$, $$frac{5}{- 8}$$, $$frac{2}{- 3}$$ xa0in ascending order is :
$$frac{5 × 1.6 - 2 × 1.4}{1.3}$$ xa0 = ?
0.3 + 3 + 3.33 + 3.3 + 3.03 + 333 = ?
(0.05 × 6.25) ÷ 2.5 = ?
If 2994 ÷ 14.5 = 172, then 29.94 ÷ 1.45 = ?
0.5 × 0.5 + 0.5 ÷ 5 is equal to :
If 1 3 + 2 3 + 3 3 + .... + 9 3 = 2025, then the value of (0.11) 3 + (0.22) 3 + .... + (0.99) 3 is close to :
The value of $$left( {frac{{8.6 imes 5.3 + 8.6 imes 4.7}}{{4.3 imes 9.7 - 4.3 imes 8.7}}}
ight)$$ xa0 xa0 is :
$$frac{3.25 × 3.20 - 3.20 × 3.05}{0.064}$$ xa0 xa0 xa0is equal to :
Solve : $$frac{17292}{33}$$xa0 ÷ 8 = ?
Solve this : $$frac{1.6 × 3.2}{0.08}$$ xa0= ?
Solve $$frac{3}{5}$$ of $$frac{4}{7}$$ of $$frac{5}{12}$$ of 1015 = ?
The value of $$frac{241.6 × 0.3814 × 6.842}{0.4618 × 38.25 × 73.65}$$ xa0 xa0 is close to :
$${ ext{Evalute}}:frac{{{{left( {2.39}
ight)}^2} - {{left( {1.61}
ight)}^2}}}{{2.39 - 1.61}}$$
What decimal of an hour is a second ?
$${ ext{The}},{ ext{value}},{ ext{of}},frac{{{{left( {0.96}
ight)}^3} - {{left( {0.1}
ight)}^3}}}{{{{left( {0.96}
ight)}^2} + 0.096 + {{left( {0.1}
ight)}^2}}},{ ext{is}}:$$
$$eqalign{
& { ext{The}},{ ext{value}},{ ext{of}} cr
& frac{{0.1 imes 0.1 imes 0.1 + 0.02 imes 0.02 imes 0.02}}{{0.2 imes 0.2 imes 0.2 + 0.04 imes 0.04 imes 0.04}}, ext{is}: cr} $$
When 0.232323..... is converted into a fraction, then the result is:
The expression (11.98 × 11.98 + 11.98 × X + 0.02 × 0.02) will be a perfect square for X equal to:
$$frac{{left( {0.1667}
ight)left( {0.8333}
ight)left( {0.3333}
ight)}}{{left( {0.2222}
ight)left( {0.6667}
ight)left( {0.1250}
ight)}}$$ xa0 xa0 is approximately equal to:
3889 + 12.952 - ? = 3854.002
32.4 × 11.5 × 8.5 = ?
The value of $$frac{{3.6 imes 0.48 imes 2.50}}{{0.12 imes 0.09 imes 0.5}}$$ xa0 xa0is :
Which of the following are in descending order of their values ?
3927 + 5526 ÷ 12.5 = ?
Which of the following numbers does not lie between $$frac{4}{5}$$ and $$frac{7}{13}$$ = ?
383 × 38 × 3.8 = ?
What is the difference between the biggest and the smallest fraction among $$frac{2}{3}$$, $$frac{3}{4}$$, $$frac{4}{5}$$ and $$frac{5}{6}$$ ?
The number 0.121212 ..... in the form $$frac{p}{q}$$ is equal to :
Find the value of the following expression upto four places of decimals. $$left[ {1 + frac{1}{{1 imes 2}} + frac{1}{{1 imes 2 imes 4}} + frac{1}{{1 imes 2 imes 4 imes 8}} + frac{1}{{1 imes 2 imes 4 imes 8 imes 16}}}
ight]$$
$$frac{{{{left( {0.013}
ight)}^3} + 0.000000343}}{{{{left( {0.013}
ight)}^2} - 0.000091 + 0.000049}} = ,?$$
$$left[ {frac{{8{{left( {3.75}
ight)}^3} + 1}}{{{{left( {7.5}
ight)}^2} - 6.5}}}
ight]$$ xa0 is equal to :
The vulgar fraction of $${ ext{0}}{ ext{.39}}overline {{ ext{39}}} $$ xa0 is ?
