Simplification - Study Mode

[#61] Two fractions are such that their product is -4 and sum is $$frac{{ - 32}}{{15}}.$$ xa0Find the two fractions.
Correct Answer

(B) $$frac{6}{5},,frac{{ - 10}}{3}$$

Explanation

Solution: $$eqalign{
& { ext{Let the fractions }}a{ ext{ and }}b cr
& ab = - 4 cr
& a + b = frac{{ - 32}}{{15}}{ ext{ }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}left( 1
ight) cr
& {left( {a + b}
ight)^2} = {left( {a - b}
ight)^2} + 4ab cr
& {left( {frac{{ - 32}}{{15}}}
ight)^2} = {left( {a - b}
ight)^2} + 4left( { - 4}
ight) cr
& {left( {a - b}
ight)^2} = frac{{1024}}{{225}} + 16 cr
& a - b = sqrt {frac{{4624}}{{225}}} cr
& a - b = frac{{68}}{{15}}{ ext{ }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}left( 2
ight) cr
& { ext{Equate equation }}left( 1
ight){ ext{ and }}left( 2
ight){ ext{,}},{ ext{we get}} cr
& a = frac{{18}}{{15}} = frac{6}{5} cr
& b = frac{{ - 100}}{{30}} = frac{{ - 10}}{3} cr} $$

[#62] Simplify the expression, $$25 - left[ {16 - left{ {14 - left( {18 - overline {8 + 3} }
ight)}
ight}}
ight]$$
Correct Answer

(D) 16

Explanation

Solution: $$eqalign{
& 25 - left[ {16 - left{ {14 - left( {18 - overline {8 + 3} }
ight)}
ight}}
ight] cr
& = 25 - left[ {16 - left{ {14 - left( {18 - 8 - 3}
ight)}
ight}}
ight] cr
& = 25 - left[ {16 - left{ {14 - left( {18 - 11}
ight)}
ight}}
ight] cr
& = 25 - left[ {16 - left{ {14 - 7}
ight}}
ight] cr
& = 25 - left[ {16 - 7}
ight] cr
& = 25 - 9 cr
& = 16{ ext{ Answer}} cr} $$

[#63] Find the value of $$a$$ in the following equation. (Given: $$a$$ < 10.) $$frac{{left( {187 div 17 imes a - 3 imes 3}
ight)}}{{left( {{8^2} - 9 imes 7 + {a^2}}
ight)}} = 1$$
Correct Answer

(B) 1

Explanation

Solution: $$eqalign{
& frac{{left( {187 div 17 imes a - 3 imes 3}
ight)}}{{left( {{8^2} - 9 imes 7 + {a^2}}
ight)}} = 1 cr
& 11a - 9 = 64 - 63 + {a^2} cr
& 11a - 9 = 1 + {a^2} cr
& {a^2} - 11a + 10 = 0 cr
& {a^2} - left( {10 + 1}
ight)a + 10 = 0 cr
& {a^2} - 10a - a + 10 = 0 cr
& aleft( {a - 10}
ight) - 1left( {a - 10}
ight) = 0 cr
& left( {a - 10}
ight)left( {a - 1}
ight) = 0 cr
& a = 1,,10{ ext{ Answer}} cr} $$

[#64] The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:
Correct Answer

(C) 5.2

Explanation

Solution: 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 = 15.2 + 2 × 2 - 3.5 × 9 = 15.2 + 4 - 14 = 5.2

[#65] The value of $$frac{{27 imes {{left( {0.25}
ight)}^3} + 125{{left( {0.05}
ight)}^3}}}{{{{left( {0.75}
ight)}^2} - 0.25 imes 0.5}}$$ xa0 xa0 is:
Correct Answer

(A) 1

Explanation

Solution: $$eqalign{
& frac{{27 imes {{left( {0.25}
ight)}^3} + 125{{left( {0.05}
ight)}^3}}}{{{{left( {0.75}
ight)}^2} - 0.25 imes 0.5}} cr
& = frac{{frac{{27 imes 25 imes 25 imes 25}}{{1000000}} + frac{{125 imes 5 imes 5 imes 5}}{{1000000}}}}{{frac{{75 imes 75}}{{10000}} - frac{{25 imes 5}}{{1000}}}} cr
& = frac{{frac{{27 imes 25 imes 25 imes 25}}{{1000}} + frac{{125 imes 5 imes 5 imes 5}}{{1000}}}}{{frac{{75 imes 75 - 25 imes 5 imes 10}}{{10}}}} cr
& = frac{{frac{{27}}{8} + frac{1}{8}}}{{frac{{45 - 10}}{{10}}}} cr
& = frac{{280}}{{8 imes 35}} cr
& = 1 cr} $$