Simplification - Study Mode
[#166] If 56 × 75 × 60 × 84 × 210 = 2 p × 3 q × 5 r × 7 s , then what is the value of $$left[ {frac{{left( {{ ext{p}} + { ext{q}}}
ight)}}{{ ext{s}}}}
ight] + { ext{r}}?$$
Correct Answer
(B) 8
Explanation
Solution: 56 × 75 × 60 × 84 × 210 = 2 p × 3 q × 5 r × 7 s 7 × 2 3 × 5 2 × 3 × 2 2 × 3 × 5 × 2 2 × 3 × 7 × 7 × 3 × 2 × 5 = 2 p × 3 q × 5 r × 7 s 2 8 × 3 4 × 5 4 × 7 3 = 2 p × 3 q × 5 r × 7 s ∴ p = 8 q = 4 r = 4 s = 3 $$eqalign{
& herefore left[ {frac{{left( {{ ext{p}} + { ext{q}}}
ight)}}{{ ext{s}}}}
ight] + { ext{r}} cr
& = left[ {frac{{left( {8 + 4}
ight)}}{3}}
ight] + 4 cr
& = frac{{12}}{3} + 4 cr
& = 4 + 4 cr
& = 8 cr} $$
[#167] Solve the following $$frac{{24 div frac{3}{8}{ ext{of}}left( {8 + 2 imes overline {7 - 3} }
ight) + left[ {frac{2}{{11}} div frac{4}{{55}} - left{ {frac{5}{8} + frac{6}{{16}}}
ight}}
ight]}}{{32 div overline {15 - 7} + 75 div left( {6 + 15 div 3 + 4}
ight)}}$$
Correct Answer
(C) $$frac{{11}}{{18}}$$
Explanation
Solution: $$eqalign{
& frac{{24 div frac{3}{8}{ ext{of}}left( {8 + 2 imes overline {7 - 3} }
ight) + left[ {frac{2}{{11}} div frac{4}{{55}} - left{ {frac{5}{8} + frac{6}{{16}}}
ight}}
ight]}}{{32 div overline {15 - 7} + 75 div left( {6 + 15 div 3 + 4}
ight)}} cr
& = frac{{left[ {24 div frac{3}{8}{ ext{of}}left( {8 + 2 imes 4}
ight) + left( {frac{2}{{11}} imes frac{{55}}{4} - left{ {frac{{16}}{{16}}}
ight}}
ight)}
ight]}}{{32 div 8 + 75 div left( {6 + 5 + 4}
ight)}} cr
& = frac{{left[ {24 div 6 + left( {frac{5}{2} - 1}
ight)}
ight]}}{{left[ {4 + 75 div 15}
ight]}} cr
& = frac{{left[ {4 + frac{3}{2}}
ight]}}{{left[ {4 + 5}
ight]}} cr
& = frac{{left[ {frac{{11}}{2}}
ight]}}{9} cr
& = frac{{11}}{{18}} cr} $$
[#168] If $$frac{{45}}{{53}} = frac{1}{{a + frac{1}{{b + frac{1}{{c - frac{2}{5}}}}}}},$$ xa0 xa0where a, b and c are positive integers, then what is the value of (4a - b + 3c)?