The value of : $$left( {frac{{0.051 imes 0.051 imes 0.051 + 0.041 imes 0.041 imes 0.041}}{{0.051 imes 0.051 - 0.051 imes 0.041 + 0.041 imes 0.041}}}
ight)$$
$$frac{10.3 × 10.3 × 10.3 + 1}{10.3 × 10.3 - 10.3 + 1}$$ xa0 xa0 is equal to :
$$frac{{{{left( {4.53 - 3.07}
ight)}^2}}}{{left( {3.07 - 2.15}
ight)left( {2.15 - 4.53}
ight)}} + , $$ xa0 xa0 $$frac{{{{left( {3.07 - 2.15}
ight)}^2}}}{{left( {2.15 - 4.53}
ight)left( {4.53 - 3.07}
ight)}} + ,, $$ xa0 xa0 $$frac{{{{left( {2.15 - 4.53}
ight)}^2}}}{{left( {4.53 - 3.07}
ight)left( {3.07 - 2.15}
ight)}}$$ xa0 xa0 is simplified to :
The value of $$left( {frac{{0.125 + 0.027}}{{0.5 imes 0.5 + 0.09 - 0.15}}}
ight)$$ xa0 xa0 is :
The fraction equivalent to $$frac{2}{5}$$% is :
The value of $$left[ {35.7 - left( {3 + frac{1}{{3 + frac{1}{3}}}}
ight) - left( {2 + frac{1}{{2 + frac{1}{2}}}}
ight)}
ight]$$ xa0 xa0 xa0 is :
The value of $$frac{{{{left( {0.06}
ight)}^2} + {{left( {0.47}
ight)}^2} + {{left( {0.079}
ight)}^2}}}{{{{left( {0.006}
ight)}^2} + {{left( {0.047}
ight)}^2} + {{left( {0.0079}
ight)}^2}}}$$ xa0 xa0 xa0 is :
Solve 3899 ÷ 11.99 - 2379 ÷ 13.97 = ?
Solve : $$1frac{1}{2}$$ + $$2frac{2}{7}$$ = $$3frac{1}{2}$$ + ?
Solve this : $$frac{0.0203 × 2.92}{0.0073 × 14.5 × 0.7}$$ xa0xa0 = ?
The value of $$frac{{{{left( {2.3}
ight)}^3} - 0.027}}{{{{left( {2.3}
ight)}^2} + 0.69 + 0.09}}$$ xa0 xa0is :
(99.75) 2 - 2250.0625 = ?
Simplify : $$frac{0.2 × 0.2 + 0.2 × 0.02}{0.044}$$
$$2frac{1.5}{5}$$ + 2$$frac{1}{6}$$ - $$1frac{3.5}{15}$$ = $$left( {frac{{{{(?)}^{frac{1}{3}}}}}{4}}
ight)$$xa0 + $$1frac{7}{30}$$
$$frac{4.41 × 0.16}{2.1 × 1.6 × 0.21}$$ xa0 is simplified to :
The value of (0.98) 3 + (0.02) 3 + 3 × 0.98 × 0.02 - 1 is :
Solve (41.99 2 - 18.04 2 ) ÷ ? = 13.11 2 - 138.99
$$frac{{.009}}{?} = .01$$
The least among the following is:
Answer Key
& left( {8.3x08ar 1 + 0.x08ar 6 + 0.00x08ar 2}
ight) cr
& = 8 + frac{{31 - 3}}{{90}} + frac{6}{9} + frac{2}{{900}} cr
& = frac{{7200 + 280 + 600 + 2}}{{900}} cr
& = frac{{8082}}{{900}} cr
& = 8frac{{882}}{{900}} cr
& = 8 + frac{{979 - 97}}{{900}} cr
& = 8.97x08ar 9 cr} $$
ight)}^2} - {{left( {3.46}
ight)}^2}}}{x} = 40$$ x = $$frac{{{{left( {36.54}
ight)}^2} - {{left( {3.46}
ight)}^2}}}{40} $$ x = $$frac{{{{left( {36.54}
ight)}^2} - {{left( {3.46}
ight)}^2}}}{{36.54 + 3.46}}$$ Let 36.54 = $$a$$ and 3.46 = $$b$$ x = $$frac{{{a^2} - {b^2}}}{{a + b}}$$ x = (a - b) x = (36.54 - 3.46) x = 33.08
& 0.125125.... cr
& = 0.overline {125} cr
& = frac{{125}}{{999}} cr} $$
& frac{2}{3} = 0.666 cr
& frac{3}{5} = 0.6 cr
& frac{2}{5} = 0.4 cr
& frac{1}{3} = 0.333 cr
& frac{1}{{15}} = 0.066 cr
& frac{{31}}{{50}} = 0.62 cr} $$ Clearly, 0.62 lies between 0.6 and 0.666 So, $$frac{31}{50}$$ lies between $$frac{2}{3}$$ and $$frac{3}{5}$$
