Average - Study Mode

[#96] What is the average of all natural numbers from 21 to 39?
Correct Answer

(A) 30

Explanation

Solution: $$eqalign{
& 21,,22,,23,........,39 cr
& n = left( {39 - 21}
ight) + 1 cr
& n = 18 + 1 cr
& n = 19 cr
& {S_{19}} = frac{n}{2}left( {{ ext{first}} + { ext{last}}}
ight) cr
& = frac{{19}}{2}left( {21 + 39}
ight) cr
& = frac{{19}}{2} imes 60 cr
& = 19 imes 30 cr
& { ext{Average}} = frac{{{ ext{sum}}}}{n} cr
& = frac{{19 imes 30}}{{19}} cr
& = 30 cr} $$

[#97] In a class of 50 students there are 22 girls who scored an average of 35 marks in the test. What is the average marks of the boys if the class average is 42 marks?
Correct Answer

(C) 47.5

Explanation

Solution: By using alligation method Total students = 50 Girls = 22 (Given) Boys = 50 - 22 = 28 [x08egin{array}{*{20}{c}}
{}&{28}&:&{22} \
{{ ext{Ratio}} Rightarrow }&{14}&:&{11}
end{array}] 14 units = 7 1 unit = $$frac{7}{{14}}$$ 1 unit = $$frac{1}{2}$$ 11 units = 11 × $$frac{1}{2}$$ = 5.5 Average marks of boys x = 42 + 5.5 = 47.5

[#98] If the average of 5 consecutive integers is x then, find the average of next to next 5 consecutive integers.
Correct Answer

(C) x + 10
(G) x + 10

Explanation

Solution: $$eqalign{
& { ext{Let numbers be }}a,,a + 1,,a + 2,,a + 3,,a + 4 cr
& { ext{Next, }}a + 5,,a + 6,,a + 7,,a + 8,,a + 9 cr
& { ext{Next to Next, }}a + 10,,a + 11,,a + 12,,a + 13,,a + 14 cr
& {{ ext{I}}^{{ ext{st}}}}{ ext{ condition}} = frac{{5a + 10}}{5} = x cr
& 5a + 10 = 5x,......,left( { ext{i}}
ight) cr
& { ext{I}}{{ ext{I}}^{{ ext{nd}}}}{ ext{ condition}} cr
& = frac{{a + 10 + a + 11 + a + 12 + a + 13 + a + 14}}{5} cr
& = frac{{5a + 50 + 10}}{5} cr
& { ext{From equation }}left( { ext{i}}
ight) cr
& = frac{{5x + 50}}{5} cr
& = x + 10 cr} $$

[#99] The average of 8 consecutive natural numbers is 38.5. What is the largest of these 8 numbers?
Correct Answer

(C) 42

Explanation

Solution: $$eqalign{
& x + x + 1 + ,......., + x + 7 = 38.5 imes 8 = 308 cr
& 8x + 28 = 308 cr
& x = frac{{308 - 28}}{8} cr
& x = frac{{280}}{8} = 35 cr
& { ext{Lergest number}} = x + 7 cr
& = 35 + 7 cr
& = 42 cr} $$

[#100] What is the average of all numbers between 11 and 80 which are divisible by 6?
Correct Answer

(D) 45

Explanation

Solution: $$eqalign{
& { ext{11 to 80 divisible by 6}} cr
& { ext{12, 18, }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. 78}} cr
& { ext{First term}}left( a
ight) = 12 cr
& { ext{Last term}}left( l
ight) = 78 cr
& { ext{Total number of terms}} cr
& = frac{{{ ext{Last term }} - { ext{ First term}}}}{{{ ext{Common difference}}}} + 1 cr
& = frac{{78 - 12}}{6} + 1 cr
& = 12 cr
& { ext{Average of 12 numbers}} cr
& = frac{n}{2}frac{{left( {a + l}
ight)}}{n} cr
& = frac{{12 + 78}}{2} cr
& = 45 cr} $$