Simplification - Study Mode

[#221] A man has some hens and cows. If the number of heads be 48 and the number of feet equals 140, then the number of hens will be:
Correct Answer

(D) 26

Explanation

Solution: Let the number of hens be x and the number of cows be y. Then, x + y = 48 . . . . . (i) and 2x + 4y = 140 ⇒ x + 2y = 70 . . . . . (ii) Solving (i) and (ii) we get: x = 26, y = 22 ∴ The required answer = 26

[#222] $${{{{(469 + 174)}^2} - {{(469 - 174)}^2}} over {(469 imes 174)}} = ?$$
Correct Answer

(B) 4

Explanation

Solution: $$eqalign{
& frac{{{{left( {469 + 174}
ight)}^2} - {{left( {469 - 174}
ight)}^2}}}{{left( {469 imes 174}
ight)}} = ? cr
& { ext{We}},{ ext{ know that}}, cr
& 4ab = {left( {a + b}
ight)^2} - {left( {a - b}
ight)^2} cr
& herefore frac{{{{left( {469 + 174}
ight)}^2} - {{left( {469 - 174}
ight)}^2}}}{{left( {469 imes 174}
ight)}} cr
& = frac{{4 imes 469 imes 174}}{{469 imes 174}} cr
& = 4 cr} $$

[#223] David gets on the elevator at the 11 th floor of a building and rides up at the rate of 57 floors per minute. At the same time, Albert gets on an elevator at the 51st floor of the same building and rides down at the rate of 63 floors per minute. If they continue travelling at these rates, then at which floor will their paths cross ?
Correct Answer

(C) 30 th floor

Explanation

Solution: Suppose their paths cross after x minutes Then, 11 + 57x = 51 - 63x ⇒ 57x + 63x = 51 - 11 ⇒ 120x = 40 ⇒ x = $$frac{1}{3}$$ Number of floors covered by David in $$frac{1}{3}$$ min. $$eqalign{
& = {frac{1}{3} imes 57} cr
& = 19 cr} $$ So, their paths cross at (11 + 19) i.e., 30 th floor

[#224] Simplify : $$1 + {1 over {1 + {2 over {2 + {3 over {1 + {4 over 5}}}}}}}$$
Correct Answer

(A) $$1frac{{11}}{{17}}$$

Explanation

Solution: $$eqalign{
& 1 + frac{1}{{1 + frac{2}{{2 + frac{3}{{1 + frac{4}{5}}}}}}} cr
& = 1 + frac{1}{{1 + frac{2}{{2 + frac{3}{{frac{9}{5}}}}}}} cr
& = 1 + frac{1}{{1 + frac{2}{{2 + frac{5}{3}}}}} cr
& = 1 + frac{1}{{1 + frac{2}{{frac{{11}}{3}}}}} cr
& = 1 + frac{1}{{1 + frac{6}{{11}}}} cr
& = 1 + frac{1}{{frac{{17}}{{11}}}} cr
& = 1 + frac{{11}}{{17}} cr
& = 1frac{{11}}{{17}} cr} $$

[#225] Simplify : $$1 + {2 over {1 + {3 over {1 + {4 over 5}}}}}$$
Correct Answer

(A) $$frac{7}{4}$$

Explanation

Solution: $$eqalign{
& 1 + frac{2}{{1 + frac{3}{{1 + frac{4}{5}}}}} cr
& = 1 + frac{2}{{1 + frac{3}{{frac{9}{5}}}}} cr
& = 1 + frac{2}{{1 + frac{5}{3}}} cr
& = 1 + frac{2}{{frac{8}{3}}} cr
& = 1 + frac{3}{4} cr
& = frac{7}{4} cr} $$