& = frac{{{{left( {0.943}
ight)}^2} - left( {0.943 imes 0.057}
ight) + {{left( {0.057}
ight)}^2}}}{{{{left( {0.943}
ight)}^3} + {{left( {0.057}
ight)}^3}}} cr
& = frac{{{a^2} - ab + {b^2}}}{{{a^3} + {b^3}}} cr
& = frac{1}{{a + b}} cr
& = frac{1}{{0.943 + 0.057}} cr
& = 1 cr} $$
ight)}^2}}}{{125}} = 3.56$$ Then, $$eqalign{
& {x^2} + 324 = 125 imes 3.56 = 445 cr
& Rightarrow {x^2} = 121 cr
& Rightarrow x = 11 cr} $$
& {left( {frac{{18}}{4}}
ight)^2} imes left( {frac{{455}}{{19}}}
ight) div left( {frac{{61}}{{799}}}
ight) cr
& = frac{{324}}{{16}} imes frac{{455}}{{19}} imes frac{{799}}{{61}} cr
& = 6350 cr} $$
& frac{{294 div 14 imes 5 + 11}}{x} = {8^2} div 5 + 1.7 cr
& Rightarrow frac{{frac{{294}}{{14}} imes 5 + 11}}{x} = frac{{64}}{5} + 1.7 cr
& Rightarrow frac{{21 imes 5 + 11}}{x} = 12.8 + 1.7 cr
& Rightarrow frac{{105 + 11}}{x} = 12.8 + 1.7 cr
& Rightarrow frac{{116}}{x} = 14.5 cr
& Rightarrow x = frac{{116}}{{14.5}} cr
& Rightarrow x = frac{{116 imes 10}}{{145}} cr
& Rightarrow x = 8 cr} $$ Hence, the number is 8
ight)}
ight}}
ight]$$ $$ = frac{{15}}{2} - $$ xa0 $$left[ {frac{9}{4} div left{ {frac{5}{4} - frac{1}{2}left( {frac{3}{2} - frac{1}{3} - frac{1}{6}}
ight)}
ight}}
ight]$$ $$ = frac{{15}}{2} - $$ xa0 $$left[ {frac{9}{4} div left{ {frac{5}{4} - frac{1}{2}left( {frac{{9 - 2 - 1}}{6}}
ight)}
ight}}
ight]$$ $$eqalign{
& = frac{{15}}{2} - left[ {frac{9}{4} div left{ {frac{5}{4} - frac{1}{2}}
ight}}
ight] cr
& = frac{{15}}{2} - left[ {frac{9}{4} div left{ {frac{{5 - 2}}{4}}
ight}}
ight] cr
& = frac{{15}}{2} - left[ {frac{9}{4} div frac{3}{4}}
ight] cr
& = frac{{15}}{2} - left[ {frac{9}{4} imes frac{4}{3}}
ight] cr
& = frac{{15}}{2} - 3 cr
& = frac{{15 - 6}}{2} cr
& = frac{9}{2} cr
& = 4frac{1}{2} cr} $$
& frac{{11}}{{14}} = 0.785 cr
& frac{{16}}{{19}} = 0.842 cr
& frac{{19}}{{21}} = 0.904 cr} $$ Now, 0.785 < 0.842 < 0.904 So, $$frac{11}{14}$$ < $$frac{16}{19}$$ < $$frac{19}{21}$$
& = frac{{4.036}}{{0.04}} cr
& = frac{{403.6}}{4} cr
& = 100.9 cr} $$
& frac{1}{3} = 0.333 cr
& frac{3}{4} = 0.75 cr
& frac{{117}}{{300}} = 0.39 cr
& frac{{287}}{{400}} = 0.7175 cr
& frac{{95}}{{300}} = 0.316 cr
& frac{{301}}{{400}} = 0.7525 cr
& frac{{99}}{{300}} = 0.33 cr
& frac{{97}}{{300}} = 0.323 cr
& frac{{299}}{{500}} = 0.598 cr} $$ Clearly, each one of 0.39 and 0.7175 lies between 0.333 and 0.75 So, $$frac{117}{300}$$ and $$frac{287}{400}$$ lie between $$frac{1}{3}$$ and $$frac{3}{4}$$
& = frac{{3.157 imes 4126 imes 3.198}}{{63.972 imes 2835.121}} cr
& approx frac{{3.2 imes 4126 imes 3.2}}{{64 imes 2835}} cr
& = frac{{32 imes 4126 imes 32}}{{64 imes 2835}} imes frac{1}{{100}} cr
& = frac{{66016}}{{2835}} imes frac{1}{{100}} cr
& = frac{{23.28}}{{100}} cr
& = 0.23 cr
& approx 0.2 cr} $$