Correct Answer
(C) 5
Explanation
Solution: $$eqalign{
& frac{{45}}{{53}} = frac{1}{{a + frac{1}{{b + frac{1}{{c - frac{2}{5}}}}}}} cr
& frac{{53}}{{45}} = 1 + frac{8}{{45}},,frac{{45}}{8} = 5 + frac{5}{8},,frac{8}{5} = 2 - frac{2}{5} cr
& a = 1 cr
& b = 5 cr
& c = 2 cr
& 4a - b + 3c cr
& = 4 imes 1 - 5 + 3 imes 2 cr
& = 4 - 5 + 6 cr
& = 5 cr} $$
[#169] The value of $$left( {1frac{1}{3} div 2frac{6}{7}{ ext{ of }}5frac{3}{5}}
ight) div left( {6frac{2}{5} div 4frac{1}{2}{ ext{ of}},5frac{1}{3}}
ight) imes left( {frac{3}{4} imes 2frac{2}{3} div frac{5}{9}{ ext{ of }}1frac{1}{5}}
ight) = 1 + { ext{k}},$$ xa0 xa0 xa0 xa0 xa0 xa0xa0where k lies between:
Correct Answer
(A) -0.07 and -0.06
Explanation
Solution: $$eqalign{
& left( {1frac{1}{3} div 2frac{6}{7}{ ext{ of }}5frac{3}{5}}
ight) div left( {6frac{2}{5} div 4frac{1}{2}{ ext{ of}},5frac{1}{3}}
ight) imes left( {frac{3}{4} imes 2frac{2}{3} div frac{5}{9}{ ext{ of }}1frac{1}{5}}
ight) = 1 + { ext{k}} cr
& left( {frac{4}{3} div frac{{20}}{7}{ ext{ of}}frac{{28}}{5}}
ight) div left( {frac{{32}}{5} div frac{9}{2}{ ext{ of}},frac{{16}}{3}}
ight) imes left( {frac{3}{4} imes frac{8}{3} div frac{5}{9}{ ext{ of }}frac{6}{5}}
ight) = 1 + { ext{k}} cr
& left( {frac{4}{3} div 4 imes 4}
ight) div left( {frac{{32}}{5} div 24}
ight) imes left( {frac{3}{4} imes frac{8}{3} div frac{2}{3}}
ight) = 1 + { ext{k}} cr
& left( {frac{4}{3} imes frac{1}{{16}}}
ight) div left( {frac{4}{5} imes frac{1}{3}}
ight) = 1 + { ext{k}} cr
& frac{1}{{12}} div frac{4}{{15}} imes 3 = 1 + { ext{k}} cr
& frac{1}{{12}} imes frac{{15}}{4} imes 3 = 1 + { ext{k}} cr
& frac{{15}}{{16}} - 1 = { ext{k}} cr
& frac{{ - 1}}{{16}} = { ext{k}} cr
& - 0.0625 = { ext{k}} cr
& - 0.07 < { ext{k}} > - 0.0625 cr} $$
[#170] $$frac{6}{{5 - frac{5}{3}}} div frac{{4 - frac{2}{{4 - frac{1}{2}}}}}{{5 - frac{3}{2}}} - frac{2}{5},$$ xa0 xa0 $$ ext{of}$$ $$left{ {frac{6}{9} + frac{2}{3}{ ext{ of }}frac{1}{2}}
ight}$$ xa0 $$ = ?$$
Correct Answer
(C) $$1frac{7}{{16}}$$
Explanation
Solution: $$eqalign{
& { ext{Given expression,}} cr
& frac{6}{{5 - frac{5}{3}}} div frac{{4 - frac{2}{{4 - frac{1}{2}}}}}{{5 - frac{3}{2}}} - frac{2}{5}, imes ,left{ {frac{6}{9} + frac{2}{3} imes frac{1}{2}}
ight}{ ext{ }} cr
& = frac{6}{{left( {frac{{10}}{3}}
ight)}} div frac{{4 - frac{2}{{left( {frac{7}{2}}
ight)}}}}{{left( {frac{7}{2}}
ight)}} - frac{2}{5}, imes ,left{ {frac{6}{9} + frac{1}{3}}
ight} cr
& = frac{{6 imes 3}}{{10}} div frac{{4 - frac{{2 imes 2}}{7}}}{{left( {frac{7}{2}}
ight)}} - frac{2}{5} imes 1 cr
& = frac{9}{5} div frac{{left( {4 - frac{4}{7}}
ight)}}{{left( {frac{7}{2}}
ight)}} - frac{2}{5} cr
& = frac{9}{5} div left( {frac{{24}}{7} imes frac{2}{7}}
ight) - frac{2}{5} cr
& = frac{9}{5} imes frac{{49}}{{48}} - frac{2}{5} cr
& = frac{{147}}{{80}} - frac{2}{5} cr
& = frac{{147 - 32}}{{80}} cr
& = frac{{115}}{{80}} cr
& = frac{{23}}{{16}} cr
& = 1frac{7}{{16}} cr} $$