Algebra - Study Mode

[#251] If $$p imes q = p + q + frac{p}{q}{ ext{,}}$$ xa0xa0 then the value of 8 × 2 is?
Correct Answer

(C) 14

Explanation

Solution: $$eqalign{
& { ext{8}} imes { ext{2}} cr
& = 8 + 2 + frac{8}{2} cr
& = 10 + 4 cr
& = 14 cr} $$

[#252] The value of $$left( {{ ext{1 + }}frac{1}{x}}
ight)$$ $$left( {{ ext{1 + }}frac{1}{{x + 1}}}
ight)$$xa0 $$left( {{ ext{1 + }}frac{1}{{x + 2}}}
ight)$$xa0 $$left( {{ ext{1 + }}frac{1}{{x + 3}}}
ight)$$ xa0 is?
Correct Answer

(D) $$frac{{x + 4}}{x}$$

Explanation

Solution: $$left( {{ ext{1 + }}frac{1}{x}}
ight)$$ $$left( {{ ext{1 + }}frac{1}{{x + 1}}}
ight)$$xa0 $$left( {{ ext{1 + }}frac{1}{{x + 2}}}
ight)$$xa0 $$left( {{ ext{1 + }}frac{1}{{x + 3}}}
ight)$$ Taking L.C.M of each term $$ Rightarrow left( {frac{{x + 1}}{x}}
ight)$$ $$left( {frac{{x + 1 + 1}}{{x + 1}}}
ight)$$ xa0$$left( {frac{{x + 2 + 1}}{{x + 2}}}
ight)$$ xa0$$left( {frac{{x + 3 + 1}}{{x + 3}}}
ight)$$ $$eqalign{
& Rightarrow frac{1}{x} imes left( {x + 4}
ight) cr
& Rightarrow frac{{x + 4}}{x} cr} $$

[#253] If $$frac{a}{b}{ ext{ = }}frac{2}{3}$$ xa0 and $$frac{b}{c}{ ext{ = }}frac{4}{5}{ ext{,}}$$ xa0 then the ration $$frac{{a + b}}{{b + c}}$$ xa0 equal to?
Correct Answer

(A) $$frac{{20}}{{27}}$$

Explanation

Solution: $$eqalign{
& frac{a}{b}{ ext{ = }}frac{2}{3}{ ext{ and }}frac{b}{c}{ ext{ = }}frac{4}{5},,left( {{ ext{Given}}}
ight) cr
& or,frac{c}{b} = frac{5}{4} cr
& frac{{a + b}}{{b + c}} cr
& = frac{{bleft( {frac{a}{b} + 1}
ight)}}{{bleft( {frac{c}{b} + 1}
ight)}} cr
& = frac{{frac{a}{b} + 1}}{{frac{c}{b} + 1}} cr
& = frac{{left( {frac{2}{3} + 1}
ight)}}{{left( {frac{5}{4} + 1}
ight)}} cr
& = frac{{frac{2 + 3}{3}}}{{frac{{5 + 4}}{4}}} cr
& = frac{{5 imes 4}}{{3 imes 9}} cr
& = frac{{20}}{{27}} cr
& herefore frac{{a + b}}{{b + c}} = frac{{20}}{{27}} cr
& {x08f{Alternate:}} cr
& a{ ext{ }},,,{ ext{ }},{ ext{ }},,,,{ ext{ }}b{ ext{ }},,,,{ ext{ }},,,,{ ext{ }},,,{ ext{ }}c cr
& {2_{ imes left( 4
ight)}},,,,,{ ext{ }}{3_{ imes left( 4
ight)}} cr
& ,{ ext{ }},{ ext{ }},,,,,,,,,,,,,{4_{ imes left( 3
ight)}}{ ext{ }},,,,,,{5_{ imes left( 3
ight)}} cr
& overline {underline {8{ ext{ }},,,,,,,,,,,{ ext{ }}12{ ext{ }},,,,,,,,,,{ ext{ }}15,,,,} } cr
& herefore frac{{a + b}}{{b + c}} = frac{{8 + 12}}{{12 + 15}} cr
& ,,,,,,,,,,,,,,,,,,,,, = frac{{20}}{{27}} cr} $$

[#254] If $$frac{{2a + b}}{{a + 4b}} = 3{ ext{,}}$$ xa0 then find the value of $$frac{{a + b}}{{a + 2b}} = ?$$
Correct Answer

(C) $$frac{{10}}{9}$$

Explanation

Solution: $$eqalign{
& frac{{2a + b}}{{a + 4b}} = 3{ ext{ }}left( {{ ext{Given}}}
ight) cr
& Rightarrow 2a + b = 3left( {a + 4b}
ight) cr
& Rightarrow 2a + b = 3a + 12b cr
& Rightarrow - a = 11b cr
& Rightarrow a = - 11b cr
& herefore frac{{a + b}}{{a + 2b}} cr
& Rightarrow frac{{ - 11b + b}}{{ - 11b + 2b}} cr
& Rightarrow frac{{ - 10b}}{{ - 9b}} cr
& Rightarrow frac{{10}}{9} cr} $$

[#255] If a * b = a + b + ab, then 3 * 4 - 2 * 3 is equal to?
Correct Answer

(B) 8

Explanation

Solution: $$eqalign{
& a*b = a + b + ab cr
& 3*4 cr
& = 3 + 4 + 3 imes 4 cr
& = 19 cr
& 2*3 cr
& = 2 + 3 + 2 imes 3 cr
& = 11 cr
& herefore 3*4 - 2*3{ ext{ }} cr
& = 19 - 11 cr
& = 8 cr} $$