& = frac{{5.376}}{{16.8}} cr
& = frac{{53.76}}{{168}} cr
& = left( {frac{{5376}}{{168}} imes frac{1}{{100}}}
ight) cr
& = frac{{32}}{{100}} cr
& = 0.32 cr} $$
& = frac{{{{left( {0.1}
ight)}^3} + {{left( {0.02}
ight)}^3}}}{{{2^3}left[ {{{left( {0.1}
ight)}^3} + {{left( {0.02}
ight)}^3}}
ight]}} cr
& = frac{1}{8} cr
& = 0.125 cr} $$
& = 1.overline {27} cr
& = 1 + 0.overline {27} cr
& = 1 + frac{{27}}{{99}} cr
& = 1 + frac{3}{{11}} cr
& = frac{{11 + 3}}{{11}} cr
& = frac{{14}}{{11}} cr} $$
& = 0.4overline {23} cr
& = frac{{423 - 4}}{{990}} cr
& = frac{{419}}{{990}} cr} $$
& = frac{{5.32 imes left( {56 + 44}
ight)}}{{left( {7.66 + 2.34}
ight)left( {7.66 - 2.34}
ight)}} cr
& = frac{{5.32 imes 100}}{{10 imes 5.32}} cr
& = 10 cr} $$
& = 0.29overline {56} cr
& = frac{{2956 - 29}}{{9900}} cr
& = frac{{2927}}{{9900}} cr} $$
& = 0.overline 2 + 0.overline 3 + 0.overline {32} cr
& = left( {frac{2}{9} + frac{3}{9} + frac{{32}}{{99}}}
ight) cr
& = left( {frac{{22 + 33 + 32}}{{99}}}
ight) cr
& = frac{{87}}{{99}} cr
& = 0.overline {87} cr} $$
& = 2.8overline {768} cr
& = 2 + 0.8overline {768} cr
& = 2 + frac{{8768 - 8}}{{9990}} cr
& = 2 + frac{{8760}}{{9990}} cr
& = 2frac{{292}}{{333}} cr} $$
& = frac{{frac{1}{{100}}{ ext{ cm}}}}{{1{ ext{ km}}}} cr
& = frac{{left( {frac{1}{{100}}}
ight){ ext{ cm}}}}{{left( {1000 imes 100}
ight){ ext{ cm}}}} cr
& = frac{1}{{100 imes 1000 imes 100}} cr
& = frac{1}{{10000000}} cr
& = 0.0000001 cr} $$
ight) + left( {frac{1}{4} - frac{1}{9}}
ight) + left( {frac{1}{9} - frac{1}{{16}}}
ight)$$ xa0 xa0 xa0 $$ + ..... + $$ xa0 $$left( {frac{1}{{81}} - frac{1}{{100}}}
ight)$$ $$eqalign{
& = 1 - frac{1}{{100}} cr
& = frac{{99}}{{100}} cr
& = 0.99 cr} $$
& left( {frac{{1.49 imes 1.49 imes 10 - 0.51 imes 0.51 imes 10}}{{1.49 imes 10 - 0.51 imes 10}}}
ight) cr
& = frac{{10left[ {{{left( {1.49}
ight)}^2} - {{left( {0.51}
ight)}^2}}
ight]}}{{10left( {1.49 - 0.51}
ight)}} cr
& = left( {1.49 + 0.51}
ight) cr
& = 2 cr} $$
& = frac{1}{4} + frac{1}{{4 imes 5}} + frac{1}{{4 imes 5 imes 6}} cr
& = frac{1}{4}left( {1 + frac{1}{5} + frac{1}{{30}}}
ight) cr
& = frac{1}{4}left( {frac{{30 + 6 + 1}}{{30}}}
ight) cr
& = frac{1}{4} imes frac{{37}}{{30}} cr
& = frac{{37}}{{120}} cr
& = 0.3083 cr} $$
ight)$$ = 3.15 litres
& frac{{{{left( {5.71}
ight)}^3} - {{left( {2.79}
ight)}^3}}}{{{{left( {5.71}
ight)}^2} + 5.71 imes 2.79 + {{left( {2.79}
ight)}^2}}} cr
& = left( {frac{{{a^3} - {b^3}}}{{{a^2} + ab + {b^2}}}}
ight) cr
& = frac{{left( {a - b}
ight)left( {{a^2} + ab + {b^2}}
ight)}}{{left( {{a^2} + ab + {b^2}}
ight)}} cr
& = left( {a - b}
ight) cr
& = left( {5.71 - 2.79}
ight) cr
& = 2.92 cr} $$
ight)}}{{left( {a + b}
ight)}}$$ xa0,xa0where a = 3.63, b = 2.37 $$eqalign{
& = frac{{left( {a - b}
ight)left( {a + b}
