Simplification - Study Mode

[#201] If $$frac{1}{3} + frac{1}{2} + frac{1}{x}{ ext{ = 4,}}$$ xa0 xa0 then x = ?
Correct Answer

(B) $$frac{6}{{19}}$$

Explanation

Solution: $$eqalign{
& frac{1}{x} = 4 - left( {frac{1}{3} + frac{1}{2}}
ight) cr
& ,,,,,, = 4 - {frac{{2 + 3}}{6}} cr
& ,,,,,, = 4 - frac{5}{6} cr
& ,,,,,, = frac{{24 - 5}}{6} cr
& ,,,,,, = frac{{19}}{6} cr
& Rightarrow x = frac{6}{{19}} cr} $$

[#202] The value of $$frac{{sqrt {80} - sqrt {112} }}{{sqrt {45} - sqrt {63} }} = ?$$
Correct Answer

(C) $$1frac{1}{3}$$

Explanation

Solution: $$eqalign{
& { ext{According to the question,}} cr
& frac{{sqrt {80} - sqrt {112} }}{{sqrt {45} - sqrt {63} }} cr
& Rightarrow frac{{4sqrt 5 - 4sqrt 7 }}{{3sqrt 5 - 3sqrt 7 }} cr
& Rightarrow frac{{4left( {sqrt 5 - sqrt 7 }
ight)}}{{3left( {sqrt 5 - sqrt 7 }
ight)}} cr
& Rightarrow frac{4}{3} cr
& Rightarrow 1frac{1}{3} cr} $$

[#203] If the sum of two numbers is 22 and the sum of their squares is 404, then the product of two numbers is = ?
Correct Answer

(A) 40

Explanation

Solution: $$eqalign{
& { ext{According to the question}} cr
& x + y = 22,......(i) cr
& {x^2} + {y^2} = 404,......(ii){ ext{ }} cr
& herefore {left( {x + y}
ight)^2} = {x^2} + {y^2} + 2xy cr
& Rightarrow {left( {22}
ight)^2} = 404 + 2xy cr
& Rightarrow 484 = 404 + 2xy cr
& Rightarrow 2xy = 80 cr
& Rightarrow xy = 40 cr} $$

[#204] Simplify : $${ ext{8}}frac{1}{2} - left[ {3frac{1}{4} + left{ {1frac{1}{4} - frac{1}{2}left( {1frac{1}{2} - frac{1}{3} - frac{1}{6}}
ight)}
ight}}
ight]$$
Correct Answer

(A) $$4frac{1}{2}$$

Explanation

Solution: $$eqalign{
& { ext{According to the question,}} cr
& { ext{8}}frac{1}{2} - left[ {3frac{1}{4} + left{ {1frac{1}{4} - frac{1}{2}left( {1frac{1}{2} - frac{1}{3} - frac{1}{6}}
ight)}
ight}}
ight] cr
& Rightarrow frac{{17}}{2} - left[ {frac{{13}}{4} + left{ {frac{5}{4} - frac{1}{2}left( {frac{3}{2} - frac{1}{3} - frac{1}{6}}
ight)}
ight}}
ight] cr
& Rightarrow frac{{17}}{2} - left[ {frac{{13}}{4} + left{ {frac{5}{4} - frac{1}{2}left( {frac{{9 - 2 - 1}}{6}}
ight)}
ight}}
ight] cr
& Rightarrow frac{{17}}{2} - left[ {frac{{13}}{4} + left{ {frac{5}{4} - frac{1}{2} imes 1}
ight}}
ight] cr
& Rightarrow frac{{17}}{2} - left[ {frac{{13}}{4} + left{ {frac{{5 - 2}}{4}}
ight}}
ight] cr
& Rightarrow frac{{17}}{2} - left[ {frac{{13}}{4} + frac{3}{4}}
ight] cr
& Rightarrow frac{{17}}{2} - frac{{16}}{4} cr
& Rightarrow frac{{34 - 16}}{4} cr
& Rightarrow frac{9}{2} cr
& Rightarrow 4frac{1}{2} cr} $$

[#205] The value of $$frac{{25 - 5left[ {2 + 3left{ {2 - 2left( {5 - 3}
ight) + 5}
ight} - 10}
ight]}}{4} = ?$$
Correct Answer

(A) 5

Explanation

Solution: According to the question, $$eqalign{
& frac{{25 - 5left[ {2 + 3left{ {2 - 2left( {5 - 3}
ight) + 5}
ight} - 10}
ight]}}{4} cr
& = frac{{25 - 5left[ {2 + 3left{ {2 - 2 * 2 + 5}
ight} - 10}
ight]}}{4} cr
& = frac{{25 - 5left[ {2 + 3left{ {2 - 4 + 5}
ight} - 10}
ight]}}{4} cr
& = frac{{25 - 5left[ {2 + 3 * 3 - 10}
ight]}}{4} cr
& = frac{{25 - 5left[ {11 - 10}
ight]}}{4} cr
& = 25 - frac{5}{4} cr
& = frac{{100 - 5}}{4} cr
& = frac{{95}}{4} cr
& = 23.75 cr} $$