Simplification - Study Mode
[#111] The number of pairs of natural numbers the difference of whose squares is 45 will be ?
Correct Answer
(B) 3
Explanation
Solution: Let the numbers be x and y According to question, $$eqalign{
& left( {x > y}
ight) cr
& {x^2} - {y^2} = 45 cr
& left( {x + y}
ight)left( {x - y}
ight) = 45 cr
& { ext{Make factor of 45}} cr
& { ext{15}} imes 3 cr
& ,,9 imes 5,,,,,,,,,,(3,{ ext{pairs}}) cr
& 45 imes 1 cr} $$ These pairs gives the value of x and y which satisfy the given condition.
[#112] If $$
oot 3 of {{3^n}} { ext{ = 27,}}$$ xa0 then the value of n is = ?
Correct Answer
(A) 9
Explanation
Solution: According to question, $$eqalign{
&
oot 3 of {{3^n}} { ext{ = 27}} cr
& Rightarrow {left( {{3^n}}
ight)^{frac{1}{3}}} = {left( 3
ight)^3} cr
& Rightarrow {3^{frac{n}{3}}} = {3^3} cr
& Rightarrow frac{n}{3} = 3 cr
& Rightarrow n = 9 cr} $$
[#113] The lowest temperature in the night in a city is one third more than $$frac{1}{2}$$ the highest during the day. Sum of the lowest temperature and the highest temperature is 100 degrees. Then what is the lowest temperature?
Correct Answer
(B) 40 degrees
Explanation
Solution: Let the highest temperature be x degrees Then, lowest temperature $$eqalign{
& { ext{ = }}left[ {left( {1 + frac{1}{3}}
ight)frac{x}{2}}
ight]{ ext{ degrees }} cr
& = left( {frac{4}{3} imes frac{x}{2}}
ight){ ext{ degrees}} cr
& = frac{{2x}}{3}{ ext{ degrees}} cr
& herefore x + frac{{2x}}{3} = 100 cr
& Leftrightarrow frac{{5x}}{3} = 100 cr
& Leftrightarrow x = frac{{100 imes 3}}{5} cr
& ,,,,,,,,,,,, = 60 cr
& { ext{So, lowest temperature}} cr
& { ext{ = }}left( {frac{2}{3} imes 60}
ight){ ext{degrees}} cr
& { ext{ = 40 degrees}} cr} $$
[#114] A millionaire bought a lot of hats $$frac{1}{4}$$ of which were brown. The millionaire sold $$frac{2}{3}$$ of the including $$frac{4}{5}$$ of the brown hats. What fraction of the unsold hats were brown ?
Correct Answer
(C) $$frac{3}{{20}}$$
Explanation
Solution: Let the number of hats purchase be x, $$eqalign{
& { ext{Then, number of brown hats}} cr
& { ext{ = }}frac{x}{4} cr
& { ext{Number of hats sold}} cr
& { ext{ = }}frac{{2x}}{3} cr
& { ext{Number of hats left unsold}} cr
& { ext{ = }}left( {x - frac{{2x}}{3}}
ight) = frac{x}{3} cr
& { ext{Number of brown hats sold}} cr
& { ext{ = }}frac{4}{5}{ ext{ of }}frac{x}{4} = frac{x}{5} cr
& { ext{Number of brown hats left unsold}} cr
& { ext{ = }}left( {frac{x}{4} - frac{x}{5}}
ight) = frac{x}{{20}} cr
& herefore { ext{Required fraction}} cr
& { ext{ = }}frac{{left( {frac{x}{{20}}}
ight)}}{{left( {frac{x}{3}}
ight)}} cr
& = frac{x}{{20}} imes frac{3}{x} cr
& = frac{3}{{20}} cr} $$
[#115] A body of 7300 troops is formed of 4 battalions so that $$frac{1}{2}$$ of the first, $$frac{2}{3}$$ of the second, $$frac{3}{4}$$ of the third and $$frac{4}{5}$$ of the fourth are all composed of the same number of men. How many men are there in the second battalion?
Correct Answer
(C) 1800
Explanation
Solution: Let the number of men in the 1st, 2nd, 3rd and 4th battalions be x, y, z and t respectively. Then, $$eqalign{
& frac{1}{2}x = frac{2}{3}y = frac{3}{4}z = frac{4}{5}t cr
& Rightarrow x = frac{4}{3}y, cr
& ,,,,,,,,z = frac{8}{9}y, cr
& ,,,,,,,,,t = frac{5}{6}y{ ext{ }} cr
& { ext{Now,}} cr
& x + y + z + t = 7300 cr
& Rightarrow frac{4}{3}y + y + frac{8}{9}y + frac{5}{6}y = 7300 cr
& Rightarrow frac{{24y + 18y + 16y + 15y}}{{18}} = 7300 cr
& Rightarrow 73y = 7300 imes 18 cr
& Rightarrow y = 1800 cr} $$