ight)}}{{left( {a + b}
ight)}} cr
& = left( {a - b}
ight) cr
& = 3.63 - 2.37 cr
& = 1.26 cr} $$
& {left( {frac{{0.05}}{{0.25}} + frac{{0.25}}{{0.05}}}
ight)^3} cr
& = {left( {frac{5}{{25}} + frac{{25}}{5}}
ight)^3} cr
& = {left( {frac{1}{5} + 5}
ight)^3} cr
& = {left( {frac{{26}}{5}}
ight)^3} cr
& = {left( {5.2}
ight)^3} cr
& = 140.608 approx 140.6 cr} $$
& ext{Let } a = 4.2 ext{ and } b = 1.9 cr
& { ext{Given Expression}} cr
& = frac{{ {{a^2} - {b^2}} }}{{left( {a + b}
ight)left( {a - b}
ight)}} cr
& = frac{{ {{a^2} - {b^2}} }}{{ {{a^2} - {b^2}} }} cr
& = 1 cr} $$
& frac{{144}}{{0.144}} = frac{{14.4}}{x} cr
& Rightarrow frac{{144 imes 1000}}{{144}} = frac{{14.4}}{x} cr
& Rightarrow x = frac{{14.4}}{{1000}} cr
& ,,,,,,,,,,,,,, = 0.0144 cr} $$
& Rightarrow z = frac{{2.50}}{{0.25}} cr
& ,,,,,,,,,,,,, = frac{{250}}{{25}} cr
& ,,,,,,,,,,,,, = 10 cr} $$ ∴ X will cost 40 paise more than Y 10 years After 2001 i.e., 2011
& frac{1}{3} = 0.33 cr
& frac{2}{5} = 0.4 cr
& frac{3}{7} = 0.42 cr
& frac{4}{5} = 0.8 cr
& frac{5}{6} = 0.83 cr
& frac{6}{7} = 0.85 cr} $$ Clearly, 0.85 > 0.83 > 0.8 > 0.42 > 0.4 > 0.33 So, $$frac{6}{7}$$ > $$frac{5}{6}$$ > $$frac{4}{5}$$ > $$frac{3}{7}$$ > $$frac{2}{5}$$ > $$frac{1}{3}$$
& frac{3}{4} = 0.75,{kern 1pt} cr
& frac{5}{6} = 0.833,{kern 1pt} cr
& frac{1}{2} = 0.5, cr
& frac{2}{3} = 0.66, cr
& {kern 1pt} frac{4}{5} = 0.8, cr
& frac{9}{{10}} = 0.9 cr} $$ Clearly, 0.8 lies between 0.75 and 0.833 $$ herefore frac{4}{5}{ ext{lies between}}frac{3}{4}{ ext{and}}frac{5}{6}$$
& 617.00 cr
& ,,,,,,,6.017 cr
& ,,,,,,,0.617 cr
& + ,,,6.0017 cr
& - - - - - - cr
& ,,,629.6357 cr
& - - - - - - cr} $$
& frac{{489.1375 imes 0.0483 imes 1.956}}{{0.0873 imes 92.581 imes 99.749}} approx frac{{489 imes 0.05 imes 2}}{{0.09 imes 93 imes 100}} cr
& = frac{{489}}{{9 imes 93 imes 10}} cr
& = frac{{163}}{{279}} imes frac{1}{{10}} cr
& = frac{{0.58}}{{10}} cr
& = 0.058 approx 0.06 cr} $$
& ,,,,34.95 cr
& ,240.016 cr
& + 23.98 cr
& - - - - - cr
& 298.946 cr
& - - - - - cr} $$
& { ext{Given Expression}} cr
& = frac{{8 - 2.8}}{{1.3}} cr
& = frac{{5.2}}{{1.3}} cr
& = frac{{52}}{{13}} cr
& = 4 cr} $$
& 6.overline {46} cr
& = 6 + 0.overline {46} cr
& = 6 + frac{{46}}{{99}} cr
& = frac{{594 + 46}}{{99}} cr
& = frac{{640}}{{99}} cr} $$
& 101frac{{27}}{{100000}} cr
& = 101 + frac{{27}}{{100000}} cr
& = 101 + .00027 cr
& = 101.00027 cr} $$
& frac{{0.0203 imes 2.92}}{{0.0073 imes 14.5 imes 0.7}} cr
& = frac{{203 imes 292}}{{73 imes 145 imes 7}} cr
& = frac{4}{5} cr
& = 0.8 cr} $$
& 3.overline {87} - 2.overline {59} cr
& = left( {3 + 0.overline {87} }
ight) - left( {2 + 0.overline {59} }
ight) cr
& = left( {3 + frac{{87}}{{99}}}
ight) - left( {2 + frac{{59}}{{99}}}
ight) cr
& = 1 + left( {frac{{87}}{{99}} - frac{{59}}{{99}}}
ight) cr
& = 1 + frac{{28}}{{99}} cr
& = 1.overline {28} cr} $$
& { ext{Given,}} cr
& frac{{52416}}{{312}} = 168 cr
& Leftrightarrow frac{{52416}}{{168}} = 312 cr
& { ext{Now,}} cr
& frac{{52.416}}{{0.0168}} cr
& = frac{{524160}}{{168}} cr
& = frac{{52416}}{{168}} imes 10 cr
& = 312 imes 10 cr
& = 3120 cr} $$
ight)^3}$$ = $$frac{1}{8}$$ [∵ (a - b) 3 = a 3 - b 3 - 3a 2 b + 3ab 2 ]
& frac{{21.5}}{5} + frac{{21}}{6} - frac{{13.5}}{{15}} = frac{{{{left( x
ight)}^{frac{1}{3}}}}}{4} + frac{{17}}{{30}} cr
& frac{{21.5}}{5} + frac{{21}}{6} - frac{{13.5}}{{15}} - frac{{17}}{{30}} = frac{{{{left( x
ight)}^{frac{1}{3}}}}}{4} cr} $$ L.C.M of 5, 6, 15 and 30 is 30 $$eqalign{
& frac{{129 + 105 - 27 - 17}}{{30}} = frac{{{{left( x
ight)}^{frac{1}{3}}}}}{4} cr
&
oot 3 of x = frac{{190 imes 4}}{{30}} cr
&
oot 3 of x = 25.33 approx 25 cr
& x = {25^3} cr
& x = 15625 cr} $$ Hence, the numbers 15625
& 0.overline {142857} div 0.overline {285714} cr
& = frac{{142857}}{{999999}} div frac{{285714}}{{999999}} cr
& = left( {frac{{142857}}{{999999}} imes frac{{999999}}{{285714}}}
ight) cr
& = frac{1}{2} cr} $$
& Leftrightarrow frac{{a - a imes frac{{25}}{{100}}}}{{b + b imes frac{{250}}{{100}}}} = frac{6}{5} cr
& Rightarrow frac{{0.75a}}{{3.50b}} = frac{6}{5} cr
& Rightarrow frac{a}{b} = frac{6}{5} imes frac{{3.50}}{{0.75}} cr
& Rightarrow frac{a}{b} = frac{{6 imes 350 imes 100}}{{5 imes 75 imes 100}} cr
& Rightarrow frac{a}{b} = frac{{28}}{5} cr} $$
& {left( {0.11}
ight)^3} + {left( {0.22}
ight)^3} + .... + {left( {0.99}
ight)^3} cr
& = {left( {0.11}
ight)^3}left( {{1^3} + {2^3} + .... + {9^3}}
ight) cr
& = 0.001331 imes 2025 cr
& = 2.695275 approx 2.695 cr} $$
& = frac{{8.6 imes left( {5.3 + 4.7}
ight)}}{{4.3 imes left( {9.7 - 8.7}
ight)}} cr
& = frac{{8.6 imes 10}}{{4.3 imes 1}} cr
& = 20 cr} $$
& ext{Let } a = 2.39 ext{ and } b = 1.61 cr
& { ext{Given Expression}} cr
& = frac{{{a^2} - {b^2}}}{{a - b}} cr
& = frac{{left( {a + b}
ight)left( {a - b}
ight)}}{{ {a - b} }} cr
& = {a + b} cr
& = {2.39 + 1.61} cr
& = 4 cr} $$
& { ext{Required decimal}} cr
& = frac{1}{{60 imes 60}} cr
& = frac{1}{{3600}} cr
& = .00027 cr} $$
& { ext{Given}},{ ext{expression}} cr
& = frac{{{{left( {0.96}
ight)}^3} - {{left( {0.1}
ight)}^3}}}{{{{left( {0.96}
ight)}^2} + left( {0.96 imes 0.1}
ight) + {{left( {0.1}
ight)}^2}}} cr
& = {frac{{{a^3} - {b^3}}}{{{a^2} + ab + {b^2}}}} cr
& = frac{left(a-b
ight) left(a^2 + ab + b^2
ight)}{left(a^2 + ab + b^2
ight)} cr
& = {a - b} cr
& = {0.96 - 0.1} cr
& = 0.86 cr} $$
& { ext{Give}},{ ext{expression}} cr
& = frac{{{{left( {0.1}
ight)}^3} + {{left( {0.02}
ight)}^3}}}{{{2^3}left[ {{{left( {0.1}
ight)}^3} + {{left( {0.02}
ight)}^3}}
ight]}} cr
& = frac{1}{8} cr
& = 0.125 cr} $$
& { ext{Given}},{ ext{expression}} cr
& = frac{{ {0.3333} }}{{ {0.2222} }} imes frac{{left( {0.1667}
ight)left( {0.8333}
ight)}}{{left( {0.6667}
ight)left( {0.1250}
ight)}} cr
& = frac{{3333}}{{2222}} imes frac{{frac{1}{6} imes frac{5}{6}}}{{frac{2}{3} imes frac{{125}}{{1000}}}} { ext{ [where }} {0.3333} = {frac{1}{6}}, {0.8333} = {frac{5}{6}} { ext{ and }} {0.6667} = {frac{2}{3}} { ext{ ] }} cr
& = {frac{3}{2} imes frac{1}{6} imes frac{5}{6} imes frac{3}{2} imes 8} cr
& = frac{5}{2} cr
& = 2.50 cr} $$
& = frac{{3.6 imes 0.48 imes 2.50}}{{0.12 imes 0.09 imes 0.5}} cr
& = frac{{36 imes 48 imes 250}}{{12 imes 9 imes 5}} cr
& = 800 cr} $$
& = 3927 + frac{{5526}}{{12.5}} cr
& = 3927 + frac{{55260}}{{125}} cr} $$ = 3927 + 442.08 = 4369.08
ight) = frac{1}{6}$$
& 0.121212...... cr
& = 0.overline {12} cr
& = frac{{12}}{{99}} cr
& = frac{4}{{33}} cr} $$
& left[ {1 + frac{1}{{1 imes 2}} + frac{1}{{1 imes 2 imes 4}} + frac{1}{{1 imes 2 imes 4 imes 8}} + frac{1}{{1 imes 2 imes 4 imes 8 imes 16}}}
ight] cr
& = frac{{2 imes 4 imes 8 imes 16 + 4 imes 8 imes 16 + 8 imes 16 + 16 + 1}}{{2 imes 4 imes 8 imes 16}} cr
& = frac{{1024 + 512 + 128 + 16 + 1}}{{1024}} cr
& = frac{{1681}}{{1024}} cr
& = 1.6416 cr} $$
& frac{{{{left( {0.013}
ight)}^3} + 0.000000343}}{{{{left( {0.013}
ight)}^2} - 0.000091 + 0.000049}} cr
& a = 0.013 ext{ and } b = 0.007 cr
& = left( {frac{{{a^3} + {b^3}}}{{{a^2} - ab + {b^2}}}}
ight) cr
& = a + b cr
& = 0.013 + 0.007 cr
& = 0.020 cr} $$
& left[ {frac{{8{{left( {3.75}
ight)}^3} + 1}}{{{{left( {7.5}
ight)}^2} - 6.5}}}
ight] cr
& = frac{{{{left( {2 imes 3.75}
ight)}^3} + {1^3}}}{{{{left( {7.5}
ight)}^2} - left( {7.5 imes 1}
ight) + {1^2}}} cr
& = frac{{{{left( {7.5}
ight)}^3} + {1^3}}}{{{{left( {7.5}
ight)}^2} - left( {7.5 imes 1}
ight) + {1^2}}} cr
& = left( {frac{{{a^3} + {b^3}}}{{{a^2} - ab + {b^2}}}}
ight) cr
& = frac{{left( {a + b}
ight)left( {{a^2} - ab + {b^2}}
ight)}}{{left( {{a^2} - ab + {b^2}}
ight)}} cr
& = left( {a + b}
ight) cr
& = left( {7.5 + 1}
ight) cr
& = 8.5 cr} $$
ight)}^3} + {{left( {0.041}
ight)}^3}}}{{{{left( {0.051}
ight)}^2} - left( {0.051 imes 0.041}
ight) + {{left( {0.041}
ight)}^2}}}$$ Let 0.051 = $$a$$ and 0.041 = $$b$$ $$eqalign{
& = left( {frac{{{a^3} + {b^3}}}{{{a^2} - ab + {b^2}}}}
ight) cr
& = (a + b) cr
& = left( {0.051 + 0.041}
ight) cr
& = 0.092 cr} $$
& = frac{{{{left( {10.3}
ight)}^3} + {1^3}}}{{{{left( {10.3}
ight)}^2} - left( {10.3 imes 1}
ight) + {1^2}}} cr
& = left( {frac{{{a^3} + {b^3}}}{{{a^2} - ab + {b^2}}}}
ight) cr
& = (a + b) cr
& = left( {10.3 + 1}
ight) cr
& = 11.3 cr} $$
ight)}^3} + {{left( {3.07 - 2.15}
ight)}^3} + {{left( {2.15 - 4.53}
ight)}^3}}}{{left( {4.53 - 3.07}
ight)left( {3.07 - 2.15}
ight)left( {2.15 - 4.53}
ight)}}$$ Let (4.53 - 3.07) = $$a$$, (3.07 - 2.15) = $$b$$ and (2.15 - 4.53) = $$c$$ $$eqalign{
& = frac{{{a^3} + {b^3} + {c^3}}}{{abc}} cr
& = frac{{3abc}}{{abc}} cr
& = 3 cr} $$ [∵ If a + b + c = 0, a 3 + b 3 + c 3 = 3abc]
& frac{{0.125 + 0.027}}{{0.5 imes 0.5 + 0.09 - 0.15}} cr
& = frac{{{{left( {0.5}
ight)}^3} + {{left( {0.3}
ight)}^3}}}{{{{left( {0.5}
ight)}^2} + {{left( {0.3}
ight)}^2} - left( {0.5 imes 0.3}
ight)}} cr
& = left( {frac{{{a^3} + {b^3}}}{{{a^2} + {b^2} - ab}}}
ight) cr
& = frac{{left( {a + b}
ight)left( {{a^2} - ab + {b^2}}
ight)}}{{left( {{a^2} - ab + {b^2}}
ight)}} cr
& = (a + b) cr
& = left( {0.5 + 0.3}
ight) cr
& = 0.8 cr} $$
& = 35.7 - left( {3 + frac{1}{{frac{{10}}{3}}}}
ight) - left( {2 + frac{1}{{frac{5}{2}}}}
ight) cr
& = 35.7 - left( {3 + frac{3}{{10}}}
ight) - left( {2 + frac{2}{5}}
ight) cr
& = 35.7 - frac{{33}}{{10}} - frac{{12}}{5} cr
& = 35.7 - left( {frac{{33}}{{10}} + frac{{12}}{5}}
ight) cr
& = 35.7 - frac{{57}}{{10}} cr
& = 35.7 - 5.7 cr
& = 30 cr} $$
ight)}^2} + {{left( {frac{b}{{10}}}
ight)}^2} + {{left( {frac{c}{{10}}}
ight)}^2}}}$$ Where a = 0.06, b = 0.47 and c = 0.079 $$eqalign{
& = frac{{100left( {{a^2} + {b^2} + {c^2}}
ight)}}{{left( {{a^2} + {b^2} + {c^2}}
ight)}} cr
& = 100 cr} $$
& = frac{{{{left( {2.3}
ight)}^3} - 0.027}}{{{{left( {2.3}
ight)}^2} + 0.69 + 0.09}} cr
& = frac{{{{left( {2.3}
ight)}^3} - {{left( {0.03}
ight)}^3}}}{{{{left( {2.3}
ight)}^2} + left( {2.3 imes 0.3}
ight) + {{left( {0.3}
ight)}^2}}} cr
& = left[ {frac{{{a^3} - {b^3}}}{{{a^2} + ab + {b^2}}}}
ight] cr
& = left( {a - b}
ight) cr
& = left( {2.3 - 0.3}
ight) cr
& = 2 cr} $$
& 2frac{{1.5}}{5} + 2frac{1}{6} - 1frac{{3.5}}{{15}} = frac{{{x^{frac{1}{3}}}}}{4} + 1frac{7}{{30}} cr
& Rightarrow frac{{11.5}}{5} + frac{{13}}{6} - frac{{18.5}}{{15}} = frac{{{x^{frac{1}{3}}}}}{4} + frac{{37}}{{30}} cr} $$ ⇒ L.C.M. of 5, 6 and 15 is 30 $$eqalign{
& Rightarrow frac{{69 + 65 - 37}}{{30}} = frac{{{x^{frac{1}{3}}}}}{4} + frac{{37}}{{30}} cr
& Rightarrow frac{{97}}{{30}} = frac{{{x^{frac{1}{3}}}}}{4} + frac{{37}}{{30}} cr
& Rightarrow frac{{{x^{frac{1}{3}}}}}{4} = frac{{97}}{{30}} - frac{{37}}{{30}} cr
& Rightarrow {x^{frac{1}{3}}} = frac{{60}}{{30}} imes 4 cr
& Rightarrow {x^{frac{1}{3}}} = 8 cr
& Rightarrow x = {left( 8
ight)^3} cr
& Rightarrow x = 512 cr} $$ Hence, the number is 512
& { ext{Let}},frac{{.009}}{x} = .01
cr
& { ext{Then}},x = frac{{.009}}{{.01}} cr
& ,,,,,,,,,,,,,,,,,,,, = frac{{.9}}{1} cr
& ,,,,,,,,,,,,,,,,,,,, = .9 cr} $$
& 1 div 0.2 = frac{1}{{0.2}} = frac{{10}}{2} = 5
cr
& 0.overline 2 = 0.222...
cr
& {left( {0.2}
ight)^2} = 0.04 cr
& 0.04 < 0.2 < 0.22.... < 5 cr
& { ext{Since}},0.04,{ ext{is}},{ ext{the}},{ ext{least,}},{ ext{so}},{left( {0.2}
ight)^2},{ ext{is}},{ ext{the}},{ ext{least}}. cr